Shot noise of the conductance through a superconducting barrier in graphene
aa r X i v : . [ c ond - m a t . m e s - h a ll ] M a y Shot noise of the conductance through a superconducting barrierin graphene
Mi Liu and Rui Zhu * Department of Physics, South China University of Technology,Guangzhou 510641, People’s Republic of China
Abstract
We investigated the conductance and shot noise properties of quasiparticle-transport througha superconducting barrier in graphene. Based on the Blonder, Tinkham, and Klapwijk (BTK)formulation, the theory to investigate the transport properties in the superconductive grapheneis developed. In comparison, we considered the two cases that are the transport in the presenceand absence of the specular Andreev reflection. It is shown that the conductance and shot noiseexhibit essentially different features in the two cases. It is found that the shot noise is suppressedas a result of more tunneling channels contributing to the transport when the superconductinggate is applied. The dependence of the shot noise behavior on the potential strength and the widthof the superconducting barrier differs in the two cases. In the presence of the specular Andreevreflection, the shot noise spectrum is more sensitive to the potential strength and the width of thesuperconducting barrier. In both of the two cases, total transmission occurs at a certain parametersetting, which contributes greatly to the conductance and suppresses the shot noise at the sametime.
PACS numbers: 74.20.-z, 72.70.+m, 72.10.-d * Corresponding author. Electronic address: [email protected] . INTRODUCTION In the past decade, there has been a great deal of interest in studying the physicalproperties of graphene both theoretically and experimentally . In graphene, the low-energy excitations are massless and chiral Dirac fermions with linear energy dispersion nearthe Dirac point. Because of the unique energy dispersion in graphene, there are somespecial phenomena such as the unconventional quantum Hall effect, Klein paradox, etc .Particularly relevant to the present work, the presence of the Dirac Fermions in the graphene-based superconductor junctions results in the specular Andreev reflection .The Andreev reflection describes the tunneling phenomenon of electron excitation con-verting into hole excitation by the superconducting pair potential. The interface between ametal and a superconductor can reflect a negatively charged electron incident from the metalinto a positively charged hole, while the missing charge enters the superconductor forming aCooper pair. Usually the hole is reflected back along the path of the incident electron in theconventional materials, which is called “retro-Andreev reflection”. However, the Andreevreflection in undoped graphene is specular—the so-called ”specular Andreev reflection”, inwhich the reflected angle is inverted . Because the graphene needs to be describedby the Dirac-like equation rather than the usual Schrodinger equation, the superconduc-tive graphene needs to be described by the Dirac–Bogoliubov–de Gennes equation ratherthan the Bogoliubov–de Gennes equation for usual superconductors . However,graphene is not a natural superconductor. Recent research has shown that superconductiv-ity can be induced in a graphene layer in the presence of a superconducting electrode bymeans of the proximity effect . So far, the physical properties of the graphene-based su-perconductor junctions have been extensively studied and many important results have beenobtained. Most of them focus on the transport properties of the graphene/superconductivegraphene interface and only the conductance is considered , which prompts us toinvestigate the transport properties through a superconducting barrier in graphene and es-pecially focus on the shot noise spectrum. In this work, based on the Dirac–Bogoliubov–deGennes equation and the Blonder, Tinkham, and Klapwijk (BTK) formulation, we derivethe shot noise formula and provide numerical results of the transport properties in thegraphene/superconductive graphene/graphene (G/SG/G) heterostructure. Dependence ofthe shot noise spectrum on the structure parameters and its physical mechanisms are dis-2ussed. II. THEORY AND MODEL
