Electron transport through a strongly interacting quantum dot coupled to a normal metal and BCS superconductor

Transcription

1 INSTITUTE OFPHYSICS PUBLISHING Supercond. Sci. Technol. 7 (4 3 SUPERCONDUCTORSCIENCE AND TECHNOLOGY PII: S953-48( Electron transport through a strongly interacting quantum dot coupled to a normal metal and BCS superconductor Mariusz Krawiec and Karol I Wysokiński H H Wills Physics Laboratory, University of Bristol, Tyndall Ave, Bristol BS8 TL, UK Institute of Physics, M Curie-Skłodowska University, ul. Radziszewskiego a, -3 Lublin, Poland Received 5 January 3, in final form July 3 Published 5 November 3 Online at stacks.iop.org/sust/7/3 (DOI:.88/953-48/7//8 Abstract We study electron transport through a quantum dot coupled to a normal metal and BCS-like superconductor (N QD S in the presence of the Kondo effect and Andreev scattering. The system is described by the single impurity Anderson model in the limit of strong on-dot interaction. To study electron transport, we use the recently proposed equation of motion technique for the Keldysh non-equilibrium Green s function together with the modified slave boson approach. We derive the formula for the current which contains various tunnelling processes and we apply it to a study of the transport through the system. We find that the Andreev conductance is strongly suppressed and there is no zero-bias (Kondo anomaly in the differential conductance. We discuss the effects of the particle hole asymmetry in the electrodes as well as the asymmetry in the couplings.. Introduction In recent years there has been much experimental and theoretical work on electron transport through nanometresize areas (metallic or semiconducting islands containing a small number of electrons. These islands (sometimes called quantum dots (QDs are coupled via tunnel barriers to several external electrodes, making it possible to adjust the current flowing through the system []. The devices make it possible to study several well-known quantum phenomena in a novel and highly controllable way. For instance, it is well known, that a QD behaves like a magnetic impurity in a metallic host and, in particular, displays the Kondo effect [ 5]. The Kondo effect is a manifestation of the quantum many-body singlet state formed by the impurity spin and conduction electron spins. This state gives rise to a quasi-particle peak at the Fermi energy in the dot spectral function [6 9] and zero-bias maximum in the differential conductance observed experimentally [ 4]. Even the single ion Kondo effect in an equilibrium situation presents a challenge for a theory. This has been extensively studied in the literature and is summarized in a number of papers [5 8]. The various extensions to exotic (multichannel Kondo effects have also been discussed [9]. Another example of the quantum phenomenon is the Andreev scattering [], according to which an electron impinging on a normal metal superconductor interface is reflected back as a hole and the Cooper pair is created in the superconductor. This effect has been shown to play a crucial role in the transport properties of various hybrid mesoscopic superconducting devices []. There are a number of papers in the literature concerning the electron transport in various realizations of such devices. Here we are interested in the study of the normal-state QD coupled to one normal and one superconducting electrode (N QD S. Such a system has been studied within the scattering matrix technique [, 3]. However, this approach is valid only for non-interacting systems and cannot take into account the effects of Coulomb interactions between electrons on the dot, which are very important in these small systems, as they lead, for example, to the Coulomb blockade phenomenon [4] or the Kondo effect [ 5]. Transport through the non-interacting QD has also been studied within the non-equilibrium Green s function (GF technique. The effect of multiple discrete levels of the dot was discussed in [5, 6], the photon-assistant transport in [7], electron transport in the weak magnetic field in [8], temperature dependence of the resonant Andreev /4/3$3. 4 IOP Publishing Ltd Printed in the UK 3

