Optical properties and electron transport in low-dimensional nanostructures
aa r X i v : . [ c ond - m a t . m e s - h a ll ] O c t Optical properties and electron transport in low-dimensional nanostructures
K. Kr´al ∗ Institute of Physics, Academy of Sciences of Czech Republic, v.v.i.Na Slovance 2, 18221 Prague 8, Czech Republic
Miroslav Menˇs´ık † Institute of Macromolecular ChemistryAcademy of Sciences of Czech Republic, v.v.i.Heyrovsk´y Sq. 1888/2, 16206 Prague 6, Czech Republic (Dated: December 10, 2018)We present the theory of the electronic transfer and the optical properties of the quasi-zerodimensional quantum nanostructures, like quantum dots or the DNA molecule. The theory is basedon the multiple scattering of the charge carriers in the quasi-zero dimensional nanostructures leadingto the manifestation of the nonadiabatic influence of the atomic lattice on the charge carriers. Thetheory is based on the nonequilibrium Green’s functions and the quantum kinetic equations. Threeexamples of the electronic motion in the small systems are presented, together with a comparison ofthe theoretical results with their experimental counterparts. The comparison with the experimentsunderlines importance of the electron-phonon interaction in nanostructures.
PACS numbers: 72., 78., 72.80.Ey, 72.80.Le, 73.21.La, 73.63.Kv, 78.67.HcKeywords: quantum dots, electron-phonon interaction, optical properties, electron relaxation, DNA molecule
INTRODUCTION
The technical problems like those with the develop-ment of the heat and in the effort to increase the speedof the information processing in the semiconductor mi-croelectronic circuits have lead to the current trendsto achieve the miniaturization of the electronic devices.With respect to the bulk solids, in the systems with asmall size the motion of the charge carriers becomes re-stricted in at least one of the three dimensions. Thequantum dots and nanoparticles differ significantly fromthe bulk and from the higher dimensional nanostructuresbecause the motion of the charge carriers is restricted inall three dimensions. The three-dimensional restrictioninfluences the orbital motion of the charge carriers, theelectrons and the holes. For the sake of simplicity we willbe speaking only about the electrons in quantum dots.In quantum dots the spectrum of the stationary boundstates of the electrons has a discrete character havingusually narrow peaks in the electronic spectral density.This can be observed in optical emission or absorptionspectra. The features appear to be seen in the quan-tum dots made of the materials in which the mean freepath of the electron is larger than the lateral size of thequantum dot. The discrete character of the quantum dotelectronic energy spectra is therefore found in the inor-ganic semiconductor materials [1] and metals [2].The experimental optical spectra sometimes [3] seemto have the form of the spectral lines different from thesimple form of the delta-function of the energy variable.The reason for this shape can be connected with the mo-tion of the atomic lattice [4]. Let us briefly show ad hocarguments supporting this viewpoint: The electrons in aquantum dot necessarily collide with the atomic lattice of the nanostructure. While in the bulk systems the elec-tron leaves the target after the collision and may finishthe scattering act upon leaving to infinity, in the nanos-tructure the process may be different. In the extremecase of the nanostructure confined in all three dimen-sions, in the quasi zerodimensional (0-D) nanostructure,the electron leaving the target meets with the boundaryof the nanostructure a reflects back again to continue themultiple scattering act with the atomic lattice. In thissense the multiple scattering processes should not be arare phenomenon in the quantum dots.The multiple scattering process should produce themultiple phonon states. This means that the numberof the phonons in a given vibrational mode is then gener-ally different from 0 and 1. The multiphonon states mayremind us the coherent photon states of the laser light.The coherent light is in some properties similar to theclassical wave of the electromagnetic field, in which, say,the electric field executes classical oscillations in time. Inthe case when the atomic lattice of the 0-D nanostruc-ture is brought to such a state we may have the system inwhich the effective Hamiltonian for an electron may, tosome extent, depend explicitly on time and the energy isnot then completely conserving. Although the discussionwe present is only a qualitative one, we expect that theelectrons in the atomic lattice of the 0-D nanostructuredo not move adiabatically and the lattice influences theirmotion in a certain way. The nonadiabaticity is man-ifested in such a way that the electron and the latticedo not exchange only heat. The execute a force on eachother. Because of these circumstances the electron andthe vibrations of the atomic lattice need not have thesame effective temperatures, in the case when we ascribea temperature to the separate subsystem