Enhancements of Andreev conductance induced by the photon/vibron scattering
aa r X i v : . [ c ond - m a t . s up r- c on ] M a r Enhancements of Andreev conductance induced by the photon/vibron scattering
J. Bara´nski , and T. Doma´nski Institute of Physics, M. Curie-Sk lodowska University, 20-031 Lublin, Poland Institute of Physics, Polish Academy of Sciences, 02-668 Warsaw, Poland (Dated: September 6, 2018)We analyze the subgap spectrum and transport properties of the quantum dot embedded betweenone superconducting and another metallic reservoirs and additionally coupled to an external bosonmode. Emission/absorption of the bosonic quanta induces a series of the subgap Andreev states,that eventually interfere with each other. We discuss their signatures in the differential conductanceboth, for the linear and nonlinear regimes.
PACS numbers: 73.63.Kv;73.23.Hk;74.45.+c;74.50.+r
I. INTRODUCTION
The bosonic modes, like photons [1] or vibrational de-grees of freedom [2], can strongly affect electron tunnelingthrough the nanoscopic systems [3]. When a level spacingof nanoobject is large in comparison to the boson energy ω and a line-broadening is sufficiently narrow, a series ofthe side-peaks [4] may appear due to emission/absorptionof the bosonic quanta. Such features (spaced by ω ) havebeen really observed in measurements of the differentialconductance for several nanojunctions [5–8].Similar bosonic modes are currently studied also in thesystems, where the quantum dots/impurities are coupledwith superconducting reservoirs [9–20]. Since the prox-imity effect spreads electron pairing onto these quantumdots, the bosonic features manifest themselves in a quitepeculiar way. They could be observed by the Josephson[9–11] and the Andreev spectroscopies [12–17], in photon-assisted subgap tunneling [18], transient phenomena [19],or in prototypes of the nano-refrigerators operating dueto the multi-phonon Andreev scattering [20].First of all, in a subgap regime (assuming ω smallerthan energy gap ∆ of superconductor) the bosonic fea-tures are expected to be more numerous than in the nor-mal state. This is a consequence the proximity effect,mixing the particle and hole excitations. Secondly, it hasbeen shown numerically [12, 16, 20] that the linear (zero-bias) Andreev conductance exhibits the bosonic featuresspaced by a half of ω . To our knowledge, this intrigu-ing theoretical result was neither clarified on physical ar-guments nor checked experimentally. Verification wouldbe feasible by the tunneling spectroscopy using e.g. low-frequency vibrations of some heavy molecules or slowly-varying ac electromagnetic field. Let us emphasize, thatsuch low-energy boson mode need not be related withany pairing mechanism of the superconducting reservoir.The purpose of our paper is to provide a simple ana-lytical argument, explaining the reduced frequency ω / FIG. 1: (color online) A scheme of the quantum dot betweenthe metallic (N) and superconducting (S) electrodes and cou-pled to the monochromatic boson (phonon or photon) mode. (low-temperature) differential conductance as a functionof the source-drain bias. We predict that the multipleAndreev states could be seen with a period, dependenton the gate voltage.For calculations we consider the setup displayed in fig-ure 1. It can be practically realized in a single elec-tron