AC quantum transport: Non-equilibrium in mesoscopic wires due to time-dependent fields
AAC quantum transport:Non-equilibrium in mesoscopic wires due totime-dependent fields
Robbert-Jan DikkenDelft University of Technology
Abstract
A model is developed describing the energy distribution of quasi-particles in a quasi-one dimensional,normal metal wire, where the transport is diffusive, connected between equilibrium reservoirs. Whenan ac bias is applied to the wire by means of the reservoirs, the statistics of the charge carriers isinfluence by the formed non-equilibrium.The proposed model is derived from Green function formalism. The quasi-particle energy distributionis calculated with a quantum diffusion equation including a collision term accounting for inelasticscattering. The ac bias, due to high frequency irradiation, drives the wire out of equilibrium. Forcoherent transport the photon absorption processes create multiple photon steps in the energy distri-bution, where the number of steps is dependent on the relation between the amplitude of the field eV and the photon energy (cid:126) ω . Furthermore we observe that for the slow field regime, ωτ D < , thephoton absorption is highly time-dependent. In the fast field regime ωτ D > this time-dependencydisappears and the photon steps in the distribution have a fixed value.When the wire is extended, the transport becomes incoherent due to interaction processes, likeelectron-electron interaction and electron-phonon interaction. These interactions give rise to a re-distribution of the quasi-particles with respect to the energy. We focused on the fast field regime andconcluded that the strong interaction limit for both mechanisms gives the expected result. Strongelectron-phonon interaction forces the distribution function on every position in the wire to become aFermi function with the bath temperature, while strong electron-electron interaction causes an effec-tive temperature profile across the wire and the distribution function on every position in the wire is aFermi function with an effective temperature.So the complicated interplay between the effect of photon absorption, diffusive transport and inelasticscattering on the quasi-particle energy distribution seems to be accurately described by our model. a r X i v : . [ c ond - m a t . m e s - h a ll ] F e b elft University of Technology Faculty of Applied Sciences Physics of NanoElectronicsKavli Institute of NanoScience AC quantum transport:Non-equilibrium in mesoscopic wires due totime-dependent fields
Master’s thesis by : R.J. DikkenResearch group : Physics of NanoElectronicsGroupleader and 1st Reviewer : Prof. dr. ir. T.M. Klapwijk2nd Reviewer : Dr. K.K. Berggren3rd Reviewer : Dr. Y.M. BlanterSupervisors : N. Vercruyssen, MSc., H.L. Hortensius, MSc. ontents
Probing the quasi-particles energy distribution 52
A Shot noise 65B MATLAB code of the simulation program 67
B.1 Script for the simulation of coherent transport . . . . . . . . . . . . . . . . . . . . . 67B.2 Functions for the used operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69B.3 Script for the simulation of incoherent transport . . . . . . . . . . . . . . . . . . . . 73
C Space dependency in the distribution function 79D Differential conductance of a NIS junction 84E MATLAB script for deconvolution 86 II bstract A model is developed describing the energy distribution of quasi-particles in a quasi-one dimensional,normal metal wire, where the transport is diffusive, connected between equilibrium reservoirs. Whenan ac bias is applied to the wire by means of the reservoirs, the statistics of the charge carriers isinfluence by the formed non-equilibrium.The proposed model is derived from Green function formalism. The quasi-particle energy dis-tribution is calculated with a quantum diffusion equation including a collision term accounting forinelastic scattering. The ac bias, due to high frequency irradiation, drives the wire out of equilibrium.For coherent transport the photon absorption processes create multiple photon steps in the energy dis-tribution, where the number of steps is dependent on the relation between the amplitude of the field eV and the photon energy (cid:126) ω . Furthermore we observe that for the slow field regime, ωτ D < , thephoton absorption is highly time-dependent. In the fast field regime ωτ D > this time-dependencydisappears and the photon steps in the distribution have a fixed value.When the wire is extended, the transport becomes incoherent due to interaction processes, likeelectron-electron interaction and electron-phonon interaction. These interactions give rise to a redis-tribution of the quasi-particles with respect to the energy. We focused on the fast field regime andconcluded that the strong interaction limit for both mechanisms gives the expected result. Strongelectron-phonon interaction forces the distribution function on every position in the wire to become aFermi function with the bath temperature, while strong electron-electron interaction causes an effec-tive temperature profile across the wire and the distribution function on every position in the wire is aFermi function with an effective temperature.So the complicated interplay between the effect of photon absorption, diffusive transport andinelastic scattering on the quasi-particle energy distribution seems to be accurately described by ourmodel. hapter 1 Introduction
The last decades the non-equilibrium in mesoscopic systems is intensively studied by a part of thenano-scientific community. Despite all the efforts the physics of this is still not fully understooddue to the complexity of these systems. The systems have length scales between microscopic andmacroscopic. On one hand the system contains many particles, but on the other hand it can stillexhibit quantum features. Because of the intermediate dimensions a specific approach is neededfor calculating the physical properties. Pure quantum mechanics can not be used because the manyparticles complicate the quantum mechanical description in a horrible way and thermodynamics cannot be used because of the significance of the quantum features in the system. Therefore often aquantum statistical approach is used which reveals the intriguing world of mesoscopic physics.Before looking at mesoscopic systems, let’s look at macroscopic and microscopic systems andthe meaning of equilibrium and non-equilibrium in this context. Consider a macroscopic resistor R placed between electron reservoirs at equilibrium, which means that the electrons in the reservoirsobey Fermi statistics and the electrons with energy E are distributed according to a Fermi function, f ( E ) = ( e ( E − µ ) / ( k b T ) + 1) − , where k b is the Boltzmann constant and T the temperature. At zerotemperature this Fermi function is just a step function at the chemical potential µ of the material. Forenergies lower than the chemical potential all energy levels are occupied and for higher energies alllevels are empty. When the temperature is increased the electrons become thermally excited, creatingholes for energies below chemical potential and electrons for higher energies. This can be seen as aquasi-equilibrium situation. When we look at the unexcited resistor between the reservoirs, we seethat the electrons are at the same equilibrium, or quasi-equilibrium, as the reservoirs. Now when adc voltage V is applied on the reservoirs a current will flow from one reservoir through the resistorto the other reservoir by the relation I = V /R . The resistance on the flowing electrons due toimpurities causes dissipation, heating the resistor. The statistics of the electrons in the resistor are nolonger the same statistics as that of the reservoirs and becomes spatial dependent. The heating of theresistor causes a local equilibrium in the resistor and the electron energy distribution is described by aneffective electron temperature [1]. The applied power P = V /R causes a temperature profile alongthe wire which is bounded by the temperature of the reservoirs. Such an effective temperature profileis shown in figure 1.1. The temperature of the reservoirs is held at 4.2 K and the effective temperatureis at maximum in the middle of the wire. Figure 1.1 also shows the local equilibrium distributionfunction at the boundary of the wire and in the middle of the wire. The effect of the dissipated energyis a thermal smearing around the Fermi energy. 1igure 1.1: A bias voltage applied to a macroscopic wire causes local equilibrium in the wire andthe electron statistics are described by Fermi functions with an effective temperature. The energydistribution in the reservoir and in the middle of the wire is given by the blue and red line, respectively.The effective temperature profile shows the effect of the potential difference across the wire [2].For the opposite case, a microscopic system, the situation is completely different. A scatterer isplaced between two reservoirs and a voltage is applied. In this microscopic situation it becomes moreconvenient to evaluate the transport using scattering theory [3], so we do not speak anymore of distri-bution functions inside the transport region. The electron approaches the scatterer as a Fermi particlewith a wave function. Because of the wave-particle duality the electron can be transmitted or reflectedwith a certain probability by the scatterer, whereafter the electron leaves the scatterer as a Fermi par-ticle with a certain wave function. The transport of the electrons through the scatterer depends on theproperties of the scatterer. These properties are described by the transmission distribution, which givesthe probability of finding a transport channel in the scatterer with a certain transmission probability.A bias voltage applied to the reservoirs will only create a potential difference across the structure andthe transport depends on this potential difference and the transmission distribution of the scatterer.The number of electrons involved in the transport is a measure of the non-equilibrium.So the non-equilibrium of macroscopic systems is described by the temperature and resistanceof the object and the non-equilibrium of microscopic systems is revealed by scattering theory. Nowthe intermediate regime between macroscopic and microscopic: mesoscopic. In this research we willfocus on a diffusive wire, which shows the most resemblance with the macroscopic situation wherea resistor was evaluated. However, the general idea that the electron energy distribution inside thewire can be described by an effective temperature appears to breaks down. Pothier et al. studied theeffect of a dc voltage on a diffusive wire between electron reservoirs [4]. From this research it wasconcluded that the electron energy distribution obeys the time-independent Boltzmann equation when2he driving term, i.e. the potential difference across the wire, is absorbed in the boundary conditions. τ D d f ( x, E ) dx + I coll ( x, E, f ) = 0 (1.1)In absence of inelastic interactions the collision integral vanishes and the solution is on everyposition in the wire a superposition of the boundary conditions which are the Fermi function of rightreservoir and that of the left reservoir. When one reservoir is held at zero potential, the other reservoiris at maximum potential which shifts this Fermi function with eU . The superposition of these twodistribution functions creates a two step function dependent on the position on the wire as shown infigure 1.2.Figure 1.2: DC biased wire showing the spatial dependent superpositions of the equilibrium distribu-tion functions of the reservoirs [4].When inelastic interactions are involved the situation becomes a bit more complicated. The col-lision integral in the Boltzmann equation has to be evaluated. The energy that electrons gained fromthe electric field is redistributed during collisions on inelastic scatterers. These inelastic scatteringprocesses are electron-electron and electron-phonon interactions. Depending on the characteristics ofthe diffusive wire and the dominant scattering processes, a relation for the energy relaxation time canbe obtained which is self-consistently used in calculating the distribution function. So far only non-equilibrium of dc quantum transport is considered. The study of non-equilibrium ofac quantum transport in mesoscopic systems is interesting for better understanding of the physics ofmany-body systems and how small electronic devices respond to high frequency irradiation. Previousstudies on ac quantum transport focused mainly on coherent structures, where the phase of electronsis preserved. Different examples of study objects of ac quantum transport are SIS junctions, quantumpoint contacts (QPC), quantum dots (QD) and resonant tunneling diodes (RTD). Tien and Gordonsuccessfully constructed a theory describing the tunneling current between two superconducting filmsseparated by an insulating layer biased with an ac voltage [5]. The electrons involved in the transport3an gain energy in discrete values from the ac field creating steps in the I − V characteristics. Thesuccess of their theory reached further than the SIS and was also successfully applied to the QPC, QDand RTD.Stimulated by the success of this theory for different structures Remco Schrijvers [6] tried toapply this theory to the reservoirs and use the Boltzmann equation to calculate the electron energydistribution in a diffusive wire excited by an ac voltage. The validity of this approach was a bitdisappointing. The model was only valid for low frequencies in a wire without inelastic scattering.This was caused by the fact that Tien-Gordon theory assumes averaging over time and therefore thecollision term of the Boltzmann equation can not be evaluated in a correct manner.Figure 1.3: AC biased wire for which the non-equilibrium description is still unknownFrom the previous research on non-equilibrium due to time-dependent fields in diffusive wires itwas concluded that the situation is still not completely understood. To avoid the deducted problemsput forward by Remco Schrijvers, we derived from the Green function formalism a quantum diffusionequation for the electron energy distribution in a quasi-one dimensional diffusive wire subject to anoscillating electric field. The model is first derived for a coherent structure with elastic impurityscattering, whereafter this model is extended to account for inelastic scattering processes such aselectron-electron and electron-phonon interactions.4 hapter 2 Phase coherent quantum transport
Phase coherent quantum transport involves the transport of charge carriers where the phase of thesecharge carriers is preserved. Generally this means that scattering inside the structure is elastic, sothat the energy of the charge carriers is not redistributed. Phase coherent transport of electrons innanostructures is usually described with scattering theory. The nanostructure is defined as a scatteringregion between reservoirs and the wave function of the electrons subject to Hamiltonian ˆ H withpotential U ( r , t ) obeys the Schrodinger equation i (cid:126) ∂ψ ( r , t ) ∂t = ˆ Hψ ( r , t ); ˆ H ≡ − (cid:126) m ∇ + U ( r , t ) . (2.1)The solution of the Schrodinger equation is a stationary space-dependent function multiplied by atime-dependent function dependent on the eigen energy E of the Hamiltonian: Ψ( r , t ) = e − iEt/ (cid:126) ψ ( r ) . (2.2)The wave function ψ ( r ) obeys the time-independent Schrodinger equation ˆ Hψ ( r ) = Eψ ( r ) .Due to the wave character of a charge carrier, an electron can contribute to the current through thescatterer between the reservoirs by either being reflected or being transmitted. The probability ofbeing reflected or transmitted is dependent on the thickness and height of the barrier whereon theelectron scatters. The potential difference across the structure is determined by the difference ofthe energy distribution of the two electron reservoirs. Landauer’s result for the current through thescatterer between reservoirs is proportional to the integral over energy of the trace of the productof the transmission matrix ˆ t and its conjugate transpose ˆ t + and the difference between the energydistribution of the left and right reservoir [7]. An insightful derivation of the Landauer formula can befound in Ref [3]. I = 2 s e π (cid:126) (cid:90) ∞ T r [ˆ t + ˆ t ][ f L ( E ) − f R ( E )] (2.3)The factor s accounts for the degeneracy of electrons with charge e . When a bias is applied tothe reservoirs, creating across the structure a potential difference V , much smaller than the scale of5nergy dependence in the transmission eigenvalues T n , equation 2.3 can be evaluated at the Fermienergy µ . Introducing the conductance quantum G Q = 2 e /h gives for the current I = G Q V (cid:88) n T n ( µ ) . (2.4)This expression for the current through a scattering structure clearly shows that the structure existsof different channels in which the electrons are transported with a certain probability from one reser-voir to the other. The type of transport structure is characterized by the distribution of the transmissionprobabilities. This distribution is constructed by taking one specific nanostructure from an ensembleof identical design and counting the number of transmission eigenvalues of the transmission matrixin the interval of T to dT . This is divided by the total number of nanostructures in the ensemble.For large enough ensembles, the result converges to P ( t ) dT , so that the transmission distribution isdefined as P ( t ) = (cid:68)(cid:80) p δ ( T − T p ( E )) (cid:69) . For very short structures, where the wavelength of the elec-tron exceeds the length of the structure, the conductance quantization is prominent present and thedistribution of the transmission probabilities is sharply peaked on certain values. When the length ofthe structure increases, the resistance due to defects in the system becomes dominant. The diffusivebehavior of the electrons in the scatterer is random and for a diffusive scatterer the distribution oftransmission probabilities is universal, i.e. independent on the details of the scatterer [3]. ρ D ( T ) = (cid:104) G (cid:105) G Q T √ − T (2.5)Here (cid:104) G (cid:105) is the average conductance due to many scattering events. Now with the increasingdimensions of the structure the describing picture becomes more and more complicated due to thefact that more charge carriers are involved and inelastic scattering processes affect the energy of thecharge carriers. Therefore one has to let go the idea that the energy of electrons is unchanged bythe scattering events. Pure scattering theory can no longer describe in an effective way the transport.Quantum statistical mechanics provides a way out as we will see later on. First we look at the statis-tical information of charge carriers that the noise due to finite transmission probabilities in scatteringprocesses provides.Figure 2.1: The transmission distribution of a diffusive wire for three different kind of disorder con-figurations [3]. 