of the electronsand to that of the vibrations. From the reason of this seri-ous lack of the complete adiabaticity between the motionof the electrons and phonons, we may quite generally ex-pect that developing the theory of the electronic proper-ties of the nanostructures with vibrations on the basis ofan assumption of an overall thermodynamic equilibriummay be sometimes not sufficiently satisfactory.We shall concentrate our attention on the manifes-tations of the interaction of electrons with the opticalphonons of quasi-zero dimensional nanostructures likethose of quantum dots made from the polar semicon-ductors [4–11] or other quasi-zero dimensional structureslike the bases of the DNA molecule. The electron-phononinteraction in quantum dots is an intrinsic mechanismwhich is always present in these objects. The questionabout what is the strongest electron scattering inter-action influencing the electron kinetics and the opticalproperties of quantum dots does not seem to be com-pletely clarified until these days. From one side, thenumber of papers containing interpretations of the ex-periment on the basis of the electron-electron interactionis extensive [12], but on the other side, there is still a con-siderable number of the experimentally detected effectswhich, besides the electron-electron interaction, can bestrongly influenced by the electron-phonon interaction aswell.Using three examples, this work aims to bring ad-ditional arguments in favor of the importance of theelectron-optical-phonon interaction in the quantum dots.The first example will deal with the dependence of theoptical line width of the quantum dot optical emission onthe lateral size of the dot [13, 14]. The second exampledeals with the long-time luminescence decay in quantumdot samples with the intra-dot electron-phonon multiplescattering [15, 16]. In the third example we deal with amechanism of the irreversible transfer of a charge carrierbetween two quasi-zero dimensional nanostructures. Wepresent an explanation of the electric conduction mea-sured earlier in the molecules of DNA [17, 18].
ELECTRON-PHONON INTERACTION IN ATWO-LEVEL MODEL OF SINGLE QUANTUMDOT
In the three examples we shall use the simple pictureof a single electron moving in an individual quantum dot.As it was done in the earlier papers [4–11] we shall assumethat the holes in the valence band states of the dots aremuch heavier than the electrons in the conduction bandstates, so that they can be considered approximately as astatic charge which contributes only to the overall poten-tial in which the conduction state electron particles move.We shall also assume that because of the larger effectivemasses the spectrum of the hole band bound states is sodense that comparing to the light electron the holes relax their energy quickly with help of certain interactions likethe electron-acoustic phonons interaction, quickly reachtheir ground state in the region of the hole states andremain occupying this ground state at low temperatures.From this reason we confine ourselves to putting an em-phasis only on the kinetics of the conduction band stateselectrons in their quantum dot bound states.As it is well known from the times of the research in theso called hot electrons in the polar bulk semiconductors,the interaction of the conduction band electrons with thelongitudinal optical (LO) optical phonons has a strong in-fluence on the electronic motion. Let us thus briefly showthe basic properties of the two-electronic level model ofan electron interacting with the longitudinal optical (LO)phonons in quantum dots. The details of this argumen-tation were shown previously [4–11] and will not be re-peated here. We shall confine ourselves to the simpleelectronic model Hamiltonian H = P n =0 , E n c + n c n withtwo nondegenerate electronic bound states energy levelsonly, neglecting spin, with two orbital motion energies E and E and corresponding states with indexes n = 0 , H = X q ,m,n =0 , A q Φ( n, m, q )( b q − b + − q ) c + n c m , (1)between the electron and longitudinal optical phonons.This operator contains the coupling constant A q , whichis A q = ( − ie/q )[ E LO ( κ − ∞ − κ − )] / (2 ε V ) − / , where κ ∞ and κ are, respectively, high-frequency and staticdielectric constants, ε is permittivity of free space, − e is the electronic charge, q = | q | is the three-dimensionalwavevector of the bulk LO phonon mode and V is vol-ume of the sample. Φ is the form-factor, Φ( n, m, q ) = R d r ψ ∗ n ( r ) e i qr ψ m ( r ), taking into account the form of thequantum dot in which the electron moves. Concerningthe modes of the optical vibrations in the quantum dot,we assume that the structure of the optical lattice vi-brations is practically not touched by the presence ofthe quantum dot in the whole three-dimensional sam-ple. The Hamiltonian operator of the free LO phononsthen is H LO = P q E LO b + q b q , in which E LO is the energyof the vibrational quantum of the LO phonons and b q is the corresponding annihilation operator of the Fock’sphonon. We shall assume that the bulk LO phonons aredispersionless. The complete Hamiltonian of the two-level single quantum dot then is H QD = H + H + H LO . (2)In the present work we shall use an approximative solu-tion of this Hamiltonian. We shall not make any assump-tion about an overall thermodynamic equilibrium in theelectron-phonon system of the dot. Single quantum dot electron kinetics