2 M Krawiec and K I Wysokiński reflections in [9] and transport in a three-terminal system (one superconducting and two ferromagnetic electrodes in [3]. In the presence of strong Coulomb interaction in the N QD N device the Kondo effect appears and influences the electron transport in the system. If one of the electrodes is superconducting, both the single electron and the Andreev current are affected by the Abrikosov Suhl resonance. This problem has been extensively studied within various techniques [6, 3 38] and there is no consensus. Some authors have predicted suppression [6, 3, 35, 36] of the conductance due to Andreev reflections while others have predicted enhancement [3, 34]. Recently, it has been shown [37] that we can obtain either suppression or enhancement of the conductance in dependence on the values of the model parameters. Other effects have also been revealed, such as the emergence of the Kondo-like peaks in the local density of states (DOS at energies equal to ± ( is the superconducting order parameter [35, 37] and the novel co-tunnelling process (Andreev normal co-tunnelling [38]. This process involves Andreev tunnelling from the QD S interface and normal tunnelling from the N QD interface. As a result, the Cooper pair directly participates in the formation of the spin singlet (Kondo effect and leads to the emergence of the additional Kondo resonances in the local DOS and enhancement of the tunnelling current. The purpose of the present work is to apply the new equation of motion technique (EOM to derive the formula for the current through the QD (in the limit of strong on-dot Coulomb interaction in terms of various tunnelling processes. The technique, which is explained in section, has previously been applied to study the current across the QD asymmetrically coupled to two normal leads [39]. Like the standard EOM [6] it gives narrower and smaller Kondo peaks than the non-crossing approximation (NCA [39]. However, positions of the Kondo peaks calculated within these approaches coincide. The standard EOM technique has been shown to provide correct qualitative physics of the Kondo effect at low temperature. In the present work we also study the interplay between the Kondo effect and Andreev reflections to give additional insight into the the problem of the suppression/enhancement of the zero-bias current voltage anomaly. Furthermore, we discuss the question of participation of the superconducting electrons in the creation of the Kondo effect. Finally, we investigate the influence of the electron hole asymmetry in the leads on tunnelling transport as well as the asymmetry in the couplings to the leads. The paper is organized as follows. In section we present the model under consideration and derive the formula for the current using the EOM for the non-equilibrium GF. In section 3 we apply the obtained formula for the current to the numerical study of the transport through the N QD S system. Section 4 is devoted to the interplay between the Kondo and Andreev scattering. In section 5 we discuss the QD asymmetrically coupled to the leads, while the effect of electron hole asymmetry in the leads is investigated in section 6. Some conclusions are given in section 7.. Model and formulation The Anderson Hamiltonian of the single impurity [4], in the Nambu representation, can be written in the form H = λkσ λkσ H λk λkσ σ σ H QD σ λkσ λkσ H I λk σ where the Nambu spinors λkσ and σ are defined as ( ( cλkσ dσ λkσ = ; σ = and c λ k σ H λk = ( ɛλk λk λk ɛ λk d σ denotes the Hamiltonian of the normal ( Nk = or superconducting ( Sk lead ( H QD Ed U d n σ = (4 (E d U d n σ is the dot Hamiltonian and ( Hλk I = Vλk (5 V λk is the dot electrode hybridization. Here λ = N and S denoting the normal metal (N or superconducting (S lead in the system. The symbols have the following meanings: c λkσ (c λkσ denotes the creation (annihilation operator for a conduction electron with the wave vector k, spin σ in the lead λ, λk is the superconducting order parameter in the lead λ ( Sk = S, Nk =, andv λk is the hybridization matrix element between conduction electron of energy ɛ λk in the lead λ and localized electron on the dot with the energy E d. d σ (d σ is the creation (annihilation operator for an electron on the dot and U d is the on-dot Coulomb repulsion. To derive the formula for the average current in the system we