6 .1.2 Shot noise A physical phenomenon that contains information about statistics of charge carriers in a mesoscopicconductor is shot noise. Shot noise is caused by the quantization of charge [8]. When a single incidentcharge in a state with occupation 1 scatters on some potential barrier it has a probability R of beingreflected and a probability T = 1 − R of being transmitted. Figure 2.2 shows how the incoming wavepacket of an electron scattering on a barrier with transmission probability T is splitted and only a partof the initial wave packet is transmitted with a probability T , causing fluctuations in the current.Figure 2.2: Shot noise arises when the wave packet of an electron is splitted due to a scattering eventand the finite transmission probability T causes fluctuations in the current [9].When the initial state is occupied by the distribution function f , an incident particle is reflectedwith probability f R and transmitted with probability f T , so the averaged occupation of the reflectedstate is (cid:104) n R (cid:105) = f R and the averaged occupation of the transmitted state is (cid:104) n T (cid:105) = f T . By lookingat many scattering processes the fluctuations from the average occupation can be determined. For theincident state the average occupation is just the Fermi distribution (cid:104) n in (cid:105) = f , so that the mean squaredfluctuations in the incident state vanishes: (cid:10) ( f − (cid:104) n in (cid:105) ) (cid:11) = 0 . The fluctuations in the reflected andtransmitted state have a finite value. The fluctuations are expressed as a deviation from the average so δn T = n T − (cid:104) n T (cid:105) and δn R = n R − (cid:104) n R (cid:105) . When we use these identities to calculate the mean squaresof the correlations between reflected and transmitted state and of the reflected and transmitted stateitself we find: (cid:104) δn T δn T (cid:105) = − T Rf (2.6) (cid:10) ( δn T ) (cid:11) = T f (1 − T f ) (2.7) (cid:10) ( δn R ) (cid:11) = Rf (1 − Rf ) . (2.8)From these expressions we can distinguish two limits. One limit is given by full transparency andthe other limit is given by full reflectance. Both limits have the same outcome in the fluctuations. In asituation where the occupation of the initial state is given by a Fermi distribution at zero temperature,the mean square fluctuations vanish. However, for finite temperature this is not the case. The meansquare fluctuations does not vanish, but fluctuates like the incident state with occupation f .7hese mean square fluctuations contribute in the current and from the current expressions de-rived in appendix A the noise power can be obtained. When a multi-channel scatterer between tworeservoirs is considered, the noise power can be evaluated at Fermi energy when the scale of energydependence of the transmission coefficients is much larger than the thermal energy and the energyassociated with the applied bias voltage on the reservoirs. The shot noise power is then [8]: S = e π (cid:126) [2 k b T (cid:88) n T n + eV coth (cid:18) eV k b T (cid:19) (cid:88) n T n (1 − T n )] . (2.9)As we will see later on, the shot noise for an ac bias has a bit different form than equation 2.9 andtherefore the non-equilibrium due to the ac transport can be seen in the shot noise. When we make the switch from dc quantum transport to ac quantum transport, the needed describingtheoretical frameworks become a bit more sophisticated. Approximately five decades ago Dayem andMartin observed interactions of electrons with photons in the tunneling current between the super-conducting films A and B separated by an insulating layer, when the structure was illuminated withmicrowave radiation, causing an ac bias across the junction [10]. Figure 2.3 shows the clear differencebetween the I − V characteristic with and without this oscillating electric field.Figure 2.3: The by Dayem and Martin measured I − V characteristic of an SIS junction biased withand without oscillating field [5].In order to explain these quantum interactions Tien and Gordon developed a describing theory forelectric fields normal and parallel to the surface of the superconductor [5]. Here we will only considerthe case where the field is normal to the surface of the superconductor.The potential difference between the superconductors A and B due to the electric field is given by V cos ( ωt ) , where the bias is applied to one reservoir and the other reservoir is held at zero potential.8hen no field is present the wave functions of the charge carriers of energy E satisfy the unperturbedHamiltonian H . ψ ( x, y, z, t ) = ψ ( x, y, z ) e − iEt/ (cid:126) (2.10)The perturbed Hamiltonian due to the oscillating electric field is given by H = H + eV cos ( ωt ) . (2.11)This interaction Hamiltonian only effects the time-dependent part of the wave function given byequation 2.10. The new wave function under influence of the oscillating electric field becomes ψ ( x, y, z, t ) = ψ ( x, y, z ) e − i (cid:126) [ Et + (cid:82) t eV cos ( ωt (cid:48) ) dt (cid:48) ]= ψ ( x, y, z ) e − iEt/ (cid:126) e eV (cid:126) ω sinωt = ψ ( x, y, z ) e − iEt/ (cid:126) ∞ (cid:88) n = −∞ J n (cid:18) eV (cid:126) ω (cid:19) e inωt . (2.12)To come to the last line the identity e zsin ( θ ) = (cid:80) ∞ n = −∞ J n ( z ) e inθ is used, where J n ( z ) is theBessel function giving the probability of the absorption of n field quanta. The wave function inequation 2.12 is normalized, since (cid:2)(cid:80) ∞ n = −∞ J n ( z ) (cid:3) = 1 . It appears that the wave function no longerhas one energy variable. The energy variable is extended in a sum of multiples of the photon energy.This means that where a charge carrier in the situation without the oscillating field could only tunnelto a state with the same energy, now also could tunnel to states with energy E ± n (cid:126) ω . Basically thedensity of states of the superconductor is modulated by the electric field. The unperturbed density ofstates of the superconductor is ρ ( E ) . In the presence of the oscillating field the density of states (cid:101) ρ ( E ) becomes (cid:101) ρ ( E ) = ∞ (cid:88) n = −∞ ρ ( E + n (cid:126) ω ) J n (cid:18) eV (cid:126) ω (cid:19) . (2.13)The tunnel current is calculated from the density of states. For an SIS junction biased with a dcvoltage V the tunnel current is I AB = C (cid:90) ∞−∞ [ f ( E − eV ) − f ( E )] ρ A ( E − eV ) ρ B ( E ) dE. (2.14)Here C is a proportionality constant depending on the junction resistance. When an additional acvoltage is applied to the SIS junction the tunnel current shows the multiple photon steps. (cid:101) I AB = c ∞ (cid:88) n = −∞ J n (cid:18) eV (cid:126) ω (cid:19) (cid:90) ∞−∞ [ f ( E − eV ) − f ( E + n (cid:126) ω )] ρ A ( E − eV ) ρ B ( E + n (cid:126) ω ) dE. (2.15)When the tunnel current is explicitly calculated it shows indeed the photon steps as measured byDayem and Martin. Figure 2.4 shows the difference between the measured tunnel current withoutoscillating electric field given by the solid lines and the calculated tunnel current with oscillatingelectric field between two superconducting films for two different ratios of eV (cid:126) ω given by the dashedlines. 9igure 2.4: The measured I − V characteristic of an SIS junction without oscillating electric field andthe calculated I − V characteristic with oscillating field for different ratios of eV (cid:126) ω [5].The energy diagram of an ac biased SIS junction in figure 2.5 shows explicitly how a photonassist the transport of an electron from the first superconductor through the insulating layer to thesecond superconductor. The gap in the density of states of the superconductor makes it impossiblefor an electron unaffected by the electric field to tunnel through the barrier to an unoccupied level inthe second superconductor. The absorption of a photon can provide the required energy to make thispossible. Figure 2.5: Photon-assisted transport in an SIS junction [11].As said in chapter 1 photon-assisted transport is, besides in SIS junctions, also observed in othernano-electronic systems. We won’t discuss all examples. Here we only have an additional look atthe transport in a quantum dot illuminated with radiation, where the driving frequency exceeds thenormal tunneling rate of electrons through the dot, since it provides great insight in the mechanism ofphoton-assisted transport. 10 quantum dot is usually some island coupled by tunnel barriers to leads, the source and drain.The electronic properties of the island and the tunnel barriers can be controlled by gates. Figure 2.6shows this schematically. Figure 2.6: A schematic of a quantum dot [12].The energy levels on the island are assumed to be discrete with a spacing ∆ E while the energyspectrum of the leads is assumed to be a continuum. The radiation is coupled to the island by the gate[13]. We will not go into detail about this, since we mainly want to focus on the transport from drainto source. The normal tunneling rates are modified by the radiation due to the modification of thewave function of the electrons given by equation 2.12. (cid:101) Γ = ∞ (cid:88) n = −∞ J n ( z )Γ( E + n (cid:126) ω ) (2.16)Here z = e (cid:101) V / (cid:126) ω and (cid:101) V is the amplitude of the oscillation. The tunneling is assisted by theabsorption of photon with energy E + n (cid:126) ω and emission of photons with energy E − n (cid:126) ω . Thepossible tunneling processes in the dot with and without radiation are shown in figure 2.7. Onlythe upper energy diagram in the middle can contribute to a current through the dot without help ofradiation. The remaining diagrams show the photon-assisted tunneling through the ground state (cid:15) and the first excited state (cid:15) of the dot. Electrons which normally do not have the right energy totunnel to an unoccupied state can now absorb or emit a photon. This modifies their energy in such away that tunneling becomes possible.Figure 2.7: Tunneling processes in a quantum dot [14].11or both the SIS junction and the quantum dot the ac bias, due to radiation coupled on the structure,modulates the electronic properties making transport possible to energy states which are not accessiblewithout the energy gain from the field. The photons from the field assist in the transport of chargecarriers through the structure. In section 2.1.2 the basic idea of shot noise in mesoscopic conductors for dc quantum transport isevaluated and we stated that the expression for the shot noise differs a bit for ac quantum transport.Here we will look how it differs and how this difference arises.A general scatterer is placed between two reservoirs and an ac voltage is applied to the scattererby the left reservoirs while the other reservoir is grounded. The transport of electrons can be dividedinto two regimes: transport of affected and unaffected electrons by the ac bias [9]. The unaffectedelectrons do not contribute to the shot noise, because the number of emitted, unaffected electronsfrom the right reservoir is the same as that of the left reservoir. Since according to the Pauli exclusionprinciple both left and right outgoing states can only be occupied by one electron, the current cancelsand so does the fluctuation in current.The affected electrons from the left reservoir can contribute to the shot noise. An electron withenergy (cid:15) ≤ (cid:126) ω below the Fermi energy can get excited to an energy (cid:126) ω − (cid:15) . At energy − (cid:15) a hole iscreated. Since only the left reservoir can excite electrons in this way (the other reservoir is grounded),there is no counter current, so that this becomes the source of the fluctuations in the current. Nowwhen also a dc voltage is applied to the scatterer, the shot noise expression becomes an extendedversion of equation 2.9 [15], where the photon-assisted features are presented by the Bessel functionslike in the tunnel current calculated by Tien and Gordon. S I = 4 G Q k b T (cid:88) n T n + 2 G (cid:88) n T n (1 − T n ) (cid:88) ± ∞ (cid:88) l =0 J l ( α )( eV ± l (cid:126) ω ) coth (cid:18) eV ± l (cid:126) ω k b T (cid:19) (2.17)Here α = eV ac / (cid:126) ω . For V ac = 0 the normal expression for shot noise is obtained. Now whenwe make the transition to a diffusive wire it appears that this description still holds. Schoelkopfet al. [16] investigated the photon-assisted shot noise experimentally for phase-coherent diffusiveconductors and compared their results to the theoretical predictions for photon-assisted shot noisestated by Lesovik and Levitov [15]. The ac bias is applied on the conductor by bending the conductorbetween the reservoirs in a loop. A time-dependent magnetic field enters the loop, which induces atime-dependent electric field in the conductor. The situation is depicted in figure 2.8.Lesovik and Levitov predicted theoretically the photon steps in the noise power for an ac biaseddiffusive conductor where the phase of electrons is preserved. The experiment of Schoelkopf verifiesthis model. Figure 2.9 shows the experimental results and the expected results from equation 2.17of the differential noise power. The photon steps are not that clear in the first derivative of the noisepower. The second derivative of the noise power however clearly shows at the expected energies thesteps, indicating the photon-assisted mechanism in the shot noise.The discrete steps in the shot noise shows the absorption of field quanta and give information aboutthe statistics of the charge carriers in the diffusive wire. It reveals that the energy distribution of thecharge carriers inside the wire is affected by the ac bias. This is a completely different point of viewin comparison to the transport in the SIS junction and the quantum dot where the electronic propertiesof the reservoirs are affected by the ac bias. So apparently there arises some interesting physics in12igure 2.8: The schematic layout of the photon-assisted shot noise measurements.Figure 2.9: Photon steps in the shot noise both calculated and measured [16].the diffusive wire. This is still a relatively simple model, where the electron transport is coherent,so that scattering theory still can be used to describe the transport. However, when the length of thediffusive wire is increased and not only diffusivity and photon absorption causes a change in statisticsin the wire, but also inelastic scattering processes induce energy redistribution, scattering theory is nolonger the most convenient describing theory. As said in section 2.1.1 we can proceed with quantumstatistical theory to determine the statistics of the charge carriers described by the energy distributionfunction. In the next chapter we will evaluate the conditions for such an approach.13 hapter 3 Diffusive transport
The model that was proposed in the 1900s by Drude describes the transport properties of electrons inmetals on a microscopic level from a classical point of view. The electronic properties of a metal arethen described by a gas of electrons bouncing on heavier positive charged ions. Because of the highermass of the ions, they are seen as static potentials and the collisions of the electrons on these ions arepurely elastic. The electrons involved in the transport are assumed to be free. Between two scatteringevents no forces act on the electron. In a situation where no electric field is applied on the metalconductor, the average velocity due to different electrons cancels, as the electrons move in a varietyof directions. When an electric field is applied the average velocity and thus the net current becomesfinite. If n electrons per unit volume with charge − e move with the average velocity v ave and movein a time dt a distance v dt , then the net charge passing through a cross-section A is − ne v ave Adt [17][18]. The current density becomes j = 1 A dQdt = − ne v ave . (3.1)Now when an electron is considered at time zero with velocity v , the velocity that this electroncan gain from the electric field in time t is − e E t/m following from Newton’s laws of motion. Theinitial velocity v of every electron does not contribute to the average velocity, due to the randomcollisions from which the electron emerges on time zero. From this it is also directly clear that theaverage time t is the average time between collision τ , so that the average velocity is v ave = − e E τ /m .Substituting this in the current density gives j = ne τm E . (3.2)Ohm’s law is given by j = σ E , where σ is the conductivity. Equating the current density ofequation 3.2 and from Ohm’s law gives the final expression of the conductivity. σ = ne τm (3.3)Based on the observation that metals conduct heat better than insulators the assumption was madethat the electrons involved in the electric conduction also carry the thermal current. The originalDrude model used the Maxwell-Boltzmann distribution to account for the probability of finding anelectron with a certain energy and thus a certain velocity. However, the ratio between thermal and14lectric conductivity observed in experiments was not explained in this way. Then the Pauli exclusionprinciple was put forward, which stated that two fermions can never occupy the same state. From thisthe conclusion was drawn that the Maxwell-Boltzmann distribution had to be replaced by the Fermi-Dirac distribution. Sommerfeld exchanged the Maxwell-Boltzmann distribution by the Fermi-Diracdistribution in the classical electron gas of Drude. This modified the expression for the electronicvelocity and gave the correct expression of the ratio between thermal and electric conductivity, theWiedemann-Franz law [17]: κσ = π (cid:18) k b e (cid:19) T. (3.4)The idea that electrons form a gas in a metal is sufficient for cases where no energy exchange ispresent in all processes involving the electrons. However, this is not always the situation. When thecollisions of the electrons are no longer purely elastic and they cause energy exchange, the Drude-Sommerfeld model breaks down. Fortunately Landau’s theory of Fermi liquids provides a strongreplacement. As said in the previous section, at a certain stage the transport of electrons can no longer be explainedin a electron gas model where the interactions are purely elastic. The effect of inelastic interactionsbecomes significant and the energy exchange processes initiate the break down of the electron gasconcept. Instead one considers the transport of electrons in a liquid model. This Fermi liquid modelis developed by Lev Landau in 1956. The transport of one electron is affected by the surroundingelectrons and its wave function is extremely complicated due to screening effects. It behaves howeverstill very like a particle with a charge e . The screening can simply be seen as the modification of therelation between energy and wave vector, so E ( k ) = (cid:126) k / m ∗ , where m ∗ deviates from the freeelectron mass m . The electrons are defined as quasi-particles which are stable near the Fermi level,but lose their stability far from the Fermi level [19].The domain of validity for excitations near the Fermi surface in the Landau theory of Fermi liquidshas its origin in the assumed one-to-one correspondence between states of a non-interacting systemand states of an interacting system when the interaction is adiabatically turned on. Since the lifetimeof a quasi-particle is proportional to ( (cid:15) − (cid:15) F ) − , the high energy quasi-particles are decayed beforethe interaction process is fully complete [20]. The adiabatic continuation leads to the assumption thatthe excited states of the interacting system are labeled with the same quantum numbers as the excitedstates of the non-interacting system. The validity of adiabatic continuation from a non-interactingsystem to a interacting system can be shown by looking at the wave function. An example is givenby a particle trapped in an one-dimensional potential V ( x, t ) = V ( t ) h ( x ) [21]. The wave functionobeys the Schrodinger equation i (cid:126) ∂ψ ( x, t ) ∂t = H ( x, t ) ψ ( x, t ) = (cid:18) p m + V ( x, t ) (cid:19) ψ ( x, t ) . (3.5)Now the potential changes slowly from initial value V to a final value V . Because the potentialvaries slowly, the solution of the Schrodinger equation can be approximated by the solution of thestatic Schrodinger equation H ( x, t ) ψ V ( t ) ( x ) = E V ( t ) ψ V ( t ) ( x ) [22]. The adiabatic solution becomes ψ adiabatic ( x, t ) ≈ ψ V ( t ) ( x ) e − iE V t ) t/ (cid:126) . (3.6)15y inserting equation 3.6 in equation 3.5 the accuracy of equation 3.6 is obtained. i (cid:126) ∂ψ adiabatic ( x, t ) ∂t = E V ( t ) ψ adiabatic ( x, t ) + i (cid:126) (cid:18) ∂ψ adiabatic ( x, t ) ∂V ( t ) (cid:19) (cid:18) ∂V ( t ) ∂t (cid:19) = H ( x, t ) ψ adiabatic ( x, t ) (3.7)The adiabatic solution is a good approximation for the wave function in an one-dimensional po-tential V ( x, t ) if the first term of equation 3.7 dominates the second term, which is true if the rate ofchange of V ( t ) is small enough. Then the solution for the new potential V = V is found fromthe old value of the potential V = V from which it adiabatically rises. This implies that when theexcited state of the initial potential is a bound state, the excited state of the final potential is also abound state. A transition from a bound state to an un-bound state will never occur from an adiabaticcontinuation, no matter how small the rate of change in V ( x, t ) , because one is a decaying functionwhile the other is an oscillatory function.As said the interactions cause a modification of the relation between energy and momentum ofa particle. The total energy of an unperturbed electron system is given by the kinetic energy of theelectrons [23]. E = (cid:126) (cid:88) k k m n ( k ) (3.8)Here n ( k ) is the occupation number of the state with momentum k . When a weak external fieldis coupled on the system, there will occur a change in occupation number and thus a change in totalenergy. δE = (cid:126) (cid:88) k k m δn ( k ) (3.9)If the system now is perturbed by a adiabatically turned on interaction, with interaction energy g ( k , k (cid:48) ) between states of wave vector k and k (cid:48) , the system is taken away from its ground state energyand a change