As it has been said in the Introduction, in the quan-tum dots the energies of the electronic bound states arediscrete and the states of the optical vibrations in theFock’s states, with an integer number of the vibrationalquanta in a single mode, have discrete energies too. If wewish to consider theoretically the irreversible processes ofthe electron energy relaxation, we should step beyond thefinite order of the perturbation calculation and considergenerally the multiple scattering processes leading to vi-brational states with a non-integer number of phononsin a single mode [19]. The multiphonon states can beviewed as coherent states of the phonon modes. Thesestates are known to have a continuum of energy[19]. Assuch they may allow for the irreversible dissipation pro-cesses of the electron energy relaxation within the boundstates of a dot connected to an environment, as it hasbeen expected to occur in the electron-phonon theoret-ical interpretation of the absence of the electron energy relaxation bottleneck in quantum dots[4–11]. Let us re-mind that in the latter references the optical phononsalso play a role of a reservoir keeping at a temperature T LO .Without assuming an overall thermodynamic equilib-rium in the quantum dot system we are led to using ki-netic equations for the electron-phonon system with theHamiltonian (2). With the help of the nonequilibriumGreen’s functions [20] or with using a similar suitabletechnique [21] we can obtain certain characteristics ofthe optical spectra and the electron kinetics in quantumdots.We consider the two states of the orbital motion of theelectron in a quantum dot, namely the state n = 0, theground state of the electron and the state n = 1, theexcited state. We are interested in the time t evolution dN /dt of the electron occupation N of the excited state n = 1. Using the so called diagonal approximation, wecome to the following formula for the rate of change [6, 8] dN /dt : dN dt = − π ¯ h α (cid:20) N (1 − N ) (cid:18) (1 + ν LO ) Z ∞−∞ dE σ ( E ) σ ( E − E LO ) (3)+ ν LO Z ∞−∞ dEσ ( E ) σ ( E + E LO ) (cid:19) − N (1 − N ) (cid:18) (1 + ν LO ) Z ∞−∞ dEσ ( E ) σ ( E − E LO )+ ν LO ) Z ∞−∞ dEσ ( E ) σ ( E + E LO ) (cid:19)(cid:21) . Here σ and σ are electronic spectral densities, ν LO isBose-Einstein distribution of LO-phonons at tempera-ture T LO of the lattice and N is the average electronicoccupation of the electronic state with the index n = 0.The coupling constant α mn is α mn = X q | A q | | Φ( n, m, q ) | , α mn = α nm . (4)It includes the bulk characteristics of the electron-phononcoupling and the form factors Φ of the quantum dot.The electronic spectral densities σ n ( E ) are given[20]by the retarded self-energy M n ( E ) which is calculatedfrom the equation for the electronic self-energy (5). Forthe electronic state n , and in the self-consistent Bornapproximation [4–11], we have: M n ( E ) = X m =0 α nm (5) × { − N m + ν LO E − E m − E LO − M m ( E − E LO ) + i + + N m + ν LO E − E m + E LO − M m ( E + E LO ) + i + } , where N m is again the electronic population of the m -thstate.The above outlined equations for the single electronkinetics in the electron-phonon system under considera-tion allow us to give the theoretical explanation of a num-ber of phenomena observed in quantum dot experiments.It appears that the kinetic processes of the electrons inquantum dots on one side, and the ”static” properties ofthe electrons, as they are given by the optical line shapeon the other side, are mutually closely related and cannotbe well separated one from another. In particular, the op-tical line shape is intimately related to the nonadiabaticeffect of the upconversion determining a stationary valueof the electronic density distributed among the energylevels in a quantum dot.Besides the effect of the fast electron energy relaxationfrom the state n = 1 to the state n = 0, the equationsallow also for the effect of the electronic up-conversion,namely, for a spontaneous promotion of the electronic F W H M ( m e V ) FIG. 1. FWHM of the main peak electronic spectral densityin InAs quantum dot, at 10 K plotted against the inverse ofthe quantum dot lateral size d. The result is calculated atthe stationary distribution of the electronic density amongthe levels. density to the upper excited state. This means thatpreparing at the beginning the dot in the state with oneelectron in at the level n = 0, one can find at later timesa nonzero occupation of the excited state n = 1. Thiseffect was presented earlier at the references e.g. [10, 11].The kinetic equation (3) can be solved to obtain the timedevelopment of the electronic occupation of the two elec-tronic levels at a given temperature of the lattice of theLO phonons, T LO . The effect of the steady state oc-cupation of the electronic excited state, different by thenature from the thermal up-conversion of the electronicdistribution, is possibly well manifested in the observedlasing of the quantum dot lasers from the higher excitedstates (see e.g. [22]). In this way the nonadiabatic influ-ence of the lattice vibrations on the electrons appears torepresent an intrinsic property of the quantum dot nanos-tructure having a possible impact on the functioning ofthe quantum dot lasers. Example 1: The optical linewidth in a single dot