transistor (SET) using e.g. the carbon nanotubesuspended between the external electrodes (like in Refs[5, 6]). Another possibility could be the scanning tun-neling microscope (STM), where the conducting tip (N)probes some vibrating quantum impurity (QD) hosted ina superconducting (S) substrate [21]. In both SET andSTM configurations such boson mode can be eventuallyrelated to external ac field.In what follows we introduce the Hamiltonian and dis-cuss the method for treating the bosonic mode. We nextinvestigate the bosonic signatures in the QD spectrumand in the subgap Andreev conductance. For clarity, wefocus on the limit Γ N ≪ ω whereas the second couplingΓ S can be arbitrary. In the last section we address thecorrelation effects. II. MICROSCOPIC MODEL
For microscopic description of the tunneling schemeshown in Fig. 1 we use the Anderson impurity modelˆ H = ˆ H N + ˆ H S + ˆ H mol + ˆ H T . (1)ˆ H N ( S ) refers to the normal (superconducting) lead, ˆ H mol describes the molecular quantum dot (i.e. the local-ized electrons coupled with the boson mode) and ˆ H T is a hybridization between the QD and itinerant elec-trons. We treat the normal electrode as a free Fermigas ˆ H N = P k ,σ ξ k N ˆ c † k σN ˆ c k σN and describe the othersuperconducting lead by the BCS Hamiltonian ˆ H S = P k ,σ ξ k S ˆ c † k σS ˆ c k σS − ∆ P k (ˆ c † k ↑ S ˆ c †− k ↓ S + ˆ c − k ↓ S ˆ c k ↑ S ). Theannihilation (creation) operators ˆ c ( † ) k σβ correspond to mo-bile β = N, S electrons with spin σ = ↑ , ↓ and energy ξ k β = ε k β − µ β measured with respect to the chemicalpotential µ β . Nonequlibrium conditions can be drivenby the bias V = µ L − µ R and/or temperature difference T L = T R . The induced currents depend qualitatively onthe hybridization ˆ H T = P k ,σ,β (cid:16) V k β ˆ d † σ ˆ c k σβ + H.c. (cid:17) andon parameters of the molecular quantum dotˆ H mol = ε X σ ˆ n † dσ + U ˆ n d ↑ ˆ n d ↓ + ω ˆ a † ˆ a + λ X σ ˆ n dσ (ˆ a † + ˆ a ) . The number operator ˆ n dσ = ˆ d † σ ˆ d σ counts the localizedelectrons with spin σ , ε is the QD energy level and U de-notes the Coulomb potential between opposite spin elec-trons. The boson field (described by ˆ a ( † ) operators) isassumed as a monochromatic mode ω and its couplingwith the QD electrons is denoted by λ . III. MULTIPLE SUBGAP STATES
There are three main obstacles in determining the ef-fective energy spectrum and the tunneling transmissionof our system: i) the electron-boson coupling λ , ii) theproximity induced on-dot pairing (due to ∆), and iii) thecorrelation effects caused by the Coulomb repulsion U .The most reliable way for studying them on equal footingwould be possible within the numerical renormalizationgroup [15] approach, however such method encountersproblems in estimating the Andreev transmission. Toget some insight into the spectrum and transport prop-erties we start by neglecting the correlations and then(in the last section) treat them using the superconduct-ing atomic limit solution.Following [9–18, 20] we apply the unitary transforma-tion e ˆ S ˆ He − ˆ S = ˆ˜ H to decouple the electron from bosonquasiparticles. With the Lang-Firsov generating opera-tor [22] ˆ S = λω X σ ˆ n dσ (cid:0) ˆ a † − ˆ a (cid:1) (2) the molecular Hamiltonian (2) is transformed toˆ˜ H mol = X σ ˜ ε ˆ˜ d † σ ˆ˜ d σ + ˜ U ˆ˜ n ↓ ˆ˜ n ↑ + ω ˆ a † ˆ a, (3)where