of occupation numbers is induced. Therefore the change of energy is δE = (cid:88) k (cid:15) k δn ( k ) + 12 V (cid:88) k , k (cid:48) g ( k , k (cid:48) ) δn ( k ) δn ( k (cid:48) ) . (3.10)Due to the interaction the electron is no longer a pure particle, but it is a quasi-particle. It behavesstill like a particle, but it arises from the interactions with its local environment. A quasi-particle withwave vector k has an energy of (cid:15) k = δEδn ( k ) = (cid:15) k + 1 V (cid:88) k , k (cid:48) g ( k , k (cid:48) ) δn ( k (cid:48) ) . (3.11)In the above we have suppressed magnetic fields, so that spin dependency can be neglected, since (cid:15) ( k , σ ) = (cid:15) ( k ) in absence of magnetic fields.A fundamental parameter in the Landau theory of Fermi liquids is the effective mass. The interac-tion experienced by a quasi-particle changes its mass with respect to the mass in an environment freeof interactions. The velocity and density of states at the Fermi surface can be calculated using thiseffective mass. 16 F = p F m ∗ , N (0) = 3 N m ∗ p F (3.12)The expressions for these quantities are similar to that of a non-interacting system which confirmsthe one-to-one correspondence between the states of a non-interacting system and an interacting sys-tem. So concluding this section, we can take interactions into account in calculating the electronicproperties in quantum transport by considering the charge carriers being quasi-particles for low excitedstates. Therefore the total energy of the system is not the sum of the energy of the individual parti-cles, but is function of the energy distribution among the quasi-particles. Also due to the one-to-onecorrespondence between the states of a non-interacting system and an interacting system, the energydistribution of the quasi-particles can be calculated from a diffusion equation, like the semi-classicalBoltzmann equation. The Landau theory of Fermi liquids, discussed in the previous section, provides the justification ofusing a semi-classical Boltzmann equation to calculate the energy distribution of the quasi-particlesin a diffusive wire. In this work we focus on a quasi-one dimensional metallic wire of mesoscopicdimensions where the transport of the quasi-particles is diffusive. We will first explain what we exactlyunderstand when we talk about quasi-one dimensional, mesoscopic and diffusive. Then we discussthe non-equilibrium in such a system biased with a dc voltage by looking at the energy distribution ofthe quasi-particle involved in the transport.Mesoscopic structures are defined by the relation between length scales defining the geometricsof the structure and defining microscopic processes in the structure.The length scales defining the microscopic processes involving an quasi-particle are: • The Fermi wavelength λ F = 2 π/k F , where k F is the Fermi wave vector, • The elastic mean free path l e , which is the average distance between elastic collisions on impu-rities for instance, • The phase coherence length l φ , which is the distance that the phase of a quasi-particle is pre-served, • The energy relaxation length l E , which is the distance the energy of a quasi-particle is preserved.The length scales defining the geometrics of the structure are given by: • The length of the structure L , • The cross-section of the structure S .When the length of the structure is significantly larger than the cross-section it is more natural totalk about the structure as being a wire. The wire is said to be diffusive if the length L of the wire issignificantly larger than the elastic mean free path l e of an quasi-particle in the wire. The wire is quasi-one dimensional for λ F << S << l e provided that the width and the thickness are of the same orderof magnitude. When we want to be able to apply the Landau theory of Fermi liquids we are boundto at least quasi-one dimensional systems. For purely one dimensional systems the Landau theory of17ermi liquids is no longer valid. This has its origin in the nesting property of the Fermi surface, whichmeans that a part of the Fermi surface can be matched onto an other part by a translation of k F .Therefore there arises a divergence in calculating physical properties. A more detailed explanationcan be found in Ref. [24].Figure 3.1: Diffusive wire biased with a potential difference U [25].Pothier et al. studied the quantum transport in dc biased diffusive wires by looking at the effect ofthe induced non-equilibrium on the quasi-particle energy distribution [4]. The diffusive wire is placedbetween large electron reservoirs where the electron energy distribution is described by an equilibriumFermi function. A dc voltage U is applied on one reservoir while the other reservoir is held at zeropotential, creating a potential difference U over the wire.The distribution function can be calculated by using semi-classical kinetic theory, which is theBoltzmann equation extended with an interaction term. For wires where the diffusion time τ D isshorter than the relaxation time τ E the transport is coherent and the distribution is described by aBoltzmann equation without interaction term. When the driving term eU due to the potential differ-ence U across the wire is absorbed in the boundary conditions at the reservoirs, one Fermi function isunchanged, while the other is shifted by eU . This leads to the equation: ∂f ( x, E ) ∂t + D ∂ f ( x, E ) ∂x = 0 (3.13)The explicit boundary conditions for this equation are given by f (0 , E ) = f F ( E ) and f ( L, E ) = f F ( E + eU ) , where f F ( E ) = (1 + e E/k b T ) − is just the Fermi distribution. The stationary solutionon every position in the wire is a superposition of the two boundary conditions. f ( x, E ) = (cid:16) − xL (cid:17) f F ( E ) + xL f F ( E + eU ) (3.14)Figure 3.2 shows the two step function of the electron energy distribution on every position in adc biased wire with no interactions present.If inelastic scattering is introduced the situation becomes a bit more sophisticated. Two main phasebreaking mechanisms can be distinguished: electron-electron interactions and electron-phonon inter-actions, where the electrons are considered to be quasi-particles. We first consider electron-electron18igure 3.2: The two step distribution for a dc biased wire without interactions [25].interactions and neglect electron-phonon interactions. Strong scattering induces a local equilibriumwith temperature T e ( x ) and the distribution is described by f ( x, E ) = f F ( E − µ ( x ) , T e ( x )) (3.15)where µ ( x ) = − eU xL [25]. The effective temperature T e ( x ) in a wire with cross-section S andresistance R is calculated from the heat equation [25]. ∂∂x (cid:18) κ ∂T e ∂x (cid:19) + 1 SL U R = 0 (3.16)The boundary conditions of this equation are T e (0) = T e ( L ) = T and using the Wiedemann-Franz law (equation 3.4) for the heat conductivity κ the effective temperature is [25] T e ( x ) = (cid:115) T + xL (cid:16) − xL (cid:17) π (cid:18) ek b (cid:19) U . (3.17)Now the electron-electron interactions are negligible and the electron-phonon scattering is thedominant phase breaking mechanism. For strong scattering the electrons thermalize with the tem-perature of the phonons. The distribution function is given by f ( x, E ) = f F ( E − µ ( x ) , T ) where µ = − eU xL and T is the phonon bath temperature [25]. The space dependence of the distributionfunctions is shown for both situations in figure 3.3.For intermediate regimes where neither electron-electron scattering nor electron-phonon scatter-ing is strong, but still present, the interaction term in the Boltzmann equation has to be evaluated. Theinteraction term can be calculated from the Fermi golden rule and the belonging kernel follows froma microscopic derivation [26]. We will come back to this later in chapter 4 where we calculate theinteractions in a diffusive wire due to electron-electron scattering and electron-phonon scattering.19igure 3.3: Left: Strong electron-electron scattering, right: strong electron-phonon scattering [25]. On quantum scale the conductance of a diffusive wire is not simply given by the Drude result of theconductivity in equation 3.3. Because an electron has a wave-character, the electron is not localized.Therefore, when no phase-breaking processes are present, an electron can interfere with itself whenit returns to a certain initial position after multiple elastic scattering events. This modification of theconductance is called localization and is depicted in figure 3.4.Figure 3.4: Feynman diagrams showing on the left classical trajectories and on the right trajectoriesresulting in weak localization [27]The probability for an electron of passing between A and B is given by a classical probability andadditionally an interference term W = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:88) i A i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:88) i | A i | + (cid:88) i (cid:54) = j A i A ∗ j . (3.18)The phase gained by an electron while traveling through the diffusive medium is ∆ φ = (cid:126) − (cid:82) BA p d l . For most of the trajectories this phase gain will be much larger than one and there-fore vanish in the interference term. The self-crossings have the same phase gain when the directionof the traveled trajectory is reversed, i.e. p → − p and d l → − d l . This results in two paths and theprobability of self-crossing is W = | A + A | = | A | + | A | + 2 A A ∗ = 4 | A | . (3.19)20he quantum interference doubles the result. So the probability of scattering is increased, whichresults in a decrease of conductance. To determine qualitatively the effect of weak localization on theconductance we shall follow a heuristic derivation which can be found in Ref. [27]. The de Brogliewavelength λ F = 2 π/k F of the electron determines the scattering cross-section on site O . In time t it travels diffusively a distance √ Dt , where D is the diffusion coefficient. The interference volume in d dimensions becomes ( Dt ) d/ r − d , where r is the thickness of the system. The electron has to enterthe interference volume to experience interference, which occurs with a probability of v F λ F dt ( Dt ) d/ r − d .This leads to a relative correction to the conductivity of ∆ σσ ∝ − (cid:90) τ φ τ e v F λ F dt ( Dt ) d/ r − d . (3.20)The phase coherence time in the upper limit of the integral shows the condition for phase preser-vation. Now when we focus on the one dimensional situation for our quasi-one dimensional wire, theevaluation of the integral gives ∆ σσ ∝ − v F λ F D / r ( √ τ φ − √ τ e ) = − v F λ F Dr ( l φ − l e ) . (3.21)For the last expression we used l φ ∝ (cid:112) Dτ φ , l e ∝ v F τ e , D ∝ v F l e . (3.22)If the elastic mean free path is much smaller than the phase coherence length we can neglect thisterm in the conductivity correction. ∆ σσ ∝ − v F λ F Dr l φ (3.23)The Drude conductivity can be expressed in terms of the elastic mean free path and the Fermimomentum. σ ∝ ne τ e m ∝ ne l e p F ∝ e p F l e (cid:126) (3.24)Substituting this in the relative correction expression, where we use the identities 3.22 and λ F ∝ (cid:126) /p F leads to ∆ σ ∝ − e (cid:126) r l φ . (3.25)To get the correction to the conductance we introduce ∆ G = ∆ σr /L and arrive at ∆ G ∝ − e (cid:126) l φ L . (3.26)This quantum correction to the conductance is known as weak localization and arises due to a self-crossing in the diffusive transport of an electron. However, when l φ << L the correction becomesnegligible.At zero temperature the phase of an electron is not broken ( l φ → L ) , so that the correction to theconductance is no longer negligible [27]. This is known as strong, or Anderson, localization. Whenagain the conductance is implemented, the expression for this correction is obtained [6].21 GG ∝ − e / (cid:126) e p F l e r (cid:126) L = − (cid:126) Lp F l e r ∝ − Ll e ( r /λ F ∝ − LN ⊥ l e (3.27)The number of transverse channels available for conduction is determined by the ratio of Fermiwavelength and cross-section. Now the correction is negligible if ∆ G/G << , which is true fora large number of open conduction channels. Since we consider a diffusive wire, we can look atthe distribution of transmission probabilities in equation 2.5, and see that if the average conductanceincreases the number of open channels increases.We can conclude that for our quasi-one dimensional diffusive wire, we can neglect the quantumcorrection to the conductance due to interference effects when we consider wires with length muchlarger than the phase coherence length and a conductance significantly larger than the conductancequantum. 22 hapter 4 Photon absorption and other energyexchange processes in diffusive wires
The model proposed by Remco Schrijvers had the aim to describe the electron energy distribution in adiffusive wire subject to high frequency irradiation with energy relaxation present inside the wire [6].Unfortunately this aim was not fully achieved. The assumption was made that the path traveled bythe electron inside the wire is of no influence to the energy distribution, so that Tien-Gordon theorycould be applied to the reservoirs and the distribution inside the wire was described by the Boltzmannequation. However, this turned out to be incorrect since Tien-Gordon theory assumes averaging overtime and therefore the collision integral can not be evaluated in the correct manner. Therefore adifferent approach is required.A.V. Shytov developed a theoretical framework to calculate the electron energy distribution forwires where the phase coherence time and energy relaxation time exceeds the diffusion time, so thatthe transport is fully coherent. We derive from Green function formalism an equivalent model. Theinsight we gain from this derivation is helpfull in the extension of the theoretical framework of Shytovwith a term accounting for inelastic scattering, breaking the phase of the electrons. Since Greenfunction formalism is not basic knowledge, the most important parts for our derivation are first shortlyexplained.
The Green function formalism provides a strong calculation method which can be used to calculatea variety of properties of many-particle systems. In mathematics Green functions obey a inhomoge-neous differential equation, where the inhomogeneity is singular. As we have seen in the previouschapters, the Schrodinger equation is the central equation in quantum mechanics. Since this is a dif-ferential equation the Green functions apply in describing many-body physics in both equilibrium andnon-equilibrium situations. The basis of the formalism is the definition of the single-particle Greenfunction by the wave function [28]. G ( x, t ; x (cid:48) , t (cid:48) ) = − i (cid:126) (cid:10) ψ | T [ ψ H ( x, t ) ψ + H ( x (cid:48) , t (cid:48) )] | ψ (cid:11) (cid:104) ψ | ψ (cid:105) (4.1)23o the Green function is based on the wave function ψ of the ground state of the system withHamiltonian H and the time-evolving wave function ψ H of the system which evolves like e iHt/ (cid:126) ψ ( t =0) e − iHt/ (cid:126) . The time-ordening operator T is defined in such a way that it always moves the operatorwith the earlier time-argument to the right. T [ A ( t ) B ( t (cid:48) )] = θ ( t − t (cid:48) ) A ( t ) B ( t (cid:48) ) ∓ θ ( t (cid:48) − t ) A ( t (cid:48) ) B ( t ) (4.2)The sign in the time-ordening is dependent on the nature of the considered particle. For fermionsthe sign is negative, so that the Pauli exclusion principle is not violated, and for bosons the sign ispositive. In the following we shall only consider fermions. The equation of motion is now derived bydifferentiating the equation for the single particle Green function with respect to t . i (cid:126) ∂G ( x, t ; x (cid:48) , t (cid:48) ) ∂t = δ ( t − t (cid:48) ) (cid:10) ψ | [ ψ H ( x, t ) , ψ + H ( x (cid:48) , t (cid:48) )] + | ψ (cid:11) (cid:104) ψ | ψ (cid:105) − i (cid:126) (cid:68) ψ | i (cid:126) ∂ψ H ( x,t ) ∂t ψ + H ( x (cid:48) , t (cid:48) ) | ψ (cid:69) (cid:104) ψ | ψ (cid:105) (4.3)From second quantization it is know that the anticommutation of a wave function in the Heisenbergpicture with its conjugate gives a delta-function, so that the first term on the right side of the equationof motion is a multiplication of a spatial and a temporal delta-function. For the second term we use theHeisenberg equation of motion i (cid:126) ∂ψ H ∂t = [ ψ H , H ] . When we consider a particle free of interactionssubject to a Hamiltonian H = − (cid:126) m ( − i ∇ − e (cid:126) A ( t )) , where the vector potential A ( t ) representing anelectric field is integrated in the momentum operator by principle of minimal substitution, the equationof motion for the Green function G of a free particle becomes (cid:26) i (cid:126) ∂∂t − (cid:126) m (cid:16) − i ∇ − e (cid:126) A ( t ) (cid:17) (cid:27) G ( x, t ; x (cid:48) , t (cid:48) ) = (cid:126) δ ( t − t (cid:48) ) δ ( x − x (cid:48) ) . (4.4)Because the Hamiltonian is time-dependent in the vector potential we are already consideringnon-equilibrium. When now also a many-particle system is considered where the particles interactwith eachother, the picture becomes a bit complicated. The wave functions, and thus the Green func-tions, are subject to both an external potential and an internal potential. To ease the calculations theoperations are contour-ordered. This replaces the time-ordening operator T in equation 4.1 with thecontour-ordening operator T C which has the same properties, only not in time, but on the defined con-tour. Because in non-equilibrium the final state does not have to return to the initial state the contour,on which the particle is defined, lies in the complex plane depicted in figure 4.1. We won’t go intodetail on this, but a insightful derivation can be found in Ref.[28] and Ref.[29].The derivation in Ref.[28] and Ref.