Let us remind once again that we ignore the contribu-tion of the holes in the valence band states of the quan-tum dot exciton and ascribe the linewidth properties ofthe exciton emission peaks as solely due to the electronsin the conduction band states of the dot. By doing thiswe shall avoid the complications connected with consider-ing the whole exciton particle containing the electron andthe hole. According to the presently used theoretical ap-prach to the electron-LO-phonon interaction and to theeffect of the electron occupation upconversion, it seemsthat this effect has a direct impact on the properties ofthe optical transitions linewidth, as it varies with the quantum dot lateral size. From the experimental point ofview, about a decade ago measurements were publisheddue to which the full width at half maximum (FWHM)of the lowest energy optical transitions in the quantumdots depends linearly on the inverse value of the quan-tum dot lateral size d . The observed linear dependencegoes through the beginning of the coordinates, namelythrough the point of the width equal zero and the limitof 1 /d equal zero too. The experiments were performedon the nanoparticles of CdSe, CuCl and CuBr. The mea-surement technique was the accumulated photon echo ex-periment [13, 14]. The optical linewidth represented bythe electronic spectral density line width has been cal-culated recently to obtain the dependence on the lateralsize d (see e.g. Reference [23]). The calculated linewidthdependence was found for the nanoparticles of InAs semi-conductor and also for CdSe [24]. In the case of InAs thetheory clearly gives the linear dependence found in exper-iments and goes through the beginning of the coordinatesas well (the linewidth is directly proportional to 1 /d ). Inthe course of the theoretical evaluation it has appearedthat a key step to be made in the calculation is to fullyrespect the tendency of the system to achieve the steadystate electronic level occupation upconversion. The cal-culated dependence has come out as a linear function. Inthe case of the material CdSe the calculation shows thatone gets approximately a linear dependence, and the linehas the tendency to go through the beginning of coordi-nates, nevertheless the linear dependence is modified bya certain noise. This computational noise structure is atpresent attributed to the used approximation. Namely,the noise in the calculated dependence is likely due tothe fact that while in the CdSe material the strength ofelectron-LO-phonon interaction is not small, the calcula-tion uses an approach which is perhaps well suited onlyfor a material with a weak electron-LO-phonon coupling,like for example InAs. The recalculation of the resultsfor the materials with an improved approximation is tobe performed.The Fig. 1 shows a certain fine structure of the widthdependence on 1 /d . These features, which are proba-bly difficult to detect experimentally, are numerically ob-tained at such values of d , at which there is a resonancebetween the electron energy level difference E − E andan integer multiple of the optical phonon energy.The direct proportionality of the linewidth to the quan-tity 1 /d has not probably been theoretically explainedwith using another theoretical mechanism so far. Thepresent agreement between experiment and theory pro-vides therefore a supporting argument in favor of animportant role of the optical phonons and their multi-phonon states in the quasi-zero dimensional nanostruc-tures. It should be emphasized that without taking intoaccount the upconversion effect of the electronic distri-bution in the dot states, the agreement with experimentin the form of the straight line going through the originof coordinates would not be obtained. Example 2: Long-time luminescence decay