start from the time derivative of the charge (for convenience, we perform calculations in the normal electrode [4] ( ( (3 J = e d dt N N = iē h [N N,H], (6 where N N = kσ c Nkσ c kσ is the total electron number operator in the lead N and e is the elementary charge. The above formula can be written in terms of the GFs J = e h kσ d π [ H I Nk G < Nkσ,d (], (7 where G < Nk,dσ ( is the Fourier transform of the Keldysh matrix GF [4] G < Nk,dσ (t = i σ ( Nkσ (t. Now we have to calculate the GF G < Nk,dσ (. We can do this in the usual way, i.e. using the Keldysh equation [4, 4] G < ( = (G r ( r (G < (( a (G a ( G r ( < (G a ( (8 (superscripts r and a are for retarded and advanced GFs, respectively and make more or less justified approximations for the lesser self-energy < (. Usually, we use the approximation due to Ng [43] which states that full 4

3 Electron transport through a strongly interacting QD lesser self-energy is proportional to the non-interacting one ( < ( < (. This approximation is widely used in the literature [3, 33]. However, we wish to use another approach based on the recently proposed EOM technique for non-equilibrium GFs [44]. The usual EOM derived from the Heisenberg equation yields undefined singularities, which depend on the initial conditions. The advantage of this new technique, based on Schwinger Keldysh perturbation formalism, is that it explicitly determines these singular terms. Moreover, together with the EOM for retarded (advanced GFs it allows us to treat the problem in a very consistent way making similar approximations in the decoupling procedure for all types of GFs. Such an approach was recently proposed to calculate the charge on the QD upon non-equilibrium conditions [45]. According to [44] the equation for the lesser GFs is A B < = g< ( [A, B] ± g r ( [A, H I ] B < g < ( [A, H I ] B a (9 where g r(< ( is the free-electron GF and H I denotes the interacting part of the Hamiltonian. In general, this equation allows us to calculate the GF on the left-hand side. However, in practice we have to approximate the higher-order GFs appearing on the right-hand side (as in the EOM for equilibrium GFs. But, by performing analogous approximations in the decoupling procedure for both retarded and lesser GFs, we make the theory consistent. Applying the above equation for the GF occurring in the formula for the current (7 G < Nkσ,d (, we obtain the following expression Nkσ < σ = gr Nk (H Nk I σ < σ g Nk < (H Nk I σ a σ ( where as an interacting part of Hamiltonian H I we have taken the third term in equation (. g r(< Nk ( is the retarded ( lesser free-electron matrix GF of the normal-state electrode. ( gnk r ( = ɛ Nki ( ɛ Nki g Nk < ( ( πif( ev δ( ɛnk = πif( ev δ( ɛ Nk ( where f( is the Fermi distribution function and ev = µ N µ S corresponds to the applied voltage between the normal-state electrode with the chemical potential µ N and the superconducting electrode with µ S. In the following, we fix the chemical potential of the superconducting electrode (µ S = and use ev as a measure of the bias voltage. Expression ( is general for the Anderson model and does not depend explicitly on the form of the Hamiltonian describing the QD (H QD. The dependence enters only through the GF G <(a dσ ( = σ <(a σ. In the following we wish to study the QD in the limit of strong on-dot Coulomb repulsion (U d. In this limit the double occupancy of the dot is forbidden and it is convenient to work in the slave boson representation in which the real electron operator is replaced by the product of fermion and boson operators (d σ b f σ [46, 47]. Additionally, the fact that there is no double occupancy on the dot should be taken into account in some way. Usually such a constraint is added to the Hamiltonian by the Lagrange multiplier. There are a number of variants of this approach in the literature and here we work within the Le Guillou Ragoucy scheme [48, 49]. In this approach the constraint of no double occupancy is enforced through the modification of the commutation relations of both fermion and boson operators in comparison to the standard ones. This approach was successfully used in the study of the charge on the QD [45]. The Hamiltonian of the system in the limit U d in the slave boson representation is given in the