[29] is an approach from non-equilibrium statistical mechanicsand leads to the Dyson equation for the Green function which consists of the free particle Greenfunction G and a self energy term responsible for the interactions. G (1 , (cid:48) ) = G (1 , (cid:48) ) + 1 (cid:126) (cid:90) dx (cid:90) dx (cid:90) C dτ (cid:90) C dτ G (1 , , G (3 , (cid:48) ) (4.5)The complex contour integral in equation 4.5 is rather impractical in calculations. Fortunatelyanalytic continuation provides a method to replace the contour integrals by real time integrals. TheGreen function is defined by different Green functions on the contour, the lesser and greater Greenfunction, the time-ordered and anti-time-ordered Green function and the advanced and retarded Greenfunction, dependent on the position of the time coordinates of the Green function on the contour.When the initial time t is set to infinity and the interactions are coupled adiabatically, the complex24igure 4.1: The contour on which the particle is defined in non-equilibrium [29].part of the contour depicted in figure 4.1 vanishes. By doing this one neglect initial correlations, butin many situations the interactions in the process of reaching a steady state will wash out these initialcorrelations. In highly transient situations it can however cause problems.When we consider the lesser Green function, which contains the information on the energy distri-bution, the first time coordinate is on the first half of the contour and the second time coordinate onthe second half. The contour can be deformed to form two contours in the limit of initial time goingto infinity as indicated in figure 4.2.Figure 4.2: Deformation of the contour [28].When we look at the product C ( t , t (cid:48) ) = (cid:82) C dτ A ( t , τ ) B ( τ, t (cid:48) ) , the lesser function becomeson the new deformed contour C < ( t , t (cid:48) ) = (cid:82) C dτ A ( t , τ ) B < ( τ, t (cid:48) ) + (cid:82) C dτ A < ( t , τ ) B ( τ, t (cid:48) ) .The integration on the first contour can run from −∞ to t and from t to + ∞ and on the secondcontour from −∞ to t (cid:48) and from t (cid:48) to + ∞ . By doing this all functions can be expressed in lesserfunctions (for t < t (cid:48) ) and greater functions (for t > t (cid:48) ) and when the relations G a (1 , (cid:48) ) = θ ( t (cid:48) − t )[ G < (1 , (cid:48) ) − G > (1 , (cid:48) )] and G r (1 , (cid:48) ) = θ ( t − t (cid:48) )[ G > (1 , (cid:48) ) − G < (1 , (cid:48) )] are usedLangreth’s result for analytic continuation is obtained [30]. C < ( t , t (cid:48) ) = (cid:90) + ∞−∞ dt [ A r ( t , t ) B < ( t, t (cid:48) ) + A < ( t , t ) B a ( t, t (cid:48) )] (4.6)In the next section we shall derive from a simplified Dyson equation a quantum diffusion equation.25n the subsequent section equation 4.6 is used to derive from the complete Dyson equation a quantumdiffusion equation with an interaction term accounting for inelactic scattering. To calculate the energy distribution function of electrons in a mesoscopic wire biased with an acvoltage induced by THz radiation on the reservoirs whereon the wire is coupled, we derive a quantumdiffusion equation from the Dyson equation. First we derive an equation for a situation where inelasticinteractions are neglected by neglecting the self energy term in the Dyson equation and introduceinstead an elastic interaction term which will lead to a relaxation time approximation to account forthe diffusivity of the system [31]. G (1 , (cid:48) ) = G (1 , (cid:48) ) + iI [ G (1 , (cid:48) )] (4.7)Here G (1 , (cid:48) ) is the non-equilibrium Green function of a particle at coordinates x and t providedthat the particle arises from the coordinates x (cid:48) and t (cid:48) defined by equation 4.1. G (1 , (cid:48) ) is the Greenfunction of a free particle given by equation 4.4 and I [ G (1 , (cid:48) )] is the collision term for elastic impurityscattering. By substituting the equation 4.4 in the Dyson equation we can obtain the differential formconsisting of the two conjugate parts. (cid:26) i ∂∂t − (cid:126) m (cid:16) − i ∇ − e (cid:126) A (cid:17) (cid:27) G (1 , (cid:48) ) = δ ( x − x (cid:48) ) δ ( t − t (cid:48) ) + iI [ G (1 , (cid:48) )] (4.8) (cid:26) − i ∂∂t (cid:48) − (cid:126) m (cid:16) i ∇ (cid:48) − e (cid:126) A (cid:48) (cid:17) (cid:27) G (1 , (cid:48) ) = δ ( x − x (cid:48) ) δ ( t − t (cid:48) ) − iI [ G (1 , (cid:48) )] (4.9)These two conjugate parts are subtracted from each other where the two collision terms are re-defined in a single collision term which will later provide the relaxation time approximation for elasticimpurity scattering. (cid:26) i (cid:18) ∂∂t + ∂∂t (cid:48) (cid:19) − (cid:126) m (cid:20)(cid:16) − i ∇ − e (cid:126) A (cid:17) − (cid:16) i ∇ (cid:48) − e (cid:126) A (cid:48) (cid:17) (cid:21)(cid:27) G (1 , (cid:48) )= i I coll [ G (1 , (cid:48) )] (4.10)Now the quadratic terms are expanded and we can use the fact that the vector potential is takenonly time-dependent, so that according to commutation rules the operation ∇ A is equivalent to A ∇ . (cid:26) i (cid:18) ∂∂t + ∂∂t (cid:48) (cid:19) + (cid:126) m (cid:20)(cid:0) ∇ − ∇ (cid:48) (cid:1) − i e (cid:126) ( ∇ A + ∇ (cid:48) A (cid:48) ) − e (cid:126) (cid:0) A − A (cid:48) (cid:1)(cid:21)(cid:27) G (1 , (cid:48) )= i I coll [ G (1 , (cid:48) )] (4.11)For reasons of convenience we will proceed with this equation expressed in Wigner coordinatesdefined like: 26 = t + t (cid:48) , (4.12) t = t − t (cid:48) , (4.13) R = r + r (cid:48) , (4.14) r = r − r (cid:48) . (4.15)To introduce the Wigner coordinates the quadratic parts of equation 4.11 has to be expanded.The summation and difference of the vector potential can be replaced by an representive symbols: A + ( t ) = A ( t ) + A ( t (cid:48) ) and A − ( t ) = A ( t ) − A ( t (cid:48) ) . Because we are interested in the distributionfunction we proceed with the lesser Green function in the equations. The interaction term is now justdependent on the lesser Green function. No analytic continuation procedures have to be followed,because in the end the interaction is given by a relaxation time approximation. (cid:26) i ∂∂T + (cid:126) m (cid:20) ∇ R ∇ r − i e (cid:126) ( ∇ r A − + 12 ∇ R A + ) − e (cid:126) A + A − (cid:21)(cid:27) G < ( r, R, t, T )= i I coll [ G < ] (4.16)Here we make the transition to proceed with the distribution function in a momentum representa-tion of equation 4.16. (cid:26) i ∂∂T + (cid:126) m (cid:20) ∇ R ∇ r − i e (cid:126) ( ∇ r A − + 12 ∇ R A + ) − e (cid:126) A + A − (cid:21)(cid:27) (cid:90) dp (cid:48) π e ip (cid:48) r/ (cid:126) f ( p (cid:48) , R, t, T )= i I coll [ (cid:90) dp (cid:48) π e ip (cid:48) r/ (cid:126) f ( p (cid:48) , R, t, T )] (4.17)The terms containing ∇ r operate first on the integral, so that the operator is replaced by ip (cid:48) / (cid:126) , andthe terms are rearranged. (cid:90) dp (cid:48) π (cid:40) ∂∂T + ( p (cid:48) − eA + ) m (cid:104) ∇ R − i e (cid:126) A − (cid:105)(cid:41) e ip (cid:48) r/ (cid:126) f ( p (cid:48) , R, t, T )= I coll [ (cid:90) dp (cid:48) π e ip (cid:48) r/ (cid:126) f ( p (cid:48) , R, t, T )] (4.18)Then the equation is multiplied by e − ipr/ (cid:126) and a Fourier transform is performed by integratingover all r . (cid:90) dp (cid:48) (cid:90) dr π (cid:40) ∂∂T + ( p (cid:48) − eA + ) m (cid:104) ∇ R − i e (cid:126) A − (cid:105)(cid:41) e i ( p (cid:48) − p ) r/ (cid:126) f ( p (cid:48) , R, t, T )= I coll [ (cid:90) dp (cid:48) (cid:90) dr π e i ( p (cid:48) − p ) r/ (cid:126) f ( p (cid:48) , R, t, T )] (4.19)The Fourier transform in r of the exponent creates the delta function δ ( p (cid:48) − p ) and the integralover p (cid:48) forces by means of the delta function all p (cid:48) to p .27 ∂∂T + ( p − eA + ) m (cid:104) ∇ R − i e (cid:126) A − (cid:105)(cid:41) f ( p, R, t, T ) = I coll [ f ( p, R, t, T )] (4.20)The sum of the vector potential on time t and t (cid:48) modulates the momentum of the charge car-rier. This is a second order effect so that the term in front of the momentum part of the equa-tion above can be replaced by the velocity of the charge carrier. The vector potential is defined as A ( t ) = U/ ( Lω ) cos ( ωt ) . The difference term in the vector potential is then expressed in the Wignercoordinates. A − ( t, T ) = ULω ( cos ( ω ( T + t/ − cos ( ω ( T − t/ − ULω sin ( ωT ) sin ( ωt/ − UiLω sin ( ωT )( e iωt/ − e − iωt/ ) (4.21)This vector potential is substituted in equation 4.20 and the same procedure is followed for anenergy representation as previous done for the momentum representation. A Fourier transform in t isperformed and this is integrated over E (cid:48) . For the terms without the vector potential this operation istrivial since it just replaces the variable t in the distribution function by E . For the part containing thevector potential the situation is a bit more subtle and essential in the understanding of the absorptionof energy quanta of the field by electrons. Therefore this is explicitly shown. sin ( ωT ) (cid:90) dte − iEt/ (cid:126) ( e iωt/ − e − iωt/ ) (cid:90) dE (cid:48) e iE (cid:48) t/ (cid:126) f ( p, R, E (cid:48) , T )= sin ( ωT )( (cid:90) dE (cid:48) (cid:90) dte − i ( E (cid:48) − E/ + ω (cid:126) / t/ (cid:126) f ( p, R, E (cid:48) , T ) − (cid:90) dE (cid:48) (cid:90) dte − i ( E (cid:48) − E/ − ω (cid:126) / t/ (cid:126) f ( p, R, E (cid:48) , T )= sin ( ωT )( (cid:90) dE (cid:48) δ ( E (cid:48) − E + ω (cid:126) / f ( p, R, E (cid:48) , T ) − (cid:90) dE (cid:48) (cid:90) dtδ ( E (cid:48) − E − ω (cid:126) / f ( p, R, E (cid:48) , T )= sin ( ωT ) [ f ( p, R, E − ω (cid:126) / , T ) − f ( p, R, E + ω (cid:126) / , T )]= − sin ( ωT ) [ f ( p, R, E + ω (cid:126) / , T ) − f ( p, R, E − ω (cid:126) / , T )]= − ωsin ( ωT ) D ω f ( p, R, E, T ) (4.22)When this is substituted in the kinetic equation and the operator ∇ R is replaced by a derivativewith respect to the one-dimensional space coordinate x we arrive at a form from which we can go toa diffusion equation. (cid:26) ∂∂T + v (cid:20) ∂∂x − eU (cid:126) L sin ( ωT ) D ω (cid:21)(cid:27) f ( p, x, E, T ) = I coll [ f ] (4.23)The distribution function can be divided in an odd and an even part with respect to p . f e ( p, R, E, T ) = f ( p, R, E, T ) + f ( − p, R, E, T )2 (4.24) f o ( p, R, E, T ) = f ( p, R, E, T ) − f ( − p, R, E, T )2 (4.25)28ecause the field is considered to be uniaxially symmetric the even part of the distribution functiononly depends on the absolute value of p , so that the even part of the distribution function is the distri-bution function as function of energy only: f e ( p, x, E, T ) = f ( x, E, T ) . First the kinetic equation istransformed into two equation for positive and negative momentum. (cid:26) ∂∂T + v (cid:20) ∂∂x − eU (cid:126) L sin ( ωT ) D ω (cid:21)(cid:27) f ( p, x, E, T ) = I [ f ( p, x, E, T )] (4.26) (cid:26) ∂∂T − v (cid:20) ∂∂x − eU (cid:126) L sin ( ωT ) D ω (cid:21)(cid:27) f ( − p, x, E, T ) = I [ f ( − p, x, E, T )] (4.27)The equations are added and subtracted from each other and divided by 2. ∂∂T ( f ( p, x, E, T ) + f ( − p, x, E, T )) / v (cid:20) ∂∂x − eU (cid:126) L sin ( ωT ) D ω (cid:21) ( f ( p, x, E, T ) − f ( − p, x, E, T )) / I [( f ( p, x, E, T ) + f ( − p, x, E, T )) / (4.28) ∂∂T ( f ( p, x, E, T ) − f ( − p, x, E, T )) / v (cid:20) ∂∂x − eU (cid:126) L sin ( ωT ) D ω (cid:21) ( f ( p, x, E, T ) + f ( − p, x, E, T )) / I [( f ( p, x, E, T ) − f ( − p, x, E, T )) / (4.29)The identities of the even and odd part of the distribution function can be implemented and theeven part is changed to the distribution function as function of energy only. ∂∂T f ( x, E, T ) + v (cid:20) ∂∂x − eU (cid:126) L sin ( ωT ) D ω (cid:21) f o ( p, x, E, T ) = I [ f ( x, E, T )] (4.30) ∂∂T f o ( p, x, E, T ) + v (cid:20) ∂∂x − eU (cid:126) L sin ( ωT ) D ω (cid:21) f ( x, E, T ) = I [ f o ( p, x, E, T )] (4.31)Now if we only consider inelastic impurity scattering we only have an collision integral acting onthe odd part of the distribution function. The impurity scattering can only change the momentum ofa charge carrier but can not change the energy. When for this collision integral the relaxation timeapproximation I [ f o ( p, x, E, T )] = − f o ( p, x, E, T ) /τ im is used and the impurity time is consideredto be small the time derivative of equation 4.59 can be neglected. Then f o ( p, x, E, T ) is just a functionof the momentum part times f(x,E,T) times − τ im . When this is substituted in equation 4.58 we arriveat the final form of the quantum diffusion equation, where we take D = v τ im the diffusion constant. (cid:40) ∂∂T − D (cid:20) ∂∂x − eU (cid:126) L sin ( ωT ) D ω (cid:21) (cid:41) f ( x, E, T ) = 0 (4.32)A.V. Shytov also studied the energy distribution of electrons in a diffusive, coherent wire. Theequation he used to calculate the distribution function is equivalent to that derived above.29 .4 Limit situations for the simple quantum diffusion equation The quantum diffusion equation 4.32 can be solved analytically for certain limit situations [32]. There-fore it is convenient to express the equation in dimensionless parameters. t → tω, x → x/L, E → E/eV (4.33)When we introduce the diffusion time for an electron in the wire τ D = L /D equation 4.32becomes (cid:40) ∂∂t − ωτ D (cid:20) ∂∂x − sin ( t ) D ω (cid:21) (cid:41) f ( t, E, x ) = 0 (4.34)This differential equation has for the initial and boundary conditions a Fermi distribution f ( t = 0 , E, x ) = n F ( E ) (4.35) f ( t, E, x = 0) = n F ( E ) (4.36) f ( t, E, x = 1) = n F ( E ) (4.37)The limit situations are defined by the ratio of the field frequency ω and the diffusion time τ D andthe ratio of the photon energy (cid:126) ω and the field energy eV . For ωτ D << the field oscillates slowly with respect to the time that the electron travels diffusivelythrough the wire. In equation 4.34 the time derivative can be neglected and the solution is obtained bysolving the spatial second order differential equation. The solution becomes f ( t, E, x ) = [(1 − x ) e xsin ( t ) D ω + xe ( x − sin ( t ) D ω ] n F ( E ) . (4.38)Following the approach in Ref. [32] we take the Fourier transform in energy domain to find theexponent of the finite difference operator which leads to e zD ω Φ( E ) = ∞ (cid:88) n = −∞ J (2 z )) Φ( E − nω/ . (4.39)Substituting this in the equation for the distribution equation, restoring dimensions and usingtime averaging J n (2 asin ( t )) = J n ( a ) we arrive at the final general expression for the distributionfunction. ¯ f ( E, x ) = (cid:16) − xL (cid:17) ∞ (cid:88) n> E (cid:126) ω J n (cid:18) xeVL (cid:126) ω (cid:19) n F (cid:18) E − n (cid:126) ω (cid:19) + xL ∞ (cid:88) n> E (cid:126) ω J n (cid:18) ( xL − eV (cid:126) ω (cid:19) n F (cid:18) E − n (cid:126) ω (cid:19) (4.40)So we see close resemblance with Tien-Gordon theory where the probability of absorbing n fieldquanta is also given by squared Besselfunctions. The resemblance with a dc biased wire is also visable30n the pre-factors − x/L and x/L , which gives the number of electrons that enter position x fromthe right and the left reservoir [33].When the field energy is much larger than the photon energy, (cid:126) ω << eV , the asymptotic form ofthe Bessel function at x/ω ≈ n >> may be used, which gives ¯ f ( E, x ) = (cid:16) − xL (cid:17) F ( E, x/L ) + xL F ( E, − x/L ) (4.41)where F ( E, x/L ) = π cos − ( ˜ E ) for | ˜ E | < with ˜ E = LExeV . For ˜ E < − the occupation is oneand for ˜ E > the occupation is zero. Figure 4.3 shows the distribution in slow field, strong signallimit.Figure 4.3: Left the electron energy distribution in a mesoscopic wire ac biased in the slow field,strong signal limit ( ωτ D << , (cid:126) ω << eV ), right with the blue line the distribution on position x = 0 . and with the red line the distribution on position x = 0 . . In the fast field limit the diffusion time is much larger than the reciprocal frequency of the field, ωτ D >> . This means that equation 4.34 practically becomes time-independent, since the time-derivative is proportional to /ωτ D . Averaging equation 4.34 over the field period, leads to an equationfor the time-averaged distribution function. (cid:20) ∂ ∂x + 12 D ω (cid:21) ¯ f ( E, x ) = 0 (4.42)In the limit (cid:126) ω << eV the finite difference operator D ω can be replaced by the partial energyderivative ∂/∂(cid:15) . This makes equation 4.42 become a Laplace equation in a two-dimensional stripdefined by < x < and −∞ < (cid:15) < ∞ . This strip can be conformally mapped by the function w = exp [ πi ( x + i (cid:112) (2) (cid:15) )] onto the half-plane Imw > [32] [34]. The boundary condition on the line Imw = 0 at zero temperature is set to be ¯ f ∞ ( w ) = 0 for | Rew | < and ¯ f ∞ ( w ) = 1 for | Rew | > .The imaginary part of the analytic function gives the solution of this boundary value problem. ¯ f ∞ ( w ) = Im π ln (cid:18) − w w (cid:19) (4.43)31hen the original dimensional units are restored the final expression for the time-averaged distri-bution function is ¯ f ∞ ( (cid:15), x ) = 1 π cot − (cid:32) sinh ( π √ (cid:15)/eV ) sin ( πx/L ) (cid:33) . (4.44)In the fast field the energy distribution does not have to go to zero at high energies. The energygained from the field is not limited by eV . Instead an electron has a finite probability of oscillatingseveral times back and forth with the field in the wire before leaving the wire, thereby gaining multipleenergy quanta of the field which sum exceeds eV . Figure 4.4 shows the electron energy distributionin the fast field, strong signal limit.Figure 4.4: Left the electron energy distribution in a mesoscopic wire ac biased in the fast field, strongsignal limit ( ωτ D >> , (cid:126) ω << eV ), right with the blue line the distribution on position x = 0 . and with the red line the distribution on position x = 0 . . So far only coherent transport is considered. If the length of the wire is extended in such a way that thediffusion time becomes of the same order as the phase coherence time and energy relaxation time thissimple model breaks down. Therefore this model has to be extended to account for electron-electronand electron-phonon interactions. This is done by evaluating the complete Dyson equation 4.5, wherewe isolate the collision term for the elastic impurity scattering that is treated with a relaxation timeapproximation in the same way as before. G (1 , (cid:48) ) = G (1 , (cid:48) ) + iI im [ G (1 , (cid:48) )] + 1 (cid:126) (cid:90) dx (cid:90) dx (cid:90) dτ (cid:90) dτ G (1 , , G (3 , (cid:48) ) (4.45)By substituting the equation of motion for the Green function of a free particle we obtain againtwo conjugate equations. 32 i ∂∂t − (cid:126) m (cid:16) − i ∇ − e (cid:126) A (cid:17) (cid:27) G (1 , (cid:48) ) = δ ( x − x (cid:48) ) δ ( t − t (cid:48) ) + iI [ G (1 , (cid:48) )]+ 1 (cid:126) (cid:90) dτ (cid:90) dy Σ( x , t , y, τ ) G ( y, τ, x (cid:48) , t (cid:48) ) (4.46) (cid:26) − i ∂∂t (cid:48) − (cid:126) m (cid:16) i ∇ (cid:48) − e (cid:126) A (cid:48) (cid:17) (cid:27) G (1 , (cid:48) ) = δ ( x − x (cid:48) ) δ ( t − t (cid:48) ) − iI [ G (1 , (cid:48) )]+ 1 (cid:126) (cid:90) dτ (cid:90) dyG ( x , t , y, τ )Σ( y, τ, x (cid:48) , t (cid:48) ) (4.47)As before we are interested in the distribution function, so we concentrate on the lesser Greenfunction by an analytic continuation of the above functions where we concentrate on the self energypart of the functions. The remaining part of the equations in the derivation is similar to the derivationwithout inelastic interactions. I [ G < ] = 1 (cid:126) (cid:90) dτ (cid:90) dy (cid:0) Σ r ( x , t , y, τ ) G < ( y, τ, x (cid:48) , t (cid:48) ) + Σ < ( x , t , y, τ ) G a ( y, τ, x (cid:48) , t (cid:48) ) (cid:1) (4.48) I [ G < ] = 1 (cid:126) (cid:90) dτ (cid:90) dy (cid:0) G r ( x , t , y, τ )Σ < ( y, τ, x (cid:48) , t (cid:48) ) + G < ( x , t , y, τ )Σ a ( y, τ, x (cid:48) , t (cid:48) ) (cid:1) (4.49)These two equations are subtracted from each other. I [ G ] = 1 (cid:126) (cid:90) dτ (cid:90) dy (Σ r ( x , t , y, τ ) G < ( y, τ, x (cid:48) , t (cid:48) ) + Σ < ( x , t , y, τ ) G a ( y, τ, x (cid:48) , t (cid:48) ) − G r ( x , t , y, τ )Σ < ( y, τ, x (cid:48) , t (cid:48) ) − G < ( x , t , y, τ )Σ a ( y, τ, x (cid:48) , t (cid:48) )) (4.50)Now the following identities are introduced to gain insight in the derivation [28]. A r = 12 ( A r + A a ) + 1 / A r − A a ) A a = 12 ( A a + A r ) + 1 / A a − A r )Σ = 12 (Σ r + Σ a ) G = 12 ( G r + G a ) A = i ( G r − G a )Γ = i (Σ r − Σ a ) The terms are arranged so that everything is expressed in commutators and anti-commutators. I [ G < ] = 1 (cid:126) (cid:90) dτ (cid:90) dy (cid:18) [Σ , G < ] + [Σ < , G ] + 12 (cid:8) Σ > , G < (cid:9) − (cid:8) G > , Σ < (cid:9)(cid:19) (4.51)33o simplify the calculations we assume the scattering to be local in space, so that the integraloperation over y forces the integration variable towards the central space coordinate. Also we canmake the assumption of weak interactions, so that we can apply the quasi-particle approximation.Because we also used a gradient expansion of the potential, the first two commutators of the aboverelation are second order and can be neglected. Basically this means that the density of states of thequasi-particles in the wire is not affected by the vector potential nor the interactions. I [ G < ] = 1 (cid:126) (cid:90) dτ ( 12 Σ > ( x , t ; x , τ ) G < ( x , τ ; x (cid:48) , t (cid:48) ) + 12 G < ( x , t ; x (cid:48) , τ )Σ > ( x (cid:48) , τ ; x (cid:48) , t (cid:48) ) −
12 Σ < ( x , t ; x , τ ) G > ( x , τ ; x (cid:48) , t (cid:48) ) − G > ( x , t ; x (cid:48) , τ )Σ < ( x (cid:48) , τ ; x (cid:48) , t (cid:48) )) (4.52)When we now also assume that the scattering is instantaneous, the integral over τ forces theintegration variable towards the second time variable of the self energy in the product of the selfenergy and the Green function. I [ G < ] = 1 (cid:126) ( 12 Σ > ( x , t ; x , t ) G < ( x , t ; x (cid:48) , t (cid:48) ) + 12 G < ( x , t ; x (cid:48) , t (cid:48) )Σ > ( x (cid:48) , t (cid:48) ; x (cid:48) , t (cid:48) ) −