Important applications of quantum dots are connectedwith a controlled light emission from quantum dots. Thisapplication of quantum dots is complicated by the effectof the intermittency of the light emission, or the blink-ing of the quantum dots [25], which is so far generallynot under control. Under a continuous excitation of thequantum dot sample the emission of a single dot is usu-ally not continuous. According to experimental data theemission is nearly continuous within finite time intervals.The distribution of the occurrence of the lengths of theseemission (on state) intervals obeys a power-law statistics[25]. The distribution function of the lengths of inter-vals, in which a given quantum dot does not emit light(off state intervals) similarly obeys a power-law statis-tics. Today, a convincing consensus about the mechanismof the blinking seems to be missing. The quantum dotblinking effect has a certain similarity with another phe-nomenon observed in the quantum dot samples, namely,with the long-time behavior of the decay of the lumines-cence intensity signal of a sample after an illuminationby a laser pulse [15, 16]. The experiments [15, 16] showthat the decay of the luminescence signal does not obeyan expected simple exponential law, but it has the formof a power law in a remarkably broad range of the timeseparation from the exciting laser pulse. In this exampleof the present paper we show that in a quantum dot sam-ple we can in principle expect a relatively simple intrinsicmechanism which is able to provide the experimentallydetected power law decay. As in the previous example themechanism of this intrinsic effect can be demonstratedupon using the simple two-level quantum dot model.Let us first remind that the authors of the experiments[15, 16] explain the long-time behavior of the lumines-cence decay by an assumption that after the laser pulseexcitation of the sample a part of the quantum dots cap-turing the electron and hole carriers are found in thetriplet exciton excited state. The quantum dots withtriplet a excitation do not emit light and serve a stor-age of excitation energy for the following period of time.The resulting power-law dependence of the sample lu-minescence decay is then ascribed to a combination of alarge number of channels of charge carrier tunneling fromthe triplet exciton quantum dots to such quantum dotsin which the electron-hole excitation has a singlet excitoncharacter and allows for the emission of light.In the present example we suggest an intrinsic mech-anism which allows to present, in a relatively simple an-alytical way, a possible origin of the power-law lumines-cence decay. Let us assume a quantum dot sample con-sisting of small quantum dots. The size of the dots issuch that the singlet-triplet energy separation is larger than the temperature of the experiment. After illumi-nating the sample by laser pulse, some of the dots with asmall diameter contain singlet excitons and some containthe triplet excitons. The triplet exciton dots again serveas a reservoir of the excitations in the sample. The sin-glet exciton dots relax by emitting light to their groundstate typically within about a nanosecond. Let us ignorethe processes in the triplet exciton dots which allow thedirect emission of light. Let us assume that the tripletexciton dissociates into a hole and electron particles andthat the electron is upconverted with the help of theelectron-phonon interaction to the excited state in thedot. We also assume that then the upconverted electronimmediately leaves the quantum dot and becomes im-mediately available, without a delay, for a light emissionelsewhere in the sample. In other words, after leaving thetriplet exciton quantum dot the electron can be capturedimmediately by another quantum dot in a dense sampleof the dots. In the case that the electron meets a holewith a suitable spin in the target quantum dot, then thelight is emitted. We shall therefore assume that the keymechanism, or a bottleneck mechanism, determining thetime dependence of the long-time luminescence intensitydecay is the rate at which the electrons are upconvertedfrom the triplet exciton quantum dot ground state to theexcited state in the dot.Inside a single triplet quantum dot the electron is up-converted to the excited state of the dot by the electron-phonon interaction included in the self-consistent Bornapproximation to the electronic self-energy, as mentionedabove. The key process giving the power-law decay isthe rate dN /dt , given by the formula (3) at which theelectron is being promoted to the excited state by theup-conversion. The assumption about the immediate re-lease of the upconverted electron from the triplet excitondot can be formally represented by the condition that N = 0 for all values of the time variable t . Let us notethat the effect of the immediate release of the electronfrom the dot has an important consequence, namely, thetotal amount of the electron occupation at the triplet ex-citon dot decreases with time. Because of this decrease,the influence of the total electronic occupation on themultiple phonon generation also decreases, which in turnimplies that the upconversion rate becomes slower andin the result obtains a power-law time dependence.The power-law decay of the luminescence is shown inthe Fig. (2). The model of the electron states used in thepresent paper in the quantum dot was specified earlier[10, 11]. Let us remind at this point that this model usesthe method of the effective mass. The electron states arethe bound states in an infinitely deep cubic quantum dot.We take into account the ground state of the electronin this potential and one of the triply degenerate lowestenergy excited states. In the Fig. 2 we see that after aninitial period of time the relaxation curve signal, beingassumed to be proportional to the decaying luminescence -9 -7 -5 -3 R a t e ( / p s ) Time (ps)