We consider transmission through the G/SG/G heterostructure occupying the x - y plane,the schematic of which is shown in Fig. 1. The growth direction of the graphene is takento be the x -axis. The left G region extends from x = −∞ to x = 0 and the right G regionextends from x = a to x = + ∞ . The superconductive region occupies the 0 < x < a region. The electron and hole excitations are described by the Dirac–Bogoliubov–de Gennesequation : H α − E F ∆ ( r )∆ ∗ ( r ) E F − H α Ψ α = E Ψ α . (1)Here, Ψ α = ( ψ Aα , ψ Bα , ψ ∗ A ¯ α , − ψ ∗ B ¯ α ) are the four-component wave functions for the electronand hole spinors. The index α denotes K or K ′ for the electrons or holes near the Dirac K and K ′ points. ¯ α takes values K ′ ( K ) for α = K ( K ′ ). E F denotes the Fermi energy. A and B denote the two inequivalent sites in the hexagonal lattice of graphene. The Hamiltonian H α is given by H α = − i ~ v F [ σ x ∂ x + sgn ( α ) σ y ∂ y ] + U ( r ) , (2)where v F denotes the Fermi velocity of the quasiparticles in graphene and sgn ( α ) takes thevalues of ± α = K ( K ′ ).For 0 < x < a , the superconducting electrode on the top of the graphene layer induces anonzero pair potential ∆ ( r ) via the proximity effect. We model the pair potential as∆ ( r ) = { , others , ∆ e iφ , < x < a, (3)where ∆ and φ are the amplitude and the phase of the induced superconductive orderparameter. The electrostatic potential U ( r ) in the G and SG regions can be tuned indepen-dently by a gate voltage or by doping. We take U ( r ) = { , others , − U , < x < a. (4)Eq. (1) can be solved straightforwardly to yield the wave functions Ψ in the G andSG regions, respectively. In the G region, for the electrons and holes traveling in the ± x k y = q and energy ε , the wave functions are givenby Ψ e ± N = exp ( iqy ± ikx ) √ cos α exp ( ∓ iα/ ± exp ( ± iα/ , Ψ h ± N = exp ( iqy ± ik ′ x ) √ cos α ′ exp ( ∓ iα ′ / ∓ exp ( ± iα ′ / , (5)with sin α = ~ v F qε + E F , sin α ′ = ~ v F qε − E F ,k = ε + E F ~ v F cos α,k ′ = ε − E F ~ v F cos α ′ . (6) α is the incident angle of the electron and α ′ is the reflection angle of the hole. Note thatfor the Andreev process to take place, the maximum incident angle of the electron is givenby α c = arcsin (cid:18) | ε − E F | ε + E F (cid:19) . (7)In the SG region, the quasiparticles are mixtures of the electrons and holes. The wavefunctions of these quasiparticles moving along the ± x -direction with the transverse momen-tum q and energy ε has the form ofΨ e ± S = e iqy ± i ( k − iκ ) exp ( − iβ ) ± exp ( − iβ ± iγ ) exp ( − iφ ) ± exp ( ± iγ − iφ ) , Ψ h ± S = e iqy ∓ i ( k + iκ ) exp ( iβ ) ∓ exp ( iβ ∓ iγ ) exp ( − iφ ) ∓ exp ( ∓ iγ − iφ ) . (8)4he parameters β , γ , k , and κ are defined by β = { arccos ( ε/ ∆ ) , ε < ∆ , − iarcosh ( ε/ ∆ ) , ε > ∆ ,γ = arcsin [ ~ v F q/ ( U + E F )] ,k = q ( E F + U ) / ( ~ v F ) − q ,κ = ( E F + U )∆ ( ~ v F ) k sin β, (9)Taking into account both the Andreev and normal reflection processes, the wave functionsin the left G, SG, and right G regions can be written asΨ = Ψ e + N + r c Ψ e − N + r Ac Ψ h − N , Ψ = A Ψ e + S + B Ψ e − S + C Ψ h + S + D Ψ h − S , Ψ = t c Ψ e + N + t Ac Ψ h + N , (10)respectively. Here, r c and r Ac are the amplitudes of the normal and Andreev reflections,respectively; t c and t Ac are the amplitudes of the normal and Andreev transmissions, respec-tively. A , B , C , and D are the amplitudes of electronlike and holeslike quasiparticles in theSG region. All the amplitudes in Eq. (10) can be determined by demanding wave functioncontinuity at the interfaces. These boundary conditions are given byΨ (0) = Ψ (0) , Ψ ( a ) = Ψ ( a ) . (11)The scattering amplitudes can be obtained by numerically solving these continuity equations.The electron and hole operators of the outgoing states are related to the electron andhole operators of the incoming states via the scattering matrix , b Le b Re = S eeLL S eeRL S eeLR S eeRR a Le a Re , b Lh b Rh = S ehLL S ehRL S ehLR S ehRR a Lh a Rh , (12)where the element S ee gives the outgoing electron current amplitude in response to anincoming electron current amplitude and S eh gives the outgoing hole current amplitude inresponse to an incoming electron current amplitude. The generalized current operator forthe electrons and holes in the left electrode can be written as I L ( t ) = e π ~ Z ∞ dEdE ′ e i ( E − E ′ ) t/ ~ < a + Le ( E ) a Le ( E ′ ) > − < b + Le ( E ) b Le ( E ′ ) > + < b + Lh ( − E ) b Lh ( − E ′ ) > . (13)5ubstituting the scattering matrix in Eq. (12), we can obtain I α ( t ) = e π ~ X µmγp Z ∞ dEdE ′ e i ( E − E ′ ) t/ ~ a + µem ( E ) A mpµγ ( α, E, E ′ ) a γep ( E ′ )+ P n a + µhm ( E ) S hαµnm ( E ′ ) a γhp ( E ′ ) , (14)where A mpµγ ( α, E, E ′ ) = δ µα δ γα δ mn δ pn − X n S e + αµnm ( E ) S eαγnp ( E ′ ) . (15)The general expression for the current fluctuations between contacts α and β is S αβ ( t − t ′ ) = 12 < ∆ I α ( t ) δI β ( t ′ ) + δI β ( t ′ ) ∆ I α ( t ) >, (16)with its Fourier transform2 πδ ( ω − ω ′ ) S αβ ( ω ) = < ∆ I α ( ω ) δI β ( ω ′ ) + δI β ( ω ′ ) ∆ I α ( ω ) > . (17)We restrict our consideration to coherent tunneling and neglect the Coulomb interaction.In the zero-frequency limit with ω = 0, providing all the information above, we can expressthe noise power as S αβ = e π ~ X µmγp Z ∞ dE × A mpµγ ( α, E, E ) A pmγµ ( β, E, E ) × f µe ( E ) [1 − f ηe ( E )] + f ηe ( E ) [1 − f µe ( E )]+ P nl S h ∗ αµnm ( E ) S hαγnp ( E ) S h ∗ βγlp ( E ) S hβµlm ( E ) × f µe ( − E ) [1 − f ηe ( − E )] + f ηe ( − E ) [1 − f µe ( − E )] . (18)By introducing the distribution function for the electrons f e ( E ) =[ exp [( E − eV ) /k B T ] + 1] − and that for the holes f h ( E ) = [ exp [( E + eV ) /k B T ] + 1] − ,the zero-temperature conductance can be obtained from the current operator and from theusual quantum statistical assumptions for the averages and correlations of the electron andhole operators in the normal reservoirs as G ( eV ) = G Z α c (cid:18) − | r c | + | r Ac | cos α ′ cos α (cid:19) cos αdα. (19)The shot noise power can be obtained as S (0) = 4 eG Z α c " | r c | | t c | + | r Ac | | t Ac | (cid:18) cos α ′ cos α (cid:19) cos αdα, (20)where G = 4 e N ( eV ) /h is the ballistic conductance of metallic graphene, V is the biasvoltage, and N ( ε ) = ( ε + E F ) w/ ( π ~ v F ) denotes the number of available channels for agraphene sample of width w . 6 II. NUMERICAL RESULTS AND DISCUSSION
Now we present numerical results for the Andreev reflection coefficients and the tunnelingconductance for the
G/SG/G junction with U = 0. In this condition, there is a largemismatch of Fermi surfaces on the G and SG sides. Such a mismatch is well known to actas an effective barrier . The transport properties of the two cases of E F ≫ ∆ ( E F = 0)and E F ≪ ∆ are significantly different. In the case of E F ≫ ∆ , i.e., the incident electronand the reflected hole both lie in the conduction band, which results in the “retro-Andreevreflection”, while in the case of E F ≪ ∆ , only the “specular Andreev reflection” takesplace, since the incident electron in the conduction band is converted into the reflected holein the valence band . In general, it is very difficult to reach the regime E F ≪ ∆ inexperiment . So we only consider the case of E F ≫ ∆ and the condition of comparable E F and ∆ , in the latter of which both normal Andreev reflection and specular Andreevreflection play roles and we can see that the retroreflection crosses over to specular Andreevreflection.Firstly, we consider dependence of the transport properties on the SG-barrier thickness.The tunneling conductance through the SG-barrier as a function of the thickness for differentpotential strengths U is shown in Fig. 2. Dimensionless thickness k a is used. The solidand dashed lines correspond to U /E F = 2 and 10, respectively. When the bias voltage issmall, the conductance exhibits oscillation features. Their oscillation amplitudes decay withincrease of the thickness of the SG layer. The oscillation period of the shot noise power is thesame as that of the conductance. The shot noise characterises correlations of the current. Itcan be seen from the curves that the growing trend of the shot noise is opposite to that of theconductance. When the conductance reaches the maximum value, the shot noise approachesthe minimum value. This can be interpreted by the relation between the shot noise and theproperties of the scatterer. A coherent conductor with all the transmission channels open(The open channel means that the transmission probability is close to one.) has minimalshot noise with the Fano factor approaching 0. A coherent conductor with all closed channels(The closed channel means that the transmission probability is close to zero.) has maximalshot noise with the Fano factor approaching 1. The strength of the shot noise is in the middleof the two limits when a conductor has open and closed channels coexistent. Therefore largetransmission probabilities enhance the conductance and suppress the shot noise. It can also7e seen in the panels (b) and (d) that for the case of E F = ∆ the conductance and shotnoise decrease with the increase of the thickness of the SG layer. This is because that theparameter κ is proportional to (∆ / ~ v F ) sin β for identical U and it is in the exponentialform of e ± κa in the wave function as shown in Eq. (8). It is larger for the case of E F = ∆ than for the case of E F ≫ ∆ , which results in the quick decrease of the shot noise andthe conductance. On the other hand, we can see from Fig. 2 that with the increase of thepotential strength, the tunneling conductance decreased, which illustrates that the Fermisurface mismatch between the normal and superconducting regions suppresses transmission.We also considered the dependence of the tunneling conductance on the value of U for smallbias voltages. As expected, we found that the oscillation amplitude decreases monotonicallywith the increase of U in the case of E F = ∆ and finally approaches a constant value.In Figs. 3 and 4, we provide numerical results of the tunneling conductance as a functionof the bias voltage V . Similar results to Beenakker’s are obtained. In the limit of V →
0, allthe conductances have the same value of 4 /