12 Σ < ( x , t ; x , t ) G > ( x , t ; x (cid:48) , t (cid:48) ) − G > ( x , t ; x (cid:48) , t (cid:48) )Σ < ( x (cid:48) , t (cid:48) ; x (cid:48) , t (cid:48) )) (4.53)As we assume a slow variation of the Green function induced by the vector potential and weassume the interactions to be weak, we can state that the effect of the self energy on time t is thesame as that at time t (cid:48) . So the self energies at t and t (cid:48) can be replaced by a single self energy Σ( x , t ; x (cid:48) , t (cid:48) ) . I [ G < ] = 1 (cid:126) (cid:0) Σ > ( x , t ; x (cid:48) , t (cid:48) ) G < ( x , t ; x (cid:48) , t (cid:48) ) − Σ < ( x , t ; x (cid:48) , t (cid:48) ) G > ( x , t ; x (cid:48) , t (cid:48) ) (cid:1) (4.54)By applying the Wigner transformation to this collision term, the product of the self energies withthe Green functions can be interpreted as the imaginary in- and out scattering rates i (cid:126) Γ e,h with theelectron and hole distribution [35] [36]. I [ f ] = i (cid:90) dE (cid:48) (cid:90) dp (cid:48) e i ( E (cid:48) t + p (cid:48) r ) / (cid:126) (Γ h ( R, T, E (cid:48) , p (cid:48) ) f ( R, T, E (cid:48) , p (cid:48) ) − Γ e ( R, T, E (cid:48) , p (cid:48) ) f h ( R, T, E (cid:48) , p (cid:48) )) (4.55)This is again multiplied by e − i ( Et − pr ) / (cid:126) and integrated over t and r leading to the final form ofthe total collision term due to inelastic scattering where this is multiplied by i from the rest of theequation. I tot [ f ] = Γ h ( R, T, E, p ) f ( R, T, E, p ) − Γ e ( R, T, E, p )(1 − f ( R, T, E, p )) (4.56)The two parts of the quantum diffusion equation are again connected. (cid:26) ∂∂T − v (cid:20) ∂∂x − eU (cid:126) L sin ( ωT ) D ω (cid:21)(cid:27) f ( E, p, x, T ) = I im [ f ] + I tot [ f ] (4.57)34ame procedure is followed to come to a diffusion equation as for elastic impurity scattering. Theequation is divided in an even and odd part, where the elastic impurity scattering only contributes tothe even part and the inelastic interactions contribute to the odd part. ∂∂T f ( x, E, T ) + v (cid:20) ∂∂x − eU (cid:126) L sin ( ωT ) D ω (cid:21) f o ( p, x, E, T ) = I tot [ f ( x, E, T )] (4.58) ∂∂T f o ( p, x, E, T ) + v (cid:20) ∂∂x − eU (cid:126) L sin ( ωT ) D ω (cid:21) f ( x, E, T ) = I [ f o ( p, x, E, T )] (4.59)Taking the same relaxation time approximation I [ f o ( p, x, E, T )] = − f o ( p, x, E, T ) /τ im for theimpurity scattering leads to the desired quantum diffusion equation. (cid:40) ∂∂T − D (cid:20) ∂∂x − eU (cid:126) L sin ( ωT ) D ω (cid:21) (cid:41) f ( E, x, T ) = I tot f ( E, x, T ) (4.60)So we see that the quantum diffusion equation 4.32 is extended with a term that controls the in- andoutscattering of quasi-particles at energy E due to inelastic collisions. These inelastic collisions couldbe due to the interaction between two quasi-particles or due to the interaction between a quasi-particleand a phonon. In the next section we will derive expressions for these interactions. The main energy relaxation mechanisms are electron-electron and electron-phonon scattering andthe sum of these contributions give the total interaction term. I tot [ f ] = I e − e [ f ] + I e − ph [ f ] (4.61)Both collision terms have an inscattering and outscattering term as seen in equation 4.56. A quasi-particle with energy E has an collision term I coll ( x, E, { f } ) = I incoll ( x, E, { f } ) − I outcoll ( x, E, { f } ) (4.62)The collision terms due to electron-electron scattering and electron-phonon scattering can be cal-culated independently of each other. First we will tread the interaction between electrons and phonon.Subsequently we look at the interactions between electrons. Let’s first focus on the electron-phonon interactions. To begin some assumptions have to be made.When we only want to consider acoustic phonons with a dispersion relation between energy and wavevector (cid:15) k = (cid:126) sq , with s the sound velocity, the phonon temperature T ph has to be small comparedto the Debye temperature T D . Further the electronic wave functions can be approximated by planewaves, which is justified by the fact that electron-phonon coupling is only relevant for higher energiesand from the dispersion relation it is seen that large wave vectors are associated with these energies.Then it is probable that the electronic mean free path is larger than /q . Also the electron-phonon the electron is in fact a quasi-particle | M ( q ) | = | M | q/V ,where | M | is geometry independent. This only is valid for spherical Fermi surfaces [37].The transition of an electron to a state with energy E can either be due to the absorption or theemission of a phonon. The same can be said of the transition out of the state with energy E . We candefine the transition due to absorption by W − and the transition due to emission by W + . Furtherwe know that the state from which the particle departes has to be occupied and the state in whichthe particle arrives has to be unoccupied. The latter is a direct consequence of the fact that we lookat fermions and according to the Pauli exclusion principle a state can only be occupied by a singlefermion. This leads to the following collision terms [38]. I ineph ( x, E k , [ f ]) = (cid:90) dE k (cid:48) W + ( x, E k (cid:48) , E k ) f ( x, E k − E k (cid:48) )(1 − f ( x, E k )) n ph ( E k − k (cid:48) )+ (cid:90) dE k (cid:48) W ( x, E k (cid:48) , E k ) − f ( x, E k − E k (cid:48) )(1 − f ( x, E k ))(1 + n ph ( E k (cid:48) − k ) (4.63) I outeph ( x, E k , [ f ]) = (cid:90) dE k (cid:48) W + ( x, E k (cid:48) , E k ) f ( x, E k )(1 − f ( x, E k − E k (cid:48) ))(1 + n ph ( E k − k (cid:48) ))+ (cid:90) dE k (cid:48) W − ( x, E k (cid:48) , E k ) f ( x, E k )(1 − f ( x, E k − E k (cid:48) )) n ph ( E k (cid:48) − k ) (4.64)Here n ph represents the Bose energy distribution of the phonons, n ph ( E ) = ( exp ( E/kT ) − − .The transition probabilities are given by Fermi’s Golden Rule [38]. W ± ( x, E k (cid:48) , E k ) = 2 π (cid:126) | α k (cid:48) − k | δ ( E k (cid:48) − E k ± E ± ( k − k (cid:48) ) ) (4.65)To obtain the collision rate at which an electron with wave vector k emits or absorbs a phononof energy E | k − k (cid:48) | the equations 4.63 and 4.64 have to be summed over k (cid:48) with E ( k − k (cid:48) ) fixed. Adetailed derivation can be found in Ref. [38]. I ineph ( x, E, [ f ]) = 2 π (cid:90) d(cid:15)α F ( (cid:15) ) f ( x, E − (cid:15) )(1 − f ( x, E )) n ph ( (cid:15) )+2 π (cid:90) d(cid:15)α F ( (cid:15) ) f ( x, E + (cid:15) )(1 − f ( x, E ))(1 + n ph ( (cid:15) )) (4.66) I outeph ( x, E, [ f ]) = 2 π (cid:90) d(cid:15)α F ( (cid:15) ) f ( x, E )(1 − f ( x, E − (cid:15) ))(1 + n ph ( (cid:15) )+2 π (cid:90) d(cid:15)α F ( (cid:15) ) f ( x, E )(1 − f ( x, E + (cid:15) )) n ph ( (cid:15) ) (4.67)The so called Eliashberg function α F ( (cid:15) ) is dependent on the coupling between the electrons andphonons. In Ref. [39] this function is determined to be α F ( (cid:15) ) = | M | (cid:15) π s N (0) , (4.68) | M | = πs Σ12 ζ (5) k b . (4.69)36ere | M | is the matrix element depending on the defined deformation potential and N (0) is theelectronic density of states at Fermi level. The precise microscopic form of | M | is dependent on thedetails of the lattice structure. Therefore in Ref. [39] they present this matrix element in terms ofa measurable quantity Σ related to the power dissipated to the lattice of volume V by P = Σ V T .A detailed form of the electron-phonon interactions and the temperature dependence in disorderedconductors can be found in Ref. [40]. The interaction between quasi-particles is due to the Coulomb potential of the particles. This Coulombinteraction is screened by an effective medium build from all the electrons in the metal. Altshuler etal . showed that multiple scattering events due to disorder in the system reduces the lifetime of thequasi-particle [41]. At zero temperature the lifetime of a particle obeying Fermi statistics in state | α > with energy (cid:15) α above Fermi level that interacts with a particle in state | γ > with energy (cid:15) γ directlyfollows from Fermi’s Golden Rule [42]. τ α = 4 π (cid:126) (cid:88) βγδ | (cid:104) αγ | U | βδ (cid:105) | δ ( (cid:15) α + (cid:15) γ − (cid:15) β − (cid:15) δ ) (4.70) U is the interaction potential from which the states | α > and | γ > evolve in the states | β > and | δ > . This lifetime has to be averaged over all states having energy (cid:15) in order not to single out a givestate. τ ee ( (cid:15) ) = 4 π (cid:126) ν (cid:88) αβγδ | (cid:104) αγ | U | βδ (cid:105) | δ ( (cid:15) α + (cid:15) γ − (cid:15) β − (cid:15) δ ) δ ( (cid:15) − (cid:15) α ) (4.71)When the energy of the states | γ > are denoted by (cid:15) (cid:48) and the energy exchange involved in thescattering is ω , energy conservation leads to energies of the final states | β > and | δ > of (cid:15) − ω and (cid:15) (cid:48) + ω . This is depicted in figure 4.5.Figure 4.5: The energy exchange in scattering between quasi-particles. Left the initial situation, rightthe final situation [42].Considering all possible initial states | γ > leads to integration over (cid:15) (cid:48) and ω .37 τ ee ( (cid:15) ) = 4 π (cid:126) ν (cid:90) (cid:15) dω (cid:90) − ω d(cid:15) (cid:48) (cid:88) αβγδ | (cid:104) αγ | U | βδ (cid:105) | δ ( (cid:15) − (cid:15) α ) δ ( (cid:15) (cid:48) − (cid:15) γ ) δ ( (cid:15) − ω − (cid:15) β ) δ ( (cid:15) (cid:48) + ω − (cid:15) δ ) (4.72)Now when the requirements of zero temperature and the Fermi statistics are dropped, this ap-proach still holds when we include the occupation numbers of the states in the obtained result 4.72. τ ee ( (cid:15) ) = 4 π (cid:126) ν (cid:90) (cid:15) dω (cid:90) − ω d(cid:15) (cid:48) ( f (cid:15) (cid:48) (1 − f (cid:15) − ω )(1 − f (cid:15) (cid:48) + ω ) + (1 − f (cid:15) (cid:48) ) f (cid:15) − ω f (cid:15) (cid:48) + ω ) W ( ω ) (4.73)Where W ( ω ) = (cid:88) αβγδ | (cid:104) αγ | U | βδ (cid:105) | δ ( (cid:15) − (cid:15) α ) δ ( (cid:15) (cid:48) − (cid:15) γ ) δ ( (cid:15) − ω − (cid:15) β ) δ ( (cid:15) (cid:48) + ω − (cid:15) δ ) . (4.74)To complete the collision term for electron-electron interactions it is convenient to let go thenotation of Ref. [42] and proceed with the notation used for electron-phonon interactions. We definethe kernel K ( (cid:15) ) , which follows from (4 π ) / ( (cid:126) ν ) W ( ω ) . Further the collision rate can be splitted inthe inscattering and outscattering term by multiplying the first part by f and the second part by − f . I inee ( x, E, [ f ]) = (cid:90) d(cid:15) (cid:90) dE (cid:48) K ( (cid:15) ) f ( x, E − (cid:15) ) f ( x, E (cid:48) + (cid:15) )(1 − f ( x, E ))(1 − f ( x, E (cid:48) )) (4.75) I outee ( x, E, [ f ]) = − (cid:90) d(cid:15) (cid:90) dE (cid:48) K ( (cid:15) ) f ( x, E ) f ( x, E (cid:48) )(1 − f ( x, E − (cid:15) ))(1 − f ( x, E (cid:48) + (cid:15) )) (4.76)In Ref. [41] and Ref. [37] the matrix element of the transition in a disorded medium is calculated.Here we will not follow the complete derivation, but directly look at the result for the kernel K ( (cid:15) ) . K ( (cid:15) ) = ν F π (cid:126) (cid:90) d q | U (cid:15)/ (cid:126) ( q ) | (cid:18) D q D q + ( (cid:15)/ (cid:126) ) (cid:19) (4.77)The bare Coulomb potential U ( q ) and the polarizability Π( q , (cid:15)/ (cid:126) ) of the electron fluid deter-mines the screened Coulomb potential U (cid:15)/ (cid:126) ( q ) effectively experienced by the quasi-particles. U (cid:15)/ (cid:126) ( q ) = U ( q )1 + Π( q , (cid:15)/ (cid:126) ) U ( q ) (4.78)where Π( q , (cid:15)/ (cid:126) ) = ν F D q D q − i(cid:15)/ (cid:126) . (4.79)In a metal the density of states ν F is so large (order of J − m − ) that the polarizability dom-inates the denominator in the expression of the screened Coulomb potential. Therefore equation 4.78simplifies to 38 (cid:15)/ (cid:126) ( q ) = 1Π( q , (cid:15)/ (cid:126) ) , (4.80)and the total kernel becomes K ( (cid:15) ) = 14 π ν F (cid:126) (cid:90) d q D q + ( (cid:15)/ (cid:126) ) . (4.81)If we consider a metallic wire with cross-section S = wt , where w is the width and t is thethickness of the wire, only the uniform modes in transverse dimensions contribute to K ( (cid:15) ) if theenergies (cid:15) are smaller than (cid:126) D/max ( w , t ) . This leads to K ( (cid:15) ) = (cid:16) √ Dπ (cid:126) / ν F S (cid:17) − (cid:15) − / (4.82)This derivation leads to a difference with the result for the screened Coulomb interactions obtainedby Kamanev and Andreev [43]. They found K ( (cid:15) ) to be a factor 2 larger. Experiments showed thatthe energy dependence of the collision term is accurate, but the intensity is off. A discussion can befound in Ref. [26] and Ref. [44]. In this chapter we used the fact that the electrons involved in the ac quantum transport in a diffusivewire can be described as quasi-particles according to the Fermi liquid theory. For coherent transportthe energy distribution of the quasi-particles obeys a relative simple quantum diffusion equation. Thenon-equilibrium in a mesoscopic, diffusive wire induced by a time-dependent field manifests itselfin the energy distribution. When the length of the wire is extended, the transport becomes incoher-ent and the redistribution of energy among the quasi-particles has to be evaluated. For this reasonthe relative simple quantum diffusion equation is extended with a collision integral accounting forelectron-electron and electron-phonon interactions. In the next chapter the model is evaluated usingnumerical calculation methods. 39 hapter 5
Numerical results
The model developed in chapter 4 allows the evaluation of the quasi-particle energy distribution in amesoscopic wire ac biased with irradiation. For very short wires, where the phase coherence time andenergy relaxation time exceed the diffusion time, the transport is fully coherent and the distributionfunction in the wire is never an equilibrium function. The non-equilibrium description is quite differ-ent in the two field limits, ωτ D << and ωτ D >> , as discussed in section 4.4. In the slow fieldlimit ( ωτ D << ) the quasi-particle energy distribution is varying in time, following the oscillationof the field instantaneously. In the limit eV >> (cid:126) ω this shows close resemblance with the dc biasedwire and the quasi-particle energy distribution is given by a two step function which varies in time.The fast field limit ( ωτ D >> ) is quite different. In this limit the quasi-particle energy distributionis given by a time-independent multiple step function. For energies eV >> (cid:126) ω the steps smooth outand a continuous function is obtained which provides a finite probability of finding a quasi-particlefar from the Fermi energy.To evaluate the slow field regime and fast field regime we can define some ratio (cid:126) ω/eV and varythe product ωτ D . Shytov showed that the crossover from low-frequency behavior to high-frequencybehavior occurs at ωτ D ≈ [32]. This is due to the fact that that the quasi-particle energy distri-bution relaxes at t → ∞ as exp ( − µt ) , where µ = π /τ D is the lowest non-zero eigenvalue of thediffusion operator. It is reasonable to assume that the crossover occurs when the relaxation time is ofthe order of the field period, π/ω . So the crossover is estimated to occur at ωτ D ∝ π ≈ , whichis close to 100.This theoretical research is done in an experimental research group. The strong connection withexperimental physics leads to the desire to evaluate the model for realistic situations (THz frequenciesand field amplitudes of 1-20 meV), so that when an experimental setup is realized the model canprovide the understanding of the experimental results. We apply these conditions in the evaluationof equation 4.60 using numerical calculation methods. The equation is expressed in dimensionlessparameters in the same way we did for the discussed limit situations of coherent transport. (cid:40) ∂∂t − ωτ D (cid:20) ∂∂x − sin ( t ) D ω (cid:21) (cid:41) f ( t, E, x ) = I tot f ( t, E, x ) ω . (5.1)As explained in the previous chapter, the collision term can be neglected for fully coherent trans-port. The energy of the quasi-particles in the wire is only affected by photon absorption and thediffusive transport itself. This means that the neglect of the collision term is only valid for short wires.40o make this somewhat more quantitative, we consider the phase coherence time of a quasi-particle.Two phase breaking mechanisms are distinguished, electron-phonon interaction and electron-electroninteraction. The experimental part of the research focuses on aluminum wires with a diffusion coef-ficient of about 100 cm s − measured at liquid helium temperatures, so that we first concentrate onthis material and temperature. Above temperatures of 1 K the phase breaking mechanism is electron-phonon interaction. The phase coherence time is approximated by [26]: τ ( e − ph ) φ = 7 πζ (3)9 E F N (0) k b (cid:126) ρs k F T . (5.2)Here E F is the Fermi energy, N (0) is the density of states at Fermi energy, ρ is the mass density, s is the speed of sound and k F is the Fermi wave vector. The phase coherence time at 2 K, which canbe achieved in a pumped liquid helium cryostat, is approximately 10 ns. This is equivalent to a wireof length L = (cid:112) Dτ φ = 10 µ m. So for wires shorter than this length the transport is coherent. Sincethis is an approximation we decided to use in our calculations wires of maximum length of 7 µ m, witha diffusion time of 5 ns, to be certain that the transport is coherent. We evaluate the quasi-particleenergy distribution for coherent transport at 2 K from the slow field regime to the fast field regime.For the slow field regime ωτ D = 1 we choose a wire of 56 nm and a field frequency of 0.5 THz. Inthe fast field regime ωτ D = 30000 we take a wire of 7 µ m and a field frequency of 2 THz. A wire of400 nm and a field frequency of 1 THz makes the evaluation of the intermediate regime ωτ D = 100 possible. We define the ratio (cid:126) ω/eV = 0 . for all regimes, so the field amplitude varies from 5 meVin the slow field regime to 20 meV in the fast field regime.For extended wires, the diffusion time can exceed the energy relaxation time, so that the transportis incoherent. In this report we will focus on the fast field regime for incoherent transport. The slowfield regime is already quite well understood [4] [6] and thereby the frequency of the field shouldbe extremely low to have a small product ωτ D , where τ D should be of the order of τ E the energyrelaxation time.The effect of electron-phonon interactions is evaluated at a temperature of 2 K. We determinedthe phase coherence time for electron-phonon interaction to be 10 ns. This will be our referencein defining the ratio between diffusion time and energy relaxation time, since the energy relaxationtime is of the same order of magnitude as the coherence time. The intensity of the interaction be-tween quasi-particle and phonon can be calculated from equation 4.68. In Ref. [39] the quantity Σ related to the power dissipation is given to be about 1 GWm − K − . This brings the intensity toabout 2 ns − meV − , which is close to the empirical intensity of 4 ns − meV − which followed fromexperiments done by Huard et al. [26].Below 1 K the situation becomes a bit complicated, since aluminum is no longer a normal metal,but has experienced a phase transition to the superconducting phase. We proceed below 1 K withan undefined material with the same diffusion coefficient of 100 cm s − , so that we can evaluate theeffect of electron-electron interactions on the energy distribution of the quasi-particles. The phasecoherence time for electron-electron interactions is approximated with [26]: τ eeφ = (cid:18) πk ee k b √ (cid:126) (cid:19) − / T − / . (5.3)Here k e e is the prefactor in the kernel of equation 4.82 and given by (cid:16) √ Dπ (cid:126) / N (0) S (cid:17) − . Atemperature of 500 mK and a cross-section of the wire of 400 nm gives an intensity of the interactionsof 0.8 ns − meV − / . The empirical intensity found by Huard et al. in silver is 0.4 ns − meV − / , so41his can be used as a realistic value. This leads to a phase coherence time of about 1 ns at 500 mK.The energy relaxation time is of the same order of magnitude.So the purpose of this chapter is dual. First, we want to investigate the quasi-particle energydistribution for coherent transport and how the slow field regime differs from the fast field regime.Second, we want to investigate the quasi-particle energy distribution for incoherent transport in thefast field regime and how weak interactions are distinguished from strong interactions. Numerical calculation principles allow the evaluation of the quantum diffusion equation 5.1 [45]. Weuse Euler’s method using finite difference approximations for the space, time and energy variables.By iterating the calculation a stable solution for the quasi-particle energy distribution is obtained. Thisiteration is performed on the time variable, so that every time step dt results in a new function whicharises from the old function and the non-time operation part of the equation: f new = f old + dtωτ D df (5.4)where df = ( D x ) f old + 2 sin ( mdt ) D x D E f old + sin ( mdt )( D E ) f old + τ D ( Iin − Iout ) . (5.5)Here D x,E is the finite difference operator for space and energy, respectively, and m is the numberof iteration. For the diffusion equation without inelastic scattering the terms Iin and
Iout in equation5.5 disappear. The finite difference operators are sparse matrices, which means that the percentageof zero elements greatly exceeds the percentage of non-zero elements and their distribution is suchthat it is advantageous to use this for a more efficient calculations. The MATLAB function sparse provides the possibility to exploit the sparse nature of the operator. What this function does is isolatethe non-zero elements, so that only these elements are used in the calculation. The MATLAB code ofthe simulation program can be found in the appendix.