FIG. 2. The rate of the upconversion r = dN /dt for cubic dotof InAs with the lateral size of 15 nm. N is kept equal zerothroughout the evolution in time. The full line correspondsto the temperatures 10 K and 30 K, while the dashed line iscomputed at 50 K. signal, becomes a power-law curve. Then for the verylong times the curve has a tendency to turn finally tothe exponential shape. In our calculation this effect canbe due to the damping factors added to the numericalprocedure because of the convergence of the calculation.Fitting the linear part of the obtained curves in the closevicinity of the time of 10 ps to the power law dN /dt = ξt − α , we get α = 2 . TWO INTERACTING SINGLE-LEVELQUANTUM DOTS
The charge carrier transfer between neighboring quasi-zero dimensional nanostructures is important for vari-ous applications, like the charge transfer along a sur-face with quantum dots in optoelectronics or even forour understanding the electric conduction of the DNAmolecule. In the latter case, we shall discuss the earlierexperiments performed by microwave measurements onthe DNA molecules, with the result saying that in thelimit of low temperatures, below about 100 K, the elec-tric conduction of DNA molecule becomes activationless[17, 18]. We show that the property of DNA molecule conduction of being independent of temperature at thelow temperatures can be explained by the interaction ofthe conduction electrons (holes) with the molecular vi-brations of the DNA bases in combination with the elec-tron tunneling between the individual quasi-zero dimen-sional units. The purpose of this demonstration is toshow that the interaction of the charge carriers, localizedwithin the molecular bases, with the intra-bases atomicvibrations displays a key role in the determination of thebasic properties of this basic biological molecule. In thissection we therefore do not pay attention to an electrontransfer from one quasi-zero dimensional unit to anotherwith using the presence of the wetting layer, as in theStranski-Krastanow method grown samples, or the trans-fer through the electron states in the matrix material ofthe quantum dot sample.In order to utilize the theoretical tools developed ear-lier for the quantum dot systems, we shall develop thetheoretical argumentation with using the concepts usualin the quantum dot systems. Using an analogy betweentwo individual quantum dots and two neighboring DNAbases, we shall make a numerical estimate of an electronirreversible transfer process in the DNA molecule. Forthis purpose the available values of the parameters of theHamiltonian will be used [26–28].In a sample of interacting quantum dots the electriccurrent can be transported by the free charge carriers.The electron transport process may be represented bya transfer of electrons within a pair of quantum dots.We choose a simple model consisting of two quantumdots, A and B . Each quantum dot is supposed to haveonly a single electronic orbital for a spinless electron. Weshall ignore the electron-electron interactions. Withoutconsidering the electron-phonon interaction, the effectiveHamiltonian for a conduction band electron can be asfollows: H e = H eA + H eB + V t . (6)Here H eA = E A c + A c A , (7)and H eB = E B c + B c B , (8)are the Hamiltonian operators of an electron on two elec-tronic orbitals, each being localized at a single quantumdot, A and B , with the site energies at the dots, E A and E B . The operator c A is annihilation operator of electronat the orbital localized at the dot A , while the particleoperator c B annihilates electron at the quantum dot B .The operator V t expresses a tunneling of the electron be-tween the two sites and will have the form V t = t ( c + A c B + c + B c A ) . (9)The above given Hamiltonian assumes that the Hilbertspace of the single electron is determined by two elec-tronic orbitals, each being localized at a separate quan-tum dot. These orbitals will be assumed energeticallynondegenerate and will be assumed to have zero mutualoverlap.An electron placed into the system of the two quan-tum dots is assumed to interact with the vibrations ofthe quantum dot lattice. Because we assume that eachquantum dot has only a single electronic orbital, we con-sistently choose the electron-phonon interaction in thefollowing form: H e − p = H e − p,A + H e − p,B . (10) H e − p,A is the operator of the electron-phonon interactionin the quantum dot A . In analogy with the earlier theoryof the electron energy relaxation in a single quantum dot,and taking into account that there is a single electronorbital per dot, we take this operator in the form takenover, including the notation, from the reference [4]: H e − p,A = X q ∈ Ω A A q Φ A (0 , , q )( b q − b + − q ) c + A c A . (11)Here A q is Fr¨ohlich’s coupling constant [4], given bythe material constants of the material of the