3. For the case of ∆ ≥ E F , a sharp change in theconductance occurs at eV = ∆ and all the conductances vanish at eV = E F , which is shownin Fig. 3. This is because of that no Andreev reflection occurs for all the incident angles(the critical angle of incidence α c = 0) when ε = E F . For the small bias voltages beforethe turning point eV = E F , the conductance decreases with the increase of the SG energygap. The conductance curves exhibit oscillatory behavior in the region of eV > ∆ , whichis different from the condition considered by Beenakker. By analyzing the transmissioncoefficients of the system, we found that in the condition of eV > ∆ the transmissionspectrum demonstrates oscillatory behavior and the oscillating period increases with theincrease of the SG energy gap. The oscillatory behavior originates from the effect of thequantum-mechanical interference between the electron-like and hole-like quasiparticles ? in the SG barrier. This effect gives rise to oscillations in the reflection and transmissionprobabilities for the incident energies larger than the gap energy.In Fig. 4, numerical results of the conductance for different SG-barrier thickness a areprovided. It can be seen that the conductance is sensitive to the SG-barrier thickness,especially in the condition of eV > ∆ . In this condition, the oscillation period increasesand the oscillation amplitude decreases sharply with the increase of the thickness. This isalso a result of the exponential term e ± κa in the wave function. It can be interpreted bythe a → ∞ limit. In this limit, the components of the wave function with the exponential8erm e + κa are nonphysical, therefore only two of the components of wave function in theSG-region are physical. The model reduces to that of the the G/SG-junction, in other words,the model proposed by Beenakker is obtained.We also considered the shot noise properties of the conductance through the G/SG/Gstructure. Numerical results of the shot noise and the Fano factor are provided in Fig.5. The parameters of panels (a) and (c) are the same with Fig. 4 and teh parameters of(b) and (b) are the same with Fig. 3. We can see from the curves that the shot noiseoscillates greatly in the region of eV ≥ ∆ , while it increases monotonically in the region eV < ∆ . These behaviors are similar to the tunneling conductance. In tunneling throughthe G/SG/G-structure, transmission is enhanced by the active hole channels. As a result,the values of the shot noise are small in comparison with the Poisson value, which is 2 eG corresponding to the uncorrelated transport. When the hole channels in addition to theelectron channels contribute to the transport, the interference effect is strong and the shotnoise is significantly suppressed. In the region of eV < ∆ , the impact of the proximityeffect in the SG-region is strong giving rise to strong conductance and the values of S and F approach 0. In the case of eV ≥ ∆ , the amplitude of oscillation decreases with the increaseof eV ; in the case of eV < ∆ , the amplitude of oscillation increases with the increase of eV , which originates from the same reason as the conductance. IV. CONCLUSIONS
Based on the Dirac–Bogoliubov–de Gennes equation and the scattering theory, we inves-tigated the transport properties of the relativistic electrons and holes through the G/SG/Gjunction. We have deduced the analytical formulas of the tunneling conductance and theshot noise. Numerical results of the tunneling conductance and shot noise in the systemare provided. We compared the two cases, one of which is in the presence of the specu-lar Andreev reflection and the other of which is in the absence of the specular Andreevreflection. The physical results can be summarized as follows. Firstly, the conductanceincreases with the increase of the thickness of the SG-layer and the shot noise is suppressedby the conductance in the case of eV = E F . Secondly, the potential strength significantlyaffects the transport properties. It suppresses the conductance and enhances the shot noise.Thirdly, we obtained similar results with the model of the G/SG junction in the condition9 V ≤ ∆ . In the limit of a → ∞ , the results of Beenakker for a G/SG junction can bereproduced. Fourthly, the thickness of the SG-layer affects the conductance and the shotnoise more prominently in the condition eV ≥ ∆ , causing the decrease of the oscillationamplitudes and the characteristic features of the specular Andreev reflection. In conclusion,the conductance is a combined result of the Andreev reflection and the specular Andreevreflection, which can be tuned by the system parameters; the shot noise is suppressed bythe SG-barrier because of the contribution of the hole channels in addition to the electronchannels. V. ACKNOWLEDGEMENTS
This project was supported by the National Natural Science Foundation of China (No.11004063), and the Fundamental Research Funds for the Central Universities, SCUT (No.2014ZG0044). 10
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