As explained in the previous section, the crossover from low-frequency behavior to high-frequencybehavior occurs at ωτ D = 100 . So when we want to evaluate the slow field regime it is sufficientto have a product ωτ D = 1 , which is two orders of magnitude below the crossover. As said beforethis is based on realistic values, but throughout this section we will only work with relative values.The amplitude of the field is such that (cid:126) ω/eV = 0 . . In the slow field regime the energy distributionis highly time-dependent. The time-averaged distribution function is given in figure 5.1 for threedifferent positions in the wire. The full space dependency is shown in the three dimensional figure inappendix C.The time-dependency in the slow field regime is evaluated by running the simulation during twofield periods and plot the normalized occupation at the photon steps. This normalization is performedby taking the value for every iteration on the first and second photon step, determine the maximumvalue during the iteration process and divide the value determined in every iteration by this maximum42igure 5.1: The quasi-particle energy distribution in the slow field regime, ωτ D = 1 , and (cid:126) ω/eV =0 . at 3 different positions in the wire at 2 K.value: | n i | = f i ( E + (cid:126) ω/ , E + 3 (cid:126) ω/ /max ( f i ( E + (cid:126) ω/ , E + 3 (cid:126) ω/ . The first photon stepimmediately follows the field, where the second photon step shows a slight delay as shown in the upleft picture in figure 5.4.For the fast field regime we take ωτ D = 30000 , two orders of magnitude above the crossoverfrom low-frequency behavior to high-frequency behavior. The amplitude of the field is again definedso that (cid:126) ω/eV = 0 . . The simulation of this situation is time-averaged depicted in figure 5.2 at threedifferent positions in the wire, where we average over a large number of periods to obtain the finaltime-independent distribution. The full space dependency is shown in the three dimensional figurein appendix C. To evaluate the time-dependence in the fast field regime the normalized value of theoccupation in the three photon steps in the distribution is plotted during the evolution of the function.It appears that in the fast field regime the energy distribution indeed becomes time-independent asshown in the up right picture in figure 5.4 and the occupation at the photon energies is maximumwhen the diffusion time is reached.The intermediate frequency regime where the crossover occurs from low-frequency behavior tohigh-frequency behavior is evaluated at ωτ D = 100 . The time-averaged distribution function at threedifferent positions is given in figure 5.3. The full space dependency is shown in the three dimensionalfigure in appendix C. The time-dependency in the distribution function is drastically decreased at thecrossover as seen in figure 5.4. When the diffusion time is reached, the occupation at the photonenergies is at maximum with a slight oscillatory deviation with the field period.43igure 5.2: The quasi-particle energy distribution in the fast field regime for ωτ D = 30000 and (cid:126) ω/eV = 0 . at 3 different positions in the wire at 2 K.Figure 5.3: The quasi-particle energy distribution in the intermediate frequency regime for ωτ D = 100 and (cid:126) ω/eV = 0 . at 3 different positions in the wire at 2 K.44igure 5.4: The time-evolution of the occupation in the photon steps for the three frequency regimes:up left the low-frequency regime ωτ D = 1 , up right the high-frequency regime ωτ D = 30000 anddown in the middle the crossover ωτ D = 100 . The blue line gives the normalized occupation at thefirst photon step and the red line at the second photon step. As seen in the previous section the absorption of field quanta in the short wire, where the transport ofquasi-particles is fully coherent, induces a staircase structure in the quasi-particle energy distribution.Now we want to investigate what happens when the wire is extended, so that photon-absorption is nolonger the only mechanism that affects the energy distribution, but also interactions between quasi-particles and between quasi-particles and phonons come into play. It appears that these interactionsredistribute the quasi-particles with respect to the energy, so that the occupation of the energy levelsis changed with respect to the occupation in the coherent situation. The effect of the two phasebreaking mechanisms is quite different. We expect that the interactions between quasi-particles causea smearing in the staircase structure, while the interactions between quasi-particles and phonons causethe annihilation of the photon steps and finally, in the strong interaction limit, leave a Fermi functionwith the bath temperature. In this section we will limit ourself to the fast field regime which is, asexplained in the previous section, the most interesting domain.45 lectron-phonon interactions
Since the first experiments are planned to be done at liquid helium temperatures, we first focuson the effect of electron-phonon interactions on the energy distribution of the quasi-particles. We candistinguish different interaction regimes. The weak interaction regime is found for τ D ≈ τ E and thestrong interaction regime is found for τ D >> τ E .Let’s first look at the weak interaction regime where the diffusion time is of the order of the energyrelaxation time, so we define ωτ D = 50000 and τ D ≈ τ E . What we expect is that the energy gainedfrom the field by a quasi-particle is redistributed, where the photon steps due to the absorption ofmultiple field quanta are first influenced. The result of this simulation is shown in figure 5.5. Theexpected disappearance of the photon steps due to the absorption of multiple field quanta is indeedobserved and the transition in the first photon step is smoothed. The space dependency is shown inthe three dimensional figure in appendix C.Figure 5.5: The quasi-particle energy distribution in the fast field, weak electron-phonon interactionwith ωτ D = 50000 , (cid:126) ω/eV = 0 . and τ D ≈ τ E at 3 different positions in the wire at 2 K.When the length of the wire is increased, the diffusion time becomes much higher than the energyrelaxation time. So we enter the strong interaction regime and define ωτ D = 10 and τ D ≈ τ E . Inthe fast field regime we expect a Fermi function at the bath temperature on every position in the wire.The result, shown in figure 5.6, indeed shows a Fermi function. There is a slight deviation from theFermi function with the bath temperature.When we look at the deviation of the calculated energy distributions from the equilibrium functionat bath temperature we see what the effect of weak and strong interactions is. For weak interactionsthe deviation is clearly defined by the photon energy, but the photon step is smoothed. For stronginteractions the deviation is no longer defined by the photon energy and the width and height of thepeak is small. The height of the peak is in both situations however for energies below Fermi energysomewhat larger. This observed deviation is probably caused by the discretization of the variables46igure 5.6: The quasi-particle energy distribution in the fast field, strong electron-phonon interactionwith ωτ D = 10 , (cid:126) ω/eV = 0 . and τ D ≈ τ E at 3 different positions in the wire at 2 K.and the fact that the interactions are calculated after n iterations, instead for each iteration, to increasecalculation speed. It is reasonable to believe that this has no physical meaning, but is just somenumerical error which can be solved by solving the equations with a program written in C. Thisshould provide a much higher calculation speed, so that the discretization can be optimized.Figure 5.7: The deviation from the equilibrium function of the bath temperature in the fast field regimefor left weak electron-phonon interactions ( τ D ≈ τ E ) and right strong electron phonon interactions( τ D ≈ τ E ). The red line gives the deviation in at x = 0 . L and the blue line at x = 0 . L . Forweak interactions the effect of the photon step is clearly visible at 0.4 which stems with the definedratio (cid:126) ω/eV . For strong interactions the width and height of this step is drastically decreased.47 lectron-electron interactions The effect of electron-electron interactions is quite different from the effect of electron-phononinteractions and dominant for lower temperature so we will evaluate this effect at a temperature of500 mK. Lets first look at the effect of weak interactions when the diffusion time is of the order ofthe relaxation time, τ D ≈ τ E and ωτ D = 10000 . Figure 5.8 shows that for weak interactions there issome smearing, but the photon steps are still good defined. The full space dependency is depicted inthe three dimensional figure in appendix C.Figure 5.8: The quasi-particle energy distribution in the fast field, weak interaction regime with ωτ D = 10000 , (cid:126) ω/eV = 0 . and τ D ≈ τ E at 3 different positions in the wire at 500 mK.When we increase the length of the wire, the time that the electron spends traveling through thewire increases also and the effect of the interactions becomes more significant. Figure 5.9 shows thatthe interactions indeed are more relevant for longer wires and the smearing in the photon steps isclearly visible.By further increasing the length of the wire, the energy relaxation rate becomes dominant withrespect to the diffusion time. The energy gained from the electric field is redistributed in a Fermifunction with an effective temperature, so that an effective temperature profile arises across the wireanalogously to the dc biased macroscopic wire evaluated in the introduction. In figure 5.10 threeFermi functions are given at different positions in the wire for ωτ D = 2000000 and τ D ≈ τ E at500 mK. Figure 5.11 shows the effective temperature profile across the wire obtained by fitting the48igure 5.9: The quasi-particle energy distribution in the fast field, weak interaction regime with ωτ D = 75000 , (cid:126) ω/eV = 0 . and τ D ≈ . τ E at 3 different positions in the wire at 500 mK.energy distribution on every position in the wire to a Fermi function using a least square method.The distribution function on every position in the wire is depicted in the three dimensional figure inappendix C.Figure 5.12 shows the theoretical prediction for the effective temperature profile given by T e ( x ) = (cid:113) T + xL (cid:0) − xL (cid:1) /π ( e/k b ) V when the voltage across the wire is taken to be V = (cid:126) ω/e . Itshows resemblance with the effective temperature profile resulting from the simulation. It seemsplausible to say that the observed difference is due to numerical inaccuracy. Another possible reasonfor this observed deviation could come from the fact that we look at the fast field regime and theposition of photon absorption is responsible for the difference. However, a closed statement on thiscalls for further study.In figure 5.13 the deviation from the equilibrium function at bath temperature is shown. It ap-pears that the smearing induced for weak interactions, τ D ≈ τ E , causes the smooth deviation of twosubtracted Fermi functions at different temperature for strong interactions, τ D >> τ E .49igure 5.10: The quasi-particle energy distribution in the fast field, strong electron-electron interactionregime with ωτ D = 2000000 , (cid:126) ω/eV = 0 . and τ D ≈ τ E at 3 different positions in the wire at500 mK.Figure 5.11: The effective temperature profile across the wire in the case of strong interactions.50igure 5.12: The theoretical predicion of the effective temperature profile across the wire in the caseof strong interactions with V = (cid:126) ω/e .Figure 5.13: The deviation from a equilibrium function at bath temperature for up left weak interac-tions ( τ D ≈ τ E ), up right weak interactions ( τ D ≈ . τ E ) and down in the middle strong interactions( τ D ≈ τ E ). The red line gives the deviation at x = 0 . L and the blue line at x = 0 . L . Theinteractions cause a smearing ending in a local equilibrium with an effective temperature.51 hapter 6 Probing the quasi-particles energydistribution
Obviously it is desirable to be able to obtain experimentally the quasi-particle energy distribution in adiffusive wire subject to high frequency irradiation to test the model. To this end a system is designedwhich should provide this possibility. An aluminum diffusive wire is connected to large aluminumreservoirs where the quasi-particle can relax to equilibrium.These equilibrium reservoirs are designed in such a way that they also function as antenna forthe high frequency radiation. This means that the thickness of the metal reservoirs should be largerthan the penetration depth of the radiation. The penetration depth can be calculated by δ = (cid:113) ωµσ [46]. Here ω is the radial frequency, µ is the permeability and σ is the conductivity. The penetrationdepth of radiation with a frequency of 1 THz is in aluminum approximately 82 nm [46]. Based on thisnumber the reservoirs are designed 100 nm thick.The antenna design is not the subject of this research and since it is a very sophisticated field ofscience, we did not spend time on calculating field profiles. Instead we used a bow-tie antenna design,so that imperfections in the design matter less due to the broadband character. The bow-tie antennais designed in such a way that it is self-complementary [47], by designing the triangles of the bow-tiewith a 45 degrees angle with respect to the wire [48]. In this way the system is frequency independentand we are not limited by a capacitive effect. The length of the triangular sides of the antenna arechosen in multiples of the wavelength of irradiation. By choosing the frequency of irradiation we takeinto account that for pronounced photon steps in the energy distribution the photon energy (cid:126) ω shouldexceed the thermal broadening k b T .We use the same probing method as Pothier et al. did for their experiments on the quasi-particledistribution function in a dc biased wire. On top of the aluminum wire, an insulating layer is posi-tioned whereon superconducting probes of niobium are placed perpendicular, so that when polarizedlight is used the probes are not affected. In this way NIS junctions are formed and from differentialconductance measurement the quasi-particle energy distribution can be obtained. The photon energyshould not exceed the gap energy of the used superconductor, otherwise cooper pairs will be brokenand trouble the probing of the energy distribution. Figure 6.1 shows a schematic overview of thesystem where two probes are connected on the wire with one connection to set the current through theNIS junction and one connection to measure the voltage drop over the junction. The THz radiationcoupled by the antennas causing the ac bias on the aluminum wire is given by V cos ( ωt ) .52igure 6.1: A schematic overview of the system existing of an antenna coupled on an aluminumwire. The niobium probes on the wire enable the differential conductance measurements by setting acurrent through the NIS junction and measure the voltage drop. The THz radiation is represented by V cos ( ωt ) .First measurement were planned at liquid helium temperature (pumped 2 K - unpumped 4.2 K).The phase coherence time can be calculated using equation 5.2 since the dominant phase breakingmechanism is electron-phonon interaction. This approximation is used to define the length of thewires ranging from 500 nm for coherent transport to 100 µ m for fully incoherent transport. When wecompare these values with the values stated in the introduction of the previous chapter, we see thatwe are not measuring in the slow field regime. This has its origin in the fact that we are interested infields with THz frequencies. The fabrication of the samples is not yet proven to be fully successful. The samples to test the NISjunctions however are functioning well enough to conclude that with that recipe the NIS junctionsprovide the ability to probe the distribution function in a wire driven out of equilibrium. Figure 6.2shows a scanning electron microscope (SEM) picture of a NIS junction.The fabrication of the structures described is done in three steps, in which the nanowires, the an-tenna and the tunneling probes are defined. The different structures are aligned by means of markers.To define the Al nanowires a double resist layer is patternd with electron beam lithography. The Pmma950k/Pmma 495k resist is developed in MIBK:IPA for 60 seconds and rinsed for 30 seconds in IPA.A 20 nm thick Al film is evaporated at a rate of 1 angstrom/sec at a pressure of 1e-7 mbar, and lift offis done in hot acetone. The procedure for the definition of the antennas is similar to the nanowires,except for a cleaning step prior to the deposition of the antennas, and an Al film which is 90 nm thickinstead of 20 nm. To fabricate the tunneling probes the sample is cleaned for six minutes in an Argonplasma, after which it is oxidized for 40 minutes in a pure Oxygen atmosphere of 1 mbar. A 80 nmthick Niobium film is sputtered in situ, at a pressure of 8e-3 mbar and a rate of 1 nm/sec. An etchmask is created using SAL resist and electron beam lithography, using MF-322 for developing. The53igure 6.2: A SEM picture of a NIS junction. Vertical the aluminum wire and horizontal the niobiumprobe.Niobium is subsequently etched in an SF /O plasma for 5 minutes with end-point detection. Theremaining resist is removed in PRS 3000 resist stripper and the sample is cleaned in acetone.Figure 6.3 shows SEM pictures of two samples with wires of different length. Measurements onmost recent samples showed an improvement of performance. The outlook for experimental resultsbecomes more promising.Figure 6.3: SEM pictures of two samples with wires of different length. The energy distribution function of the quasi-particles in the wire can be obtained from conductancemeasurements of the NIS junctions of the superconducting wires on