quantumdots. The integration over q covers the range Ω A of thephonon wavevector in the quantum dot A . The quantityΦ A (0 , , q ) is the form-factor of the quantum dot A , tak-ing into account the quasi-zero dimensional shape of thequantum dot. Let us remind at this place that we ap-proximate the phonon system of a given dot by the bulkphonon modes of the whole sample [10]. The sum over q represents the sum over the bulk longitudinal optical(LO) modes of the lattice vibrations of the sample. Theoperator b q is annihilation operator the LO phonon inthe mode with the wavevector q .Similarly, for the quantum dot B we have the electron-phonon operator as follows: H e − p,B = X q ′ ∈ Ω B A q ′ Φ B (0 , , q ′ )( b q ′ − b + − q ′ ) c + B c B . (12)Here again Φ B (0 , , q ′ ) expresses the form-factor of thequantum dot B at the value of the phonon wavevector q ′ . We consider the phonon modes with the wavevectors q and q ′ as simply two different and independent phononmode systems.To make the total Hamiltonian complete, we have thenoninteracting LO phonon modes Hamiltonian as follows: H p = X q ∈ Ω A E ( A ) LO b + q b q + X q ′ ∈ Ω B E ( B ) LO b + q ′ b q ′ . (13)This free phonon Hamiltonian operator consists of twoparts, one is for the phonons of the quantum dot A , andthe other for the phonons of the quantum dot B . We R e l a x a t i on r a t e ( / p s ) FIG. 3. The generation rate of the electronic occupation ofthe quantum dot A , dN /dτ at the state at which the electronoccupies the quantum dot B with the lower orbital energy. T is the temperature of the lattice. The quantum dot A has thelateral size of 21 nm, while the quantum dot B has the lateralsize of 19 nm. The material parameters of the crystallineGaAs are used for the electron-LO-phonon interaction. shall assume that the optical phonon energies, being dis-persionless in both the quantum dots, are simply equal, E ( A ) LO = E ( B ) LO = E LO .The whole Hamiltonian operator H QD of the systemof two interacting quantum dots then is: H QD = H e + H p + H e − p . (14)Because of the boson nature of the phonons we can expectthat even if the phonon subsystem is noninteracting withthe electrons, the whole system can have a continuum inits energy spectrum. This expectation is obviously dueto the possibility of the coherent (multiphonon) states ofthe individual vibrational modes[19]. In the present casethe multiphonon states will be present as virtual statesbrought about by using self-consistent Born approxima-tion to the electronic self-energy [4, 10].The Hamiltonian (14) is actually one of the forms of thespin-boson Hamiltonian[29]. We shall obtain the quan-tum kinetic equations in a perturbative way, using thenonequilibrium Green’s functions [20], without makingan assumption about the overall thermodynamic equilib-rium. Example 3: Electron transfer between twoquasi-zero dimensional nanostructures
Although the purpose of this third example is topresent a mechanism of the charge transfer between twoquasi-zero dimensional elements of a more general type,in choosing the parameters for the Hamiltonian we shallbe also motivated by the known electronic structure ofthe DNA molecule [26–28]. We shall assume that thebottleneck part of the DNA molecule, from the pointof view of the electron transport, will be such a pairof the neighboring DNA bases, in which an electron isto overcome a potential barrier of the large differenceof the electronic site energies on the neighboring bases, E A − E B ≈ . | ( E A − E B ) /t |≫
1, and the inter-bases parameteris taken as t = 0 .
03 eV. The molecular bases are repre-sented here approximately by quantum dots which aregiven by the same model as in the first example, withthe cubic shape and the lateral size of 19 nm. The elec-tron wave functions localized within individual molecularbases are chosen to be the ground state wave function inthe cubic dot. The electron-phonon interaction in themolecular basis is taken over from the quantum dot sys-tem. The details of the parameters of the electron-LO-phonon interaction in quantum dots can be found in ref-erences [4–11]. Assuming the electron-phonon interactionas a small perturbation, we perform a canonical trans-formation of the Hamiltonian diagonalizing the purelyelectronic part of it. After this operation the Hamil-tonian becomes formally identical with the previouslyused [4–11] two level quantum dot Hamiltonian with theelectron-LO-phonon interaction. The electron transfer isthus numerically solved in the same way as it has beendone in the case of the single two-level quantum dot,namely including the electron-phonon coupling again inthe self-consistent Born approximation to the electronicself-energy. The details of this procedure will be pub-lished elsewhere.We assume that the electric conduction measured inthe experimental paper [17] is determined mainly by thebottleneck basis pairs specified above. Calculating thetemperature dependence of the rate in time τ , dN /dτ ,of the electronic transfer from a basis with the lower elec-tron orbital energy E B to