the energy level is lowered by the polaronic shift˜ ǫ = ε − λ /ω and the effective potential ˜ U = U − λ /ω .Boson operators are shifted ˆ˜ a ( † ) = ˆ a ( † ) − λω P σ ˆ d † σ ˆ d σ whereas fermions are dressed with the polaronic cloudˆ˜ d ( † ) σ = ˆ d ( † ) σ ˆ X ( † ) , ˆ X = e − ( λ/ω )(ˆ a † − ˆ a ) . (4)Reservoirs ˆ H β are invariant on the unitary transforma-tion (2) but the operator ˆ X appears in the hybridizationterm ˆ˜ H T . For simplicity we absorb it into the effectivecoupling constants Γ β = 2 π P k | V k β | h ˆ X † ˆ X i δ ( ω − ξ k β )which can be defined for the wide band limit.The effective single particle excitation spectrum isgiven by the Green’s function G σ ( τ , τ ) = − i D ˆ T τ ˆ d σ ( τ ) ˆ d † σ ( τ ) E ˆ H , (5)where ˆ T τ denotes the time ordering operator. Since traceis invariant on the unitary transformations h ... i ˆ H = h ... i ˆ˜ H it is convenient to compute the statistical averages withrespect to ˆ˜ H . In particular, (5) can be expressed as G σ ( τ , τ ) = − i D ˆ T τ ˆ d σ ( τ ) ˆ d † σ ( τ ) E ˆ˜ H fer D ˆ T τ ˆ X ( τ ) ˆ X † ( τ ) E ˆ˜ H bos (6)because the fermionic and bosonic degrees of freedomare separated by the Lang-Firsov transformation. Froma standard procedure [4, 23] one obtains D ˆ T τ ˆ X ( τ ) ˆ X ( τ ) † E ˆ˜ H bos = exp (cid:8) − ( λ/ω ) × (7)[(1 − e − iω ( τ − τ ) )(1 + N p ) + (1 − e iω ( τ − τ ) ) N p ] o with the Bose-Einstein distribution N p = (cid:2) e βω − (cid:3) − .Fourier transform of the Green’s function (7) is found as G σ ( ω ) = X l g σ ( ω − lω ) e − ( λ √ N p /ω ) (8) × e lβω / I l (cid:20) λω ) q N p (1 + N p ) (cid:21) , where I l denote the modified Bessel functions and g σ ( τ , τ ) = − i D ˆ T τ ˆ d σ ( τ ) ˆ d † σ ( τ ) E ˆ˜ H fer is the fermionicpart of (6). In the ground state (8) simplifies tolim T → G σ ( ω ) = X l g σ ( ω − lω ) e − g g l l ! (9)with the adiabatic parameter g = ( λ/ω ) .Due to the proximity induced on-dot pairing the singleparticle Green’s function G ↑ ( τ , τ ) is mixed with the(anomalous) propagator F ( τ , τ ) = − i D ˆ T τ ˆ d †↓ ( τ ) ˆ d †↑ ( τ ) E ˆ H = (10) − i D ˆ T τ ˆ d †↓ ( τ ) ˆ d †↑ ( τ ) E ˆ˜ H fer D ˆ T τ ˆ X † ( τ ) ˆ X † ( τ ) E ˆ˜ H bos . This important fact has been remarked in the previousconsiderations of dc Josephson current [11] and it alsoplays significant role for the Andreev spectroscopy (seethe next section). The boson part of the anomalous prop-agator (10) takes the following form D ˆ T τ ˆ X † ( τ ) ˆ X ( τ ) † E bos = exp (cid:8) − ( λ/ω ) × (11)[(1 + e − iω ( τ − τ ) )(1 + N p ) + (1 + e iω ( τ − τ ) ) N p ] o . At zero temperature its Fourier transform simplifies tolim T → F ( ω ) = X l f ( ω − lω ) e − g ( − g ) l l ! . (12)As regards the fermion part f ( τ , τ ) = − i h ˆ T τ ˆ d †↓ ( τ ) ˆ d †↑ ( τ ) i ˆ˜ H fer it couples to the Green’sfunction g ( τ , τ ). Their Fourier components obey theDyson equation (cid:20) g ( ω ) f ( ω ) f ⋆ ( − ω ) − g ⋆ ( − ω ) (cid:21) − (13)= (cid:20) ω − ˜ ε ω + ˜ ε (cid:21) − Σ QD ( ω ) − Σ corrQD ( ω ) , where Σ d is the selfenergy matrix of uncorrelated molec-ular dot and the second contribution Σ corrd is due to theeffective Coulomb interaction ˜ U . In the wide-band limitthe selfenergy Σ QD ( ω ) can be expressed as Σ QD ( ω ) = − i Γ N (cid:18) (cid:19) − Γ S γ ( ω ) (cid:18) ∆ ω ∆ ω (cid:19) (14)with γ ( ω ) = ( ω √ ∆ − ω for | ω | < ∆ , i | ω |√ ω − ∆ for | ω | > ∆ . (15)We investigated the effective spectral function ρ ( ω ) = − π − Im G ( ω + i + ) at zero temperature, focusing on theintermediate electron-boson coupling g ∼