top of the mesoscopic wire ofinterest. In appendix D the differential conductance of a NIS junction is derived. The differentialconductance is a convolution of the distribution function in the normal wire and the density of states54n the superconducting probe. So the unknown distribution function can be obtained by deconvolvethe differential conductance with the BCS density of states. dIdV = − R t (cid:90) n BCS ( E ) f (cid:48) x ( E − eV ) dE (6.1)The differential conductance of the NIS junction is only usefull for probing the energy distributionwhen the material used for the superconductor is indeed superconducting at liquid helium tempera-tures. The phase transition of niobium is measured by doing a RT-measurement using a dipstick.The sample is mounted in the vacuum tube of the dipstick with a heating resistance connected. Byapplying a current to this heating resistance the temperature of the sample is increased from 4.2 K tothe desired temperature above the critical temperature. So when simultaneously the resistance of aniobium wire is measured using a four point measurement we can obtain the RT-characteristic. Thefour point measurement is done by setting a current bias to the niobium wire using a current sourceand measure the voltage drop across the wire. It appears that the niobium indeed becomes supercon-ducting at the expected critical temperature of 9 K. A result of a RT-measurement is shown in figure6.4. Figure 6.4: Measured phase transition of the niobium material used for the probes.To test the NIS junctions, differential conductance measurements are performed for wires (notconnected to equilibrium reservoirs) with the superconducting wires on top forming the NIS junction.The measurements are performed at liquid helium temperature (4.2 K) using a dipstick, where againthe sample is mounted in a vacuum tube. A current source is used to apply a current bias to the NISjunction. By measuring the voltage drop over the junction in a four points measurement setup, the IV55haracteristic is obtained. Using a lock-in amplifier the differential conductance of this characteristicis determined. The distribution function can be obtained from such measurements on the NIS junctionbecause of the non-linear IV behavior. Figure 6.5 shows a measurement of the IV-curve and the dI/dV-curve of a NIS junction.Figure 6.5: Measured IV and dI/dV of a NIS junction which is a convolution of the quasi-particleenergy distribution in the normal wire and the density of states of the superconducting probe.The deconvolution is executed using a steepest descent method. This method is commonly knownfor the use in minimizing functions [49]. This is used to deduce the distribution function from thedifferential conductance using equation 6.1. First the effective density of states of the superconductorin the NIS junction is deduced from a dI/dV measurement on a wire in equilibrium, so that fits forthe energy gap, tunneling resistance and electron temperature can be implemented. Then an initialdistribution function can be chosen in such a way that the calculation converges in a relative shorttime [50]. The initial distribution function is used to calculate the differential conductance and this iscompared to the measured differential conductance for a wire out of equilibrium. If the difference isequal or smaller than the desired precision the deconvolution is completed and the initial distributionis the distribution of the electrons in the wire out of equilibrium. If the difference between the two dI/dV ’s is larger than the desired precision a new distribution is calculated using the square deviation: χ = (cid:88) k (cid:32) ∂I∂V (cid:12)(cid:12)(cid:12)(cid:12) kcalc − ∂I∂V (cid:12)(cid:12)(cid:12)(cid:12) kmeas (cid:33) (6.2)The occupation probability at each energy in the distribution is incremented by the partial deriva-tive of the square deviation with respect to the occupation factor f k at that energy: f kit +1 = f kit + λ ∂χ ∂f k = f kit + λ (cid:48) n (cid:48) BCS (cid:32) ∂I∂V (cid:12)(cid:12)(cid:12)(cid:12) kcalc − ∂I∂V (cid:12)(cid:12)(cid:12)(cid:12) kmeas (cid:33) (6.3)This calculation is iterated until the desired precision is achieved. The procedure is illustrated infigure 6.6. The deconvolution is executed in MATLAB. The script that is used is given in appendix E.56igure 6.6: Scheme of the deconvolution procedure. The fabrication of samples with wires connected to antennas and superconducting probes on top ofthe wires is not yet proven to be successful. Therefore the proposed model of chapter 4 cannot yetbe verified. We can however calculate what the expected differential conductance measurements willlook like when we measure it while driving the wire out of equilibrium with a time-dependent electricfield. We do this by taking a calculated distribution function from the model and calculated dI/dVwith equation 6.1. This is done for the fast field regime where ωτ D = 30000 and (cid:126) ω/eV = 0 . at a temperature of 2 K. So we look at a coherent transport situation comparable to that discussedin chapter 5, where we take into account that the photon energy should be below the gap energy ofniobium. The distribution function from the model and the calculated differential conductance of theNIS junction on the wire are given in figure 6.7.Figure 6.7: The calculated differential conductance left for the distribution function right.The experiments with the described system will proceed and hopefully lead to a satisfying resultthat can be used to verify the theoretical model. 57 hapter 7 Conclusion and discussion
In this project we studied the ac quantum transport in a quasi-one dimensional, normal metal wire,where the transport is diffusive, connected between equilibrium reservoirs. For coherent transport,where the phase of a charge carrier is preserved, photon absorption and the diffusive character of thetransport influence the energy distribution of the quasi-particles inside the wire. When the diffusiontime, i.e. the time that a quasi-particle spends in the wire, exceeds the energy relaxation time, themutual interaction of quasi-particles and the interaction between quasi-particles and phonons causesincoherent transport and influences the energy distribution.Often scattering theory is used to describe the transport of charge carriers through nano-structures.However, in our situation many processes involving the energy of the charge carriers come into playin the scattering region, i.e. the diffusive wire. Therefore we studied the effect of an ac bias appliedto a diffusive wire by looking at the energy distribution of the quasi-particles inside the wire.Previous work on this subject was independently done by R. Schrijvers [6] and A.V. Shytov [32].R. Schrijvers approached the situation by applying Tien-Gordon theory to the reservoirs and calcu-lated the energy distribution in the wire with a semi-classical diffusion equation, which is a Boltzmannequation extended for inelastic interactions. He concluded that in this way only the slow field limitwithout inelastic interactions is adequately described. This is due to the fact that the assumption ismade that the path traveled by the charge carrier in the wire is of no influence on the energy distribu-tion. Also Tien-Gordon theory assumes averaging over time, which troubles the correct evaluation ofthe collision integral.A.V. Shytov calculated the energy distribution in a diffusive wire where the energy relaxationtime exceeds the diffusion time with a quantum diffusion equation. This approach seems valid in allfrequency regimes, but only for coherent transport.Stimulated by these approaches we derived from Green function formalism a quantum diffusionequation equivalent to that of Shytov and we did a second derivation from the full Dyson equationto extend this model to account for inelastic scattering processes. This approach seems successful,as the quasi-particle energy distribution can be calculated in every frequency regime and the limitsituations of strong interaction processes provide the correct distribution. This approach is justifiedby the Landau theory of Fermi liquids. This states that there is an one-to-one correspondence betweenthe states of a non-interacting particle system and the states of an interacting particle system providedthat the excitations are near the Fermi level. This means that the excited states of an interacting systemare labeled with the same quantum numbers as those of a non-interacting system. The interactions58ave the effect that the electrons are treated as quasi-particles, particles which are closely related totheir environment.For wires where the energy relaxation time exceeds the diffusion time, the transport is fully elasticand the energy distribution is calculated by a quantum diffusion equation without an inelastic collisionterm. The distribution function behaves differently in the two field limits, ωτ D << and ωτ D >> .For the slow field, strong signal limit ( ωτ D << , (cid:126) ω/eV << ) the dc situation is approached andthe energy distribution is given by a time-varying two step function. In the fast field, strong signallimit ( ωτ D >> , (cid:126) ω/eV << ) a quasi-particle can oscillate multiple times with the field in thewire before leaving the wire, therefore gaining more energy quanta. This results in a time-independentelectron energy distribution which does not go to zero at higher energies.Numerical simulations for a finite ratio (cid:126) ω/eV shows the photon steps in the energy distribution.In the slow field regime the distribution is highly time-dependent. The energy distribution directlyfollows the field and the photon steps oscillate from zero photon absorption to maximum photonabsorption. In the fast field limit the energy distribution becomes completely time-independent. Themaximum photon absorption is reached when the diffusion time is exceeded. In the crossover the twoeffects are both observed. There is a slight oscillation around the maximum photon absorption valueof the fast field regime.When the energy relaxation time becomes comparable to the diffusion time, the transport is nolonger coherent and scattering is inelastic. Numerical simulations show how the energy exchangeprocesses of mutual quasi-particles and between quasi-particles and phonons influence the energydistribution in the fast field regime, which is of interest for experimental situations. The interactionbetween quasi-particles and phonons annihilates first the photon steps in the distribution. In the strongelectron-phonon interaction limit the Fermi function at bath temperature is found on every position inthe wire.The interaction between quasi-particles is quite different. It causes a smearing in the photon steps.In the strong electron-electron interaction limit a local equilibrium is reached on every position in thewire. The photon absorption, diffusive transport and the interactions cause an effective temperatureprofile across the wire. The effective temperature profile is determined by fitting the distribution fromthe simulation on every position with a Fermi function. The obtained temperature profile deviatesfrom the calculated profile. It is not yet fully understood whether this is caused by a numerical erroror that it is caused by some physical effect, like for instance the position in the wire that the photonabsorption takes place.So the complicated interplay between the effect of photon absorption, diffusive transport andinelastic scattering on the quasi-particle energy distribution seems to be accurately described by ourmodel. The model developed in this work is not yet verified by experiments. The fabrication of the requiredsamples is not yet proven to be completely succesfull due to the failure of fabrication apparatus,however the outlook is promising and the experimental work will be continued by members of thisresearch group.To calculate the collision term, the interactions are assumed to be instantaneous and local. Anexperiment can verify this assumption, so it is desirable to proceed with the experimental part of thisproject. The comparison between experiment and theoretical model has to provide a full insight in theac quantum transport in diffusive quasi-one dimensional wires and how the non-equilibrium is shown59n the quantum statistics of the quasi-particles in the wire.While our MATLAB code seems to calculate the quasi-particle energy distribution influencedby photon absorption, diffusive transport and inelastic collisions in a correct manner, it also appearsthat the MATLAB code is not very efficient. A program written in C should in principle work moreefficient. This provides the opportunity to optimize the discretization of the variables, so that a moreaccurate result is obtained. 60 cknowledgements
The phrase
Tempus fugit is the first thing that comes to mind when I look back at the time spent inthis group working on my master thesis project. It was certainly an interesting and challenging time.The project did not go as planned. One of the main goals was to experimentally obtain data on thesubject. However, reality seems to play tricks on you when you want to control things. Apparatusfailure caused such delay in the fabrication of the samples that I did not get to experience the beautyof experimental success. This pushed me further to theoretical research and I have to say, I didn’tmind. It was really intriguing how the physics on small scale revealed itself to me by doing the mathand combining concepts. In addition I learned different experimental techniques in preparation of thebig experiment which didn’t come.I would like to thank prof. Teun Klapwijk for giving me the opportunity of doing this challengingwork. It really gave me the chance to evolve in different disciplines. I also would like to expressmy gratitude towards my daily supervisors, Nathan and Rik. There were times that I was not certainwhether everything would work out, but they gave me the confidence that leaded to this satisfyingresult and educated me in the necessary basics of experimental and theoretical research.Furthermore, it was just a fun time. I really experienced to be a part of the group. I especially wantto thank Nathan, Rik, Eduard and David for the interesting conversations on everything and nothing.Also I want to thank Reinier for his company during the whole year in the huge students office. I wantto thank the whole group for this fun time and for their part in my education. I am confident that thegained experience in this group will be of great help in the continuation of my career.61 ibliography [1] D.V. Averin Y. Naveh and K.K. Likharev. Shot noise in diffusive conductors: A quantitativeanalysis of electron-phonon interaction effects.
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Shot noise
Shot noise, the phenomenon that arises because of charge quantization, provides information about thestatistics of charge carriers involved in quantum transport. In chapter 2 the mean square fluctuationsin the occupation of incident, reflected and transmitted state are derived to be: (cid:104) δn T δn T (cid:105) = − T Rf (A.1) (cid:10) ( δn T ) (cid:11) = T f (1 − T f ) (A.2) (cid:10) ( δn R ) (cid:11) = Rf (1 − Rf ) (A.3)It appears that at zero temperature, when the distribution is given by a step function at chemicalpotential, the shot noise disappears for full reflectance or full transmittance. At finite temperatures themean square fluctuations fluctuate like the incident state with occupation f .Now when we proceed with this simplified model of a single incident charge carrier to investigatethe fluctuations in the current, we can consider a perfect conductor with three separated channels.One for the incident state, one for the reflected state and one for the transmitted state. The carriersmove in one directions with a velocity v ( E ) dependent on the energy of the charge carrier. Foran energy interval dE , the incident current is given by dI in = ev ( E ) dρ ( E ) . ρ ( E ) is the energydependent density of carriers per unit length. It is given by ρ ( E ) = n in ( E ) ν ( E ) dE , where ν ( E ) is the density of states per unit length. In a perfect conductor ν ( E ) = (2 π (cid:126) v ( E )) − . This leads to dI in = e (2 π (cid:126) ) − n in ( E ) dE . For the transmitted and reflected channel the same procedure can befollowed leading to dI T = e (2 π (cid:126) ) − n T ( E ) dE and dI R = e (2 π (cid:126) ) − n R ( E ) dE . Integrating gives theexpressions for the current. When the occupation numbers vary slowly in time, the derivation can beexpanded by just taking the occupation not only energy-dependent, but also time-dependent. I in ( E, t ) = e π (cid:126) (cid:90) n in ( E, t ) dE (A.4) I T ( E, t ) = e π (cid:126) (cid:90) n T ( E, t ) dE (A.5) I R ( E, t ) = e π (cid:126) (cid:90) n R ( E, t ) dE (A.6)For low frequency fluctuations in the current, these expressions can be Fourier transformed givingthe frequency dependent current. 65 in ( ω ) = e π (cid:126) (cid:90) n in ( E + (cid:126) ω ) dE (A.7) I T ( ω ) = e π (cid:126) (cid:90) n T ( E + (cid:126) ω ) dE (A.8) I R ( ω ) = e π (cid:126) (cid:90) n R ( E + (cid:126) ω ) dE (A.9)The fluctuations in current and occupation number are directly related. The current noise power isin the zero frequency limit S II = e (2 π (cid:126) ) − (cid:82) S nn ( E ) dE . From the current it is seen that the chargecarriers arrive at a rate of dE/ (2 π (cid:126) ) in each energy interval. This contributes to the noise with themean square fluctuations of the relevant state. Therefore S nn ( E ) = 1 / ( π (cid:126) ) (cid:104) δnδn (cid:105) . Substitution inthe current noise power relations leads to S I in I in = 2 e π (cid:126) (cid:90) f (1 − f ) dE (A.10) S I T I T = 2 e π (cid:126) (cid:90) T f (1 − T f ) dE (A.11) S I R I R = 2 e π (cid:126) (cid:90) Rf (1 − Rf ) dE (A.12) S I T I R = − e π (cid:126) (cid:90) T Rf dE. (A.13)So here we see explicitly the earlier found conclusion for the fluctuations that shot noise disappearsfor full transparency or full reflectance at zero temperature. At finite temperature the distributionfunction is thermally broadened and the shot noise will not disappear due to thermal fluctuations.When now the system under consideration is extended to a situation where a multi-channel scattereris placed between two terminals with respectively distribution f L and f R the noise power is given by S = e π (cid:126) (cid:88) n (cid:90) dET n ( E )[ f L (1 − f L ) + f R (1 − f R )] + T n ( E )[1 − T n ( E )]( f L − f R ) . (A.14)The scale of the energy dependence of the transmission coefficients is usual much bigger than thethermal and bias energy. Therefore these quantities can be taken in equation A.14 at Fermi energy.Then the noise power becomes S = e π (cid:126) [2 k b T (cid:88) n T n + eV coth (cid:18) eV k b T (cid:19) (cid:88) n T n (1 − T n )] (A.15)66 ppendix B MATLAB code of the simulationprogram