the neighboring basis with thehigh orbital electronic energy E A , in the cooperation ofthe tunneling mechanism t and the electron-phonon in-teraction on both the individual bases, we take the cal-culated temperature dependence as proportional to theelectron conduction of DNA molecule as it is measuredby Gruner et al. [17].Fig. 3 shows that at the low temperatures the temper-ature dependence of the electronic conduction of DNAmolecule comes out in the present approach as completelyactivationless, in a rather good agreement with the ex-perimental paper [17]. The activationless character ofthe conduction is given by the nonadiabatic influence ofthe atomic lattice on the conduction electron, or in otherwords, by the upconversion effect of the electron distri-bution within the system of the two bases of the DNAmolecule. We can say the the activationless shape ofthe temperature dependence of the electric conduction ofDNA at the temperature below about 40 K shows thatthis object behaves as a small system [30]. We confine ourselves to the low temperature part of the calculateddependence, not commenting the shape of the tempera-ture dependence of the conduction at the high tempera-ture region at this work. CONCLUSIONS
The multiple scattering of the electrons of the low-dimensional nanostructures manifests itself significantlyin a variety of effects in the electron system in these ob-jects. The comparison with the corresponding experi-mental data supports the importance of the charge carri-ers multiple scattering on the optical vibrations of lattice.The theoretical methods used here show that one needsto go beyond the finite degree of the perturbation calcula-tion, or beyond the Golden Rule based theoretical tools.This property of the low-dimensional nanostructures un-doubtedly brings about a not so high popularity of thepresent approach to the theory of the small systems.The second example discusses an intrinsic affect of asingle dot, which should be always present in this struc-ture, and which can give a long-time luminescence in thesample. Although the second example does not seem toresolve the quantum dot blinking problem right away, itgives a mechanism with the power-law character of theluminescence decay.The authors acknowledge the support fromthe projects ME-866, OC10007 of MˇSMT andAVOZ10100520. ∗ [email protected] † [email protected][1] N. N. Ledentsov, V. M. Ustinov, V. A. Shchukin, P. S.Kopev, and Z. I. Alferov, Semiconductors, , 343(1998).[2] V. Ray, R. Subramanian, P. Bhadrachalam, L.-C. Ma,C.-U. Kim, and S. J. Koh, Nature Nanotechnology, ,603 (2008).[3] G. Fasching, F. F. Schrey, W. Brezna, J. Smoliner,G. Strasser, and K. Unterrainer, phys. stat. sol. (c), ,3114 (2005).[4] K. Kr´al and Z. Kh´as, Phys. Rev. B, , R2061 (1998).[5] K. Kr´al and Z. Kh´as, Phys. Stat. Sol. B, , R3 (1997).[6] K. Kr´al and Z. Kh´as, Phys. Stat. Sol. B, , R5 (1998).[7] H. Tsuchiya and T. Miyoshi, J. Appl. Phys., , 2574(1998).[8] K. Kr´al and Z. Kh´as, (2001), arXiv:cond-mat/0103061.[9] K. Kr´al and P. Zdenˇek, Physica E, , 341 (2005).[10] K. Kr´al, P. Zdenˇek, and Z. Kh´as, Nanotechnology, IEEETransactions on, , 17 (2004).[11] K. Kr´al, P. Zdenˇek, and Z. Kh´as, Surf. Sci., ,321 (2004).[12] V. I. Klimov, Los Alamos Science, , 214 (2003).[13] K. Takemoto, B.-R. Hyun, and Y. Masumoto, Journ.Lumin., , 485 (2000). [14] Y. Masumoto, M. Ikezawa, B.-R. Hyun, K. Takemoto,and M. Furuya, phys. stat. sol. (b), , 613 (2001).[15] T. S. Shamirzaev, A. M. Gilinsky, A. K. Bakarov, A. I.Toropov, D. A. T´enn´e, K. S. Zhuravlev, C. von Bor-czyskowski, and D. R. T. Zahn, JETP Letters, , 389(2003).[16] T. Shamirzaev, A. Gilinsky, A. Toropov, A. Bakarov,D. Tenne, K. Zhuravlev, C. von Borczyskowski, andD. Zahn, Physica E, , 282 (2004).[17] P. Tran, B. Alavi, and G. Gruner, Phys. Rev. Lett., ,1564 (2000).[18] E. Torikai, H. Hori, E. Hirose, and K. Nagamine, PhysicaB, , 441 (2006).[19] N. Terzi, in Collective excitations in solids , Vol. 88, editedby B. DiBartolo, NATO ASI Series, Series B (PlenumPress, New York, 1983) pp. 149–182.[20] E. M. Lifshitz and L. P. Pitaevskii,
Physical kinetics (Butterworth-Heinemann, June 1981) ISBN 0750626356,reprint edition.[21] D. N. Zubarev,
Neravnovesnaya Statisticheskaya Ter-modinamika (Nauka, Moscow, 1971).[22] A. Marcus, J. X. Chen, C. Paratho¨en, and A. Fiore, Appl. Phys. Lett., , 1818 (2003).[23] K. Kr´al and M. Menˇs´ık, e-J. Surf. Sci. Nanotech., , 136(2010).[24] K. Kr´al and M. Menˇs´ık, in (Ponta Delgada University, S.Miguel, Azores, Portugal, 2009) IEEE Catalog Num-ber: CFP09485-USB, ISBN: 978-1-4244-4827-2, Libraryof Congress: 2009905508.[25] S. F. Lee and M. A. Osborne, ChemPhysChem, , 2174(2009).[26] A. Troisi and G. Orlandi, Chem. Phys. Lett., , 509(2001).[27] E. M. Conwell and S. V. Rakhmanova, Proc. Natl. Acad.Sci., , 4556 (2000).[28] E. M. Conwell, Proc. Natl. Acad. Sci., , 8795 (2005).[29] A. J. Leggett, S. Chakravarty, A. T. Dorsey, M. P. A.Fisher, A. Garg, and W. Zwerger, Rev. Mod. Phys., ,1 (1987).[30] V. ˇSpiˇcka, T. M. Nieuwenhuizen, and P. D. Keefe, Phys-ica E,29