form (, but now with ( b f σ σ = f σ b (3 and ( H QD Ed =. (4 E d Having introduced the slave boson representation, we can begin calculations of the advanced and lesser on-dot GFs appearing in equation (. To do this, we apply formula (9 together with the usual prescription for the advanced (retarded GF. The higher-order GFs appearing on the right-hand side of the equation for lesser and retarded (advanced GFs have the same structure. So the idea is to make equivalent approximations in the decoupling procedure. Explicitly, we have performed decoupling c λkσ c λ k σ A B δ λλ δ kk n λk A B (5 and neglected the other GFs. In the above formula n λk is the concentration of the electrons in the lead λ in state k and for the superconducting electrode is given by n Sk = [ ɛ ] Sk ( f(e Sk E Sk (6 with quasi-particle spectrum ESk = ɛ Sk S. However, for the normal lead this relation reduces to n Nk = f(ɛ Nk ev. (7 We want to stress here that the factorization such as c λkσ c λ k σ A B δ λλ δ k k c λkσ c λ kσ A B (8 is not allowed at the level of our approximation. This is because of the requirement of the hermiticity relation between retarded and advanced GFs, i.e. G r ( = [G a (]. The resulting advanced on-dot GF G a dσ ( can be written in the form of the Dyson equation G a dσ ( = ga dσ ( ga dσ ( a dσ (Ga dσ ( (9 where gdσ a ( is the non-perturbed QD advanced GF ( n σ gdσ a ( = E d i n σ E d i ( and self-energy dσ a ( can be written as the sum of the noninteracting d a ( and interacting Ia d ( part d a ( = a d ( Ia d ( = [ a λk ( Ia λk (]. ( λk 5

4 M Krawiec and K I Wysokiński For a superconducting electrode we have a Sk ( = V Sk ( n σ V Sk ( n σ ( u Sk E Sk i vsk E Sk i ( usk v Sk E Sk i u Skv Sk E Sk i ( V Sk usk v Sk ( n σ E Sk i u Skv Sk E Sk i V Sk ( n σ ( v Sk E Sk i u Sk E Sk i ( where we have introduced BCS factors u Sk = ( ɛ Sk E Sk, vsk = ( ɛ Sk E Sk. For the normal state, the corresponding expression is V Nk Nk a ( = ( n σ ɛ Nk i. (3 V Nk ( n σ ɛ Nk i It turns out that within the present approach the interacting part of the self-energy is simply related to the non-interacting part. Moreover, the same relation also holds for retarded as well as lesser GFs. This is a result of the consistency of the decoupling procedure and the requirement of the hermiticity relation between retarded and advanced GFs. In general, this relation can be written as λk I ( = n λkτ 3 λk (τ 3 (4 where τ 3 = ( is the Pauli matrix and nλk is the concentration of the electrons of the wave vector k in the lead λ given by equations (6 and (7. It is possible to write the equation for the lesser GF in the form of the Keldysh equation (8 with G a dσ ( given by equation (9 andg r = [G a ]. The free-electron QD lesser GF is given in the form ( πi( g dσ < ( = n σ f (δ d πi( n σ f (δ d (5 where δ ± d = δ( ± E d. As we have mentioned, the lesser self-energy has the same form as the advanced self-energy d < ( < λk ( λk I< ( ( = < d ( I< d ( = λ (6 where the non-interacting part due to the superconducting lead is Sk < ( = πif( V Sk ( ( n σ u Sk δ S v Sk S δ V Sk ( n σ u ( Skv Sk δ S δ S V Sk ( n σ u ( Skv Sk δ S δ V Sk ( S ( n σ v Sk δ S. u Sk S δ (7 For the normal lead we have Nk < ( V Nk = πi ( n σ f( ev δ N V Nk ( n σ f( ev δ N (8 where δ ± S = δ( ± E Sk and δ ± N = δ( ± ɛ Nk. Again, the interacting part of the lesser self energy is related to the non-interacting part simply through equation (4. Now we are ready to write the expression for the current (7 in terms of known GFs. First, let us rewrite the Keldysh equation for the element of the QD GF in the form G < = Gr < Ga Gr < Ga Gr < Ga Gr < Ga. (9 Note that there is no term proportional to g d < as it vanishes in our case [4]. To calculate the GF given by equation (, entering the expression for the current (7, we need the element of the advanced GF, more precisely the imaginary part of that. Note that g Nk < is purely imaginary and we need the real part of g Nk < Ga d. We can write the equation for the imaginary part of the element of G a d in a similar fashion as equation (9, i.e. Im G a = Gr Im a Ga Gr Im a Ga G r Im a Ga Gr Im a Ga. (3 By substituting equations (9 and (3 into equation ( we obtain the expression for G Nk,d, which determines the current (7. Finally the current (7 can be written as J = J