B.1 Script for the simulation of coherent transport %==========================================================================%CLEAN UP, FUNDAMENTALS%==========================================================================clear all;tic;%fundamental constantse = 1.602e-19;hbar = 6.63e-34/2/3.141592; %6.6e-16; %kb = 1.38e-23; % 8.6e-5;%==========================================================================%PARAMETERS%==========================================================================%generalsaveall=0;saverepeat=1250;%time discretizationNt0=7;dt=2*pi/Nt0;int_method=’euler’;Nt=5000*Nt0;%1000000;%energy discretizationT=2;omega=0.3e12*2*3.141592;V=3e-3;limit_E=2; %limit in multiples of eVhw=hbar*omega/e/V;Nw=16; %must be even!dE=hw/Nw; E=ceil(limit_E/dE);E=-NE*dE:dE:(NE-1)*dE+dE/2;NE=length(E);kT=kb*T/e/V;%space discretizationD_method=’lagrange_2’;Nx=100;x = linspace(0,1,Nx);dx = 1/(Nx-1);%tauD= 1e-9;%z=omega*tauD;z=30000;fprintf(’homega/eV= %f kT/eV= %f dE=%f Emax= %f omega tau= %f \n’,hw,kT,dE,NE*dE,z);%==========================================================================%INITIALIZATION%==========================================================================%INITIAL AND BOUNDARY CONDITIONSFl=1./(exp((E)/kb/T*e*V)+1);Fr=1./(exp((E)/kb/T*e*V)+1);Fold=[ones(Nx-1,1)*Fl; Fr];Feq=[ones(Nx-1,1)*Fl; Fr];Fave=[ones(Nx-1,1)*Fl; Fr];%MATRICESD_x=dx1(Nx,1,dx,D_method);D_xx=dx2(Nx,1,dx,D_method);D_E=dE1(1,NE,Nw,dE)’;D_EE=dE2(1,NE,Nw,dE)’;first_step=[ ];second_step=[];third_step=[];%==========================================================================%INTEGRATION%==========================================================================switch lower(int_method)case ’euler’%eulerfor m=1:Ntw=sin(m*dt);dF=D_xx*Fold+2*w*D_x*Fold*D_E+wˆ2*Fold*D_EE;Fnew=Fold+dF*dt/z;Fnew([1 Nx],:)=[Fl;Fr];Fold=Fnew;Fave=((m-1)*Fave+Fold)/m;first_step=[first_step Fnew(round([Nx/2])’,NE/2+Nw/2)];second_step=[second_step Fnew(round([Nx/2])’,NE/2+3*Nw/2)];third_step=[third_step Fnew(round([Nx/2])’,NE/2+5*Nw/2)];if mod(m,5000)==0 or o=1:Nxfor p=1:NEif Fnew(o,p)>1Fnew(o,p)=1;endif Fnew(o,p)<0Fnew(o,p)=0;endendendendif mod(m,1000)==0 %saverepeatclc;fprintf(’iteratie %i, time %f’,m,toc);%round(m/Nt0*100)%tic;% hold on;plot(E,Fnew(round([1 Nx/4 Nx/2])’,:));pause(.2);if saveallsave([’Fnew’ num2str(m)],’Fnew’);endendendend B.2 Functions for the used operators
First spatial derivative function S=dx1(Nx,NE,h,method)switch lower(method)case {’lagrange_1’}%lagrange 1st orderr=2;w1=[-3 4 -1];w3=[-1 0 1];n=1;case ’lagrange_2’%lagranger=12;w1=[-25 48 -36 16 -3];w2=[-3 -10 18 -6 1];w3=[1 -8 0 8 -1];n=2;case ’least’%least squaresr=70;w1=[-54 13 40 27 -26];w2=[-34 3 20 17 -6];w3=[-2 -1 0 1 2]*7;n=2;end%the boundaries of the matrix ase0=1:NE;v0=ones(1,NE);%first boundaryarray1=[];column1=[];values1=[];for k=1:length(w1)array1=[array1 base0];column1=[column1 base0+(k-1)*NE];values1=[values1 w1(k)*v0];endcolumn1=[column1 NE*Nx-column1+1];array1=[array1 NE*Nx-array1+1];values1=[values1 values1];%second boundary (if needed)array2=[];column2=[];values2=[];if n==2for k=1:length(w2)array2=[array2 base0+NE];column2=[column2 base0+(k-1)*NE];values2=[values2 w2(k)*v0];endcolumn2=[column2 NE*Nx-column2+1];array2=[array2 NE*Nx-array2+1];values2=[values2 values2];end%central partif n==1base1=1:NE*(Nx-2);v1=ones(1,NE*(Nx-2));else base1=1:NE*(Nx-4);v1=ones(1,NE*(Nx-4));endarray3=[];column3=[];values3=[];for k=1:length(w3)array3=[array3 base1+n*NE];column3=[column3 base1+(k-1)*NE];values3=[values3 w3(k)*v1];end%the total matrixS=sparse([array1 array2 array3],[column1 column2 column3],[values1 values2 values3])/r/h; Second spatial derivative function S=dx2(Nx,NE,h,method)switch lower(method) ase {’lagrange_1’}%lagrange 1st orderr=1;w1=[2 -5 4 -1];w3=[1 -2 1];n=1;case ’lagrange_2’%lagranger=12;w1=[35 -104 114 -56 11];w2=[10 -15 -4 14 -6 1];w3=[-1 16 -30 16 -1];n=2;case ’least’%least squaresr=14;w1=[9 -15 -2 13 -5]*2;w2=[11 -16 -4 12 -3];w3=[2 -1 -2 -1 2]*2;n=2;end%the boundaries of the matrixbase0=1:NE;v0=ones(1,NE);%first boundaryarray1=[];column1=[];values1=[];for k=1:length(w1)array1=[array1 base0];column1=[column1 base0+(k-1)*NE];values1=[values1 w1(k)*v0];endcolumn1=[column1 NE*Nx-column1+1];array1=[array1 NE*Nx-array1+1];values1=[values1 values1];%second boundary (if needed)array2=[];column2=[];values2=[];if n==2for k=1:length(w2)array2=[array2 base0+NE];column2=[column2 base0+(k-1)*NE];values2=[values2 w2(k)*v0];endcolumn2=[column2 NE*Nx-column2+1];array2=[array2 NE*Nx-array2+1];values2=[values2 values2];end%central partif n==1base1=1:NE*(Nx-2); First energy derivative function S=dE1(Nx,NE,Nw,h)%boundariesarray_0=[1:Nw NE-[1:Nw]+1];column_0=[ones(1,Nw) ones(1,Nw)*NE];values_0=[-ones(1,Nw) ones(1,Nw)];%central partbase=1:NE-Nw;v0=ones(1,NE-Nw);array_1=[base+Nw base];column_1=[base base+Nw];values_1=[-v0 v0];array=[array_0 array_1];column=[column_0 column_1];values=[values_0 values_1];for k=1:Nx-1array=[array array_0+NE*k array_1+NE*k];column=[column column_0+NE*k column_1+NE*k];values=[values values_0 values_1];end%the total matrixS=-sparse(array,column,values)/2/h/Nw;
Second energy derivative function S=dE2(Nx,NE,Nw,h)%boundariesarray_0=[1:2*Nw 2:2*Nw 1:2*Nw];array_0=[array_0 NE+1-array_0]; olumn_0=[ones(1,2*Nw) 2:2*Nw 2*Nw+1:4*Nw];column_0=[column_0 NE+1-column_0];values_0=[-1 ones(1,2*Nw-1) -2*ones(1,2*Nw-1) ones(1,2*Nw)];values_0=[values_0 values_0];%central partbase=1:NE-2*Nw; %4*Nwv0=ones(1,NE-2*Nw); %4*Nwarray_1=[base+1*Nw base+1*Nw base+1*Nw]; %2*Nw ; 2*Nw ; 2*Nwcolumn_1=[base base+1*Nw base+2*Nw]; %2*Nw ; 4*Nwvalues_1=[v0 -2*v0 v0];array=[array_0 array_1];column=[column_0 column_1];values=[values_0 values_1];for k=1:Nx-1array=[array array_0+NE*k array_1+NE*k];column=[column column_0+NE*k column_1+NE*k];values=[values values_0 values_1];end%the total matrixS=sparse(array,column,values)/(2*Nw*h)ˆ2; B.3 Script for the simulation of incoherent transport %==========================================================================%CLEAN UP, FUNDAMENTALS%==========================================================================clear all;tic;%fundamental constantse = 1.602e-19;hbar = 6.63e-34/2/3.141592; %6.6e-16;kb = 1.38e-23; % 8.6e-5;%==========================================================================%PARAMETERS%==========================================================================%generalsaveall=0;saverepeat=1250;%time discretizationNt0=1.1;dt=2*pi/Nt0;Nt=79000*Nt0;%20000;%energy discretizationT=0.5;omega=1.6e12*2*3.141592; =16e-3;limit_E=2; %limit in multiples of eVhw=hbar*omega/e/V;Nw=50; %must be even!dE=hw/Nw;NE=ceil(limit_E/dE);E=-NE*dE:dE:(NE-1)*dE+dE/2;NE=length(E);kT=kb*T/e/V;%interaction parametersint_mech=’eph’; %interaction mechansim, ee for electron-electron, eph for electron phononrho=2.7e3;EF=12/V; %Jdos=2e47*e*V;s=6.42e3;kf=1.75e10;D=100e14; %nmˆ2sˆ-1dos=2e47;S=20e-9*20e-9;sigma=1e9;Ke=(sqrt(2*D)*pi*hbarˆ(3/2)*dos*S)ˆ(-1);NE_ee=NE;E_ee=linspace(10,14,NE_ee)/V;dE_ee=4*e/NE_ee/V;NE_ph=NE;E_ph=linspace(-2,2,NE_ph);dE_ph=4*e/NE_ph;kph=sigma/24/zeta(5)/dos/kbˆ5*(e*V)ˆ2*dE_ph;%kph=4e12*1e-3/V; %has to be expressed in Vn_ph=(1./(exp(abs(E_ph)/kb/2*e*V)-1))’;%space discretizationD_method=’lagrange_2’;Nx=20;x = linspace(0,1,Nx);dx = 1/(Nx-1);%tauD= 1e-9;%z=omega*tauD;z=2000000;fprintf(’homega/eV= %f kT/eV= %f dE=%f Emax= %f omega tau= %f \n’,hw,kT,dE,NE*dE,z);%==========================================================================%INITIALIZATION%==========================================================================%INITIAL AND BOUNDARY CONDITIONSFl=1./(exp((E)/kb/T*e*V)+1);Fr=1./(exp((E)/kb/T*e*V)+1); old=[ones(Nx-1,1)*Fl; Fr];Feq=[ones(Nx-1,1)*Fl; Fr];Fave=[ones(Nx-1,1)*Fl; Fr];%MATRICESD_x=dx1(Nx,1,dx,D_method);D_xx=dx2(Nx,1,dx,D_method);D_E=dE1(1,NE,Nw,dE)’;D_EE=dE2(1,NE,Nw,dE)’;Iin=0;Iout=0;h=400;first_step=[ ];second_step=[];third_step=[];%==========================================================================%INTEGRATION%==========================================================================switch lower(int_mech)case ’ee’%eulerfor m=1:Ntw=sin(m*dt);dF=D_xx*Fold+2*w*D_x*Fold*D_E+wˆ2*Fold*D_EE+Iin*(z/omega)-Iout*(z/omega);Fnew=Fold+dF*dt/z;Fnew([1 Nx],:)=[Fl;Fr];Fold=Fnew;Fave=((m-1)*Fave+Fold)/m;first_step=[first_step Fnew(round([Nx/2])’,NE/2+Nw/2)];second_step=[second_step Fnew(round([Nx/2])’,NE/2+3*Nw/2)];third_step=[third_step Fnew(round([Nx/2])’,NE/2+5*Nw/2)];if mod(m,50)==0for o=1:Nxfor p=1:NEif Fnew(o,p)>1Fnew(o,p)=1;endif Fnew(o,p)<0Fnew(o,p)=0;endendendendif mod(m,h+1)==0for o=1:Nxfor p=1:NEif Fnew(o,p)>1Fnew(o,p)=1;endif Fnew(o,p)<0Fnew(o,p)=0;end ndendendif mod(m,h-1)==0for o=1:Nxfor p=1:NEif Fnew(o,p)>1Fnew(o,p)=1;endif Fnew(o,p)<0Fnew(o,p)=0;endendendendif mod(m,800)==0Fnew2=Fnew;Fnew1=Fnew;[q,r]=size(Fnew);g1=diag(ones(r-1,1),1);g2=diag(ones(r-1,1),-1);Fnew1=circshift(Fnew1,[0 1])+[Fnew(:,1), (zeros(NE-1,Nx))’];b=[];a=[];for j=1:Nx;k1=Fnew1(j,:);k2=Fnew2(j,:);for i=1:NE/2-1h1=Fnew1(j,:)*g1;h1=h1+[ones(1,1)’, zeros(NE-1,1)’];k1=[k1’ h1’]’;Fnew1(j,:)=h1;h2=Fnew2(j,:)*g2;k2=[h2’ k2’]’;Fnew2(j,:)=h2;endk=[k2’ k1’]’;E2=((E_ee).ˆ(-3/2))’;W=(1-fliplr(k’))*Fnew(j,:)’;W2=Ke*E2.*W;%In=W2’*fliplr(k’)*dE;Out=W2’*(1-k)*dE;In=fliplr(Out);b=[b Out’];a=[a In’];endIout=(Fnew.*b’);Iin=((1-Fnew).*a’);endif mod(m,1000)==0 %saverepeatclc;fprintf(’iteratie %i, time %f’,m,toc);%round(m/Nt0*100)%tic;plot(E,Fnew(round([1 Nx/4 Nx/2])’,:));pause(.2);if saveall ave([’Fnew’ num2str(m)],’Fnew’);endendendcase ’eph’%eulerfor m=1:Ntw=sin(m*dt);dF=D_xx*Fold+2*w*D_x*Fold*D_E+wˆ2*Fold*D_EE+Iin*(z/omega)-Iout*(z/omega);Fnew=Fold+dF*dt/z;Fnew([1 Nx],:)=[Fl;Fr];Fold=Fnew;first_step=[first_step Fnew(round([Nx/2])’,NE/2+Nw/2)];second_step=[second_step Fnew(round([Nx/2])’,NE/2+3*Nw/2)];third_step=[third_step Fnew(round([Nx/2])’,NE/2+5*Nw/2)];if mod(m,50)==0for o=1:Nxfor p=1:NEif Fnew(o,p)>1Fnew(o,p)=1;endif Fnew(o,p)<0Fnew(o,p)=0;endendendendif mod(m,h+1)==0for o=1:Nxfor p=1:NEif Fnew(o,p)>1Fnew(o,p)=1;endif Fnew(o,p)<0Fnew(o,p)=0;endendendendif mod(m,h-1)==0for o=1:Nxfor p=1:NEif Fnew(o,p)>1Fnew(o,p)=1;endif Fnew(o,p)<0Fnew(o,p)=0;endendendendFave=((m-1)*Fave+Fold)/m;if mod(m,400)==0h=m;Fnew2=Fnew;Fnew1=Fnew;[q,r]=size(Fnew); ppendix C Space dependency in the distributionfunction
Figure C.1: The quasi-particle energy distribution in the slow field regime, ωτ D = 1 , and (cid:126) ω/eV =0 . at all positions in the wire at 2 K. 79igure C.2: The quasi-particle energy distribution in the fast field regime, ωτ D = 30000 , and (cid:126) ω/eV = 0 . at all positions in the wire at 2 K.Figure C.3: The quasi-particle energy distribution in the intermediate regime, ωτ D = 100 , and (cid:126) ω/eV = 0 . at all positions in the wire at 2 K. 80igure C.4: The quasi-particle energy distribution in the fast field, weak electron-phonon interactionregime, ωτ D = 50000 , (cid:126) ω/eV = 0 . and τ D ≈ τ E at all positions in the wire at 2 K.Figure C.5: The quasi-particle energy distribution in the fast field, strong electron-phonon interactionregime, ωτ D = 10 , (cid:126) ω/eV = 0 . and τ D ≈ τ E at all positions in the wire at 2 K.81igure C.6: The quasi-particle energy distribution in the fast field, weak electron-electron interactionregime, ωτ D = 10000 , (cid:126) ω/eV = 0 . and τ D ≈ τ E at all positions in the wire at 500 mK.Figure C.7: The quasi-particle energy distribution in the fast field, weak electron-electron interactionregime, ωτ D = 75000 , (cid:126) ω/eV = 0 . and τ D ≈ . τ E at all positions in the wire at 500 mK.82igure C.8: The quasi-particle energy distribution in the fast field, strong electron-electron interactionregime, ωτ D = 2000000 , (cid:126) ω/eV = 0 . and τ D ≈ τ E at all positions in the wire at 500 mK.83 ppendix D Differential conductance of a NISjunction
To obtain the electron energy distribution function on a certain position in the wire, superconducting probes are used thatare positioned orthogonally on top of the wire with an insulating layer in between. Now by applying a current to this NISjunction we can measure the reciprocal value of the differential conductance. This differential conductance is a convolutionof the density of states of the used superconductor and the probed energy distribution function. This can be calculated fromFermi’s golden rule when the tunnel matrix elements are considered nearly constant for all energy states consider in themeasurement. Because the energy distribution is probed with a superconductor the tunneling current is elastic, as there isno energy dissipation in a superconductor. So from the golden rule the tunneling rate from the wire to the superconductorand from the superconductor to the wire become respectively Γ x → p ( E ) = 1 e R t n x ( E ) f x ( E ) n p ( E + eV )(1 − f p ( E + eV )) (D.1) Γ p → x ( E ) = 1 e R t n p ( E + eV ) f p ( E + eV ) n x (1 − f x ( E )) . (D.2)Here e is the elementary charge, R t is the tunnel resistance, n x and n p are respectively the density of states in the wireand in the superconductor and f x and f p are the distribution functions in respectively the wire and the superconductor. Thecurrent across the junction is calculated from these tunnel rates. I ( V ) = e (cid:90) (Γ x → p ( E ) − Γ p → x ( E ))) dE (D.3)When the tunnel rates are implemented in the formula above and the density of states of the superconductor is taken tobe the BCS density of states n BCS ( E ) = (cid:60) ( | E | / √ E − ∆ ) and as the wire is a metal the density of states in the wire istaken to be flat, we arrive at the following expression. I ( V ) = 1 eR t (cid:90) n BCS ( E + eV ) ( f x ( E ) − f p ( E + eV )) dE (D.4)A variable change of E → E − eV and taking the derivative with respect to V leads to an expression for the differentialconductance of the NIS junction. dIdV = − R t (cid:90) n BCS ( E ) f (cid:48) x ( E − eV ) dE (D.5)The distribution function shows up explicitly by integration by part and using the fact that the BCS density of states iseven. R t dIdV = 1 − (cid:90) n (cid:48) BCS ( eV − E ) f x ( E ) dE ≡ − n (cid:48) BCS ∗ f x ( eV ) (D.6)In practice the singular behaviour of the density of states of the superconductor is less sharp than the BCS theorypredicts. To avoid problems with this aspect, the effective density of states of the superconductor in the NIS junction can beprobed first for an unbiased wire. Since in this situation the energy distribution of the electrons is just a quasi-equilibriumFermi function, the effective density of states is obtained by a deconvolution of the differential conductance of the NIS unction and the distribution function. This is basically finding a fit for the gap energy ∆ , the tunneling resistance R t andthe electron temperature T . ppendix E MATLAB script for deconvolution clear all;dVdI = importdata(’testdatadIdV.txt’);dIdV1 = (1./dVdI);m=length(dIdV1);dIdV = [dIdV1’ dIdV1(m)]’;kb = 8.6e-6;%1.38*10ˆ-23;T= 4.2;e = 1.602e-19;NE = length(dIdV);NV = length(dIdV);d = 1.3*10ˆ(-3);%*e;E = linspace(-10*10ˆ(-3),10*10ˆ(-3),NE);%*e;dE = 20e-3/NE;%*e;dV = 20e-3/NV;V = linspace(-10*10ˆ(-3),10*10ˆ(-3),NV);%*e;Rt = 1;%;1.2*10ˆ4;V2=1e-3;for i=1:length(V)for j = 1:length(E)f(i,j) = 1/Rt*(exp((E(j)-V(i))/kb/T)./((exp((E(j)-V(i))/kb/T)+1).ˆ2*kb*T));endendNs=f\dIdV/dE;Ns(1:5)=Ns(6);Ns(246:m+1)=Ns(245);for i = 1:length(V)dIdV99(i) = 1/Rt.*(exp((E-V(i))/kb/T)./((exp((E-V(i))/kb/T)+1).ˆ2*kb*T)*Ns)*dE;enddIdV99(1:2)=dIdV99(3);dIdV99(250:m+1)=dIdV99(249);Diff_n=diag(ones(NE-1,1),1)-diag(ones(NE-1,1),-1);Diff_n(1:2,1)=[1 ;1];Diff_n(NE-1:NE,NE)=[1 ;1];DnBCS=Diff_n*Ns/dE; nBCS(1:2)=DnBCS(3);DnBCS(251)=DnBCS(250);E=E’;nBCS=real(abs(E)./(E.ˆ2-dˆ2).ˆ(1/2));fold=1./(exp(E/kb/T)+1);Diff_f=-diag(ones(NE-1,1),1)+diag(ones(NE-1,1),-1);Diff_f(1:2,1)=[1 ;1];Diff_f(NE-1:NE,NE)=[1 ;-1];dfold=Diff_f*fold/dV;[m,n]=size(dIdV);epsilon=3.9e-14*ones(m,1);chi2=((dIdV99-dIdV’).ˆ2)’;while sum(chi2 > epsilon) > 0fprintf(’q %i \n’,sum(chi2 > epsilon));fnew=fold+5e0*(DnBCS.*chi2.ˆ(1/2));dfnew=Diff_f*fnew/dV;g1=diag(ones(m-1,1),1);g2=diag(ones(m-1,1),-1);dfnew1=dfnew;dfnew2=dfnew;dfnew1=circshift(dfnew1,[0 1])+[dfnew(:,1), (zeros(n-1,m))’];k1=dfnew1;k2=dfnew2;for i=1:NE/2-1h1=g1*dfnew1;h1=h1+[ones(1,1)’, zeros(NE-1,1)’]’;k1=[k1 h1];dfnew1=h1;h2=g2*dfnew2;k2=[h2 k2];dfnew2=h2;enddfnew_m=[k2 k1];q=1/Rt*Ns’*dfnew_m;g=dIdV(1)/q(1);q=1/Rt*Ns’*dfnew_m*g;chi2=((q-dIdV’).ˆ2)’;fold=fnew;endplot(E,fnew)nBCS(1:2)=DnBCS(3);DnBCS(251)=DnBCS(250);E=E’;nBCS=real(abs(E)./(E.ˆ2-dˆ2).ˆ(1/2));fold=1./(exp(E/kb/T)+1);Diff_f=-diag(ones(NE-1,1),1)+diag(ones(NE-1,1),-1);Diff_f(1:2,1)=[1 ;1];Diff_f(NE-1:NE,NE)=[1 ;-1];dfold=Diff_f*fold/dV;[m,n]=size(dIdV);epsilon=3.9e-14*ones(m,1);chi2=((dIdV99-dIdV’).ˆ2)’;while sum(chi2 > epsilon) > 0fprintf(’q %i \n’,sum(chi2 > epsilon));fnew=fold+5e0*(DnBCS.*chi2.ˆ(1/2));dfnew=Diff_f*fnew/dV;g1=diag(ones(m-1,1),1);g2=diag(ones(m-1,1),-1);dfnew1=dfnew;dfnew2=dfnew;dfnew1=circshift(dfnew1,[0 1])+[dfnew(:,1), (zeros(n-1,m))’];k1=dfnew1;k2=dfnew2;for i=1:NE/2-1h1=g1*dfnew1;h1=h1+[ones(1,1)’, zeros(NE-1,1)’]’;k1=[k1 h1];dfnew1=h1;h2=g2*dfnew2;k2=[h2 k2];dfnew2=h2;enddfnew_m=[k2 k1];q=1/Rt*Ns’*dfnew_m;g=dIdV(1)/q(1);q=1/Rt*Ns’*dfnew_m*g;chi2=((q-dIdV’).ˆ2)’;fold=fnew;endplot(E,fnew)