1. Figures 2–4show the QD spectrum for ˜ U = 0, neglecting the correla-tion effects Σ corrd . Influence of the Coulomb potential ˜ U is discussed in section V.Fig 2 illustrates evolution of the bosonic features withrespect to the superconductor gap ∆. In the normal state(for ∆ = 0) such lorentzian peaks are located at ω = ˜ ε + lω (with integer l ≥
0) and their broadening is Γ N + Γ S .For finite ∆ all peaks split into the lower and upper ones -20 -10 0 10 20 30 40 0 2 4 6 8 10 0 0.05 0.1 0.15 ρ ( ω ) ω / Γ N ∆ / Γ N ρ ( ω ) FIG. 2: (color online) Energy spectrum ρ ( ω ) of the uncor-related quantum dot ( ˜ U = 0) obtained at T = 0 for ˜ ε = 0, g = 1, ω = 10Γ N . The filled triangles at lω ± ∆ are onlyguide to eye. For increasing ∆ the boson peaks split into thelower and upper states and their broadening shrinks to Γ N . -20 -10 0 10 20 30 40 0 2 4 6 8 0 0.1 0.2 0.3 ρ ( ω ) ω / Γ N ε / Γ N ρ ( ω ) FIG. 3: (color online) Spectrum of the uncorrelated quantumdot for ω = 10Γ N , g = 1, Γ S = 4Γ N , T = 0. The neighboringboson peaks are crossing at ω = ( + l ) ω for ˜ ε ≃ ω / due to the induced on-dot pairing. In the extreme limit∆ ≫ Γ S the selfenergy Σ QD ( ω ) becomes staticlim ∆ ≫ Γ S Σ QD ( ω ) = − (cid:18) i Γ N Γ S Γ S i Γ N (cid:19) (16)therefore the effective quasiparticle energies evolve to lω ± p ˜ ε + (Γ S / and their broadening shrinks to Γ N .Focusing on such superconducting atomic limit (16)we show in Fig. 4 the subgap bosonic peaks with respectto ˜ ε . In the SET configuration the energy level ˜ ε wouldbe tunable by applying the gate voltage. In particular,these peaks may overlap with each other when ˜ ε ≃ ω / lω + p ˜ ε + (Γ S / = l ′ ω − p ˜ ε + (Γ S / . (17) -20 -10 0 10 20 30 40 0 2 4 6 8 10 12 14 0 0.1 0.2 0.3 ρ ( ω ) ω / Γ N Γ S / Γ N ρ ( ω ) FIG. 4: (color online) Spectrum of the uncorrelated quantumdot for ˜ ε = 0, g = 1, ω = 10Γ N , T = 0, ∆ ≫ Γ S . Thebosonic features cross each other at ω = ( + l ) ω for Γ S = ω . The neighboring peaks ( l ′ = l + 1) overlap when ω / p ˜ ε + (Γ S / . For small Γ S such situation takes placeat ˜ ε ≃ ω . Other crossings would be eventually possiblefor the higher-order multiplications of ω / ρ ( ω ) as a func-tion of the coupling Γ S . From (17) we conclude that for˜ ε = 0 the bosonic peaks overlap at Γ S = ω . Energy ofthese crossing points is ω = ( + l ) ω . Here (for g = 1)we observe four such crossings, but for stronger electron-boson couplings a number of the in-gap states and theircrossings would increase. IV. ANDREEV CONDUCTANCE
Under nonequilibrium conditions the charge currentcan be transmitted at small voltage | eV | < ∆ via the An-dreev scattering, engaging the in-gap states. This anoma-lous transport channel occurs when electrons from themetallic lead are converted into the Cooper pairs (prop-agating in superconducting electrode) with the holes re-flected back to N electrode. The resulting current I A ( V )can be expressed by the Landauer-type formula [24] I A ( V ) = 2 eh Z dω T A ( ω ) [ f F D ( ω − eV ) − f F D ( ω + eV )] , (18)with the Fermi-Dirac function f F D ( ω ) = (cid:2) e ω/k B T + 1 (cid:3) − and the Andreev transmittance [24] T A ( ω ) = Γ N | F ( ω ) | . (19)Optimal conditions for this subgap transmittance occurwhen ω coincides with the subgap quasiparticle states.In our present case we thus expect a number of suchenhancements due the bosonic features. Let’s remarkthat T A ( − ω ) = T A ( ω ) implies the Andreev conductance G A ( V ) = ∂I A ( V ) ∂V to be an even function of the bias V . -40 -20 0 20 40 0 1 2 3 4 0 0.1 0.2 0.3G(V) [4e /h] eV/ Γ N ε / Γ N FIG. 5: (color online) The differential Andreev conductance G A ( V ) versus the source-drain voltage V and the QD level ε (tunable by the gate voltage). Results are obtained for T = 0, g = 1, Γ S / Γ N =6, ω / Γ N = 10, ˜ U = 0 and ∆ ≫ Γ S .Conductance is expressed in units of 4 e /h . Fig. 5 shows the Andreev conductance as a functionof voltage V applied between the metallic and super-conducting electrodes. We notice the differential con-ductance enhancements whenever V coincides with thein-gap quasiparticle energies. Since T A ( − ω ) = T A ( ω ) weobserve these maxima at ± p ˜ ε + (Γ S / ± lω . Theyeventually overlap when (17) is satisfied. In particular,for Γ S = 6Γ N and ω = 10Γ N the nearest bosonic peaksoverlap when ˜ ε = 4Γ N . Figure 5 clearly shows that theresulting maxima appear at | eV | = ω (1 / l ). V. CORRELATION EFFECTS
In various experimental realizations of the quantumdots (such as self-assembled InAs islands [25], carbonnanotubes [26, 27] or semiconducting nanowires [28, 29])attached to the superconducting leads the energy gap ∆was safely smaller than the repulsion potential U . Forthis reason, in the subgap Andreev spectroscopy the cor-relations hardly contributed any Coulomb blockade. In-stead of it, they can eventually induce the singlet-doubletquantum phase transition [35] and/or the Kondo physics[34]. In this paper we consider the strongly asymmet-ric coupling Γ N ≪ Γ S and focus on the deep subgapregime Γ N,S ≪ ∆, therefore the Kondo-type effects [30–34] would be rather negligible.Analysis of such singlet-doublet transition for the vi-brating quantum dot has been previously addressed [15]using the NRG technique. We revisit the same issuehere, determining the differential Andreev conductance(unavailable for the NRG calculations [15]), because thisquantity could be of interest for experimentalists. Forthe sake of simplicity, we analyze the correlation effects Γ N eV / Γ N G A (V) [4e /h] 0 10 20 30 40 -10 0 10 FIG. 6: (color online) The differential Andreev conductance G A ( V ) versus the Coulomb potential U and bias V obtainedin the superconducting atomic limit ∆ ≫ Γ S for T = 0,Γ S / Γ N = 20 in absence of the boson mode g = 0. The thick(red) line indicates the QPT at U = Γ S . in the superconducting atomic limit ∆ ≫ Γ S . Hamilto-nian of the molecular quantum dot (3) can be addition-ally updated with the pairing terms Γ S (cid:16) ˆ d †↑ ˆ d †↓ + ˆ d ↑ ˆ d ↓ (cid:17) originating from the static off-diagonal parts of the self-energy matrix (16).In absence of the boson field (i.e for λ = 0) the exactsolution of such problem has been discussed by a numberof authors (e.g. see the references cited in [36]). The ef-fective quasiparticle energies are given by ± U/ ± E d ,where E d = p ( ε + U/ + (Γ S / . In the realis-tic situations only two branches ± ( U/ − E d ) appearin the subgap regime, whereas the other high energystates ± ( U/ E d ) overlap with a continuum beyondthe gap. The quantum phase transition (QPT) from thesinglet u | i + v |↑↓i to doublet | σ i configuration occursat U/ E d [35]. In order to estimate quantitativelythe Andreev conductance we use the off-diagonal Green’sfunction f ( ω ) [35, 36], restricting to its subgap part f sub ( ω ) ≃ α uvω + i Γ N − (cid:0) U − E d (cid:1) − α uvω + i Γ N + (cid:0) U − E d (cid:1) (20)with the usual BCS coefficient uv = Γ S / E d andthe spectral weight α = h exp (cid:16) U k B T (cid:17) + exp (cid:16) E d k B T (cid:17)i / Z ,where Z = 2 exp (cid:16) U k B T (cid:17) +exp (cid:16) − E d k B T (cid:17) +exp (cid:16) E d k B T (cid:17) . Themissing part of spectral weight 1 − α belongs to the high-energy states (outside the gap). At zero temperature thissubgap weight changes abruptly from α = 1 (in the sin-glet state when U/ < E d ) to α = 0 . U/ > E d ).In figure 6 we plot the Andreev conductance obtainedfor the half-filled quantum dot ε = − U/ U = Γ S ). We notice the subgap conductance en-hancements around | eV | = U/ − E d . Yet, exactly at theQPT, both the singlet and doublet contributions canceleach other. Formally, this is due to the odd (asymmetric)structure of the Green’s function (20). Γ N eV / Γ N ~ G A (V) [4e /h] 0 10 20 30 40 -10 0 10 FIG. 7: (color online) The subgap Andreev conductance G A ( V ) as a function of the Coulomb potential ˜ U and volt-age V obtained for g = 1, ω / Γ N = 10 and the same modelparameters as in figure 6. The superconducting atomic limit solution can be gen-eralized onto g = 0 case in a straightforward way. Theunitary transformation (2) implies ε → ˜ ε , U → ˜ U andfollowing the steps (10-16) we can determine the off-diagonal Green’s function. At zero temperature, we find F sub ( ω ) ≃ α uv ∞ X l =0 e − g ( − g ) l /l ! ω + i Γ N − (cid:16) ˜ U − E d (cid:17) + s lω − e − g ( − g ) l /l ! ω + i Γ N + (cid:16) ˜ U − E d (cid:17) − s lω (21)with s ≡ sign( ˜ U − E d ).Figure 7 shows the Andreev conductance obtained forthe half-filled quantum dot using g = 1, ω / Γ N = 10,Γ S / Γ N = 20, T = 0. The bosonic side-peaks give rise toadditional subgap branches, similar to what has been re-ported for the spectral function [15]. Right at the QPT,the zero-bias conductance again vanishes G (0) → | eV | = lω (with l ≥ | eV | = | ˜ U / − E d | + lω whose spectral weights depend on ˜ U and l . VI. SUMMARY
We have investigated the subgap spectrum and trans-port properties of the quantum dot coupled between themetallic and superconducting electrodes in presence ofthe external boson mode ω . We have found that the in-duced Andreev states eventually cross each upon varyingthe gate potential (through ε ) or due to the correlations(via quantum phase transition from the singlet to doubletconfigurations). We have explored their signatures in themeasurable charge transport. The tunneling conductanceof such multilevel ’molecule’ shows a series of character-istic enhancements, dependent on: the gate voltage withfrequency ω / V applied between external leads (Fig. 5), andthe correlations (Fig. 7). External boson reservoir canthus substantially affect the anomalous Andreev currentand it can be probed experimentally using the low-energyvibrational modes or the slowly-varying ac fields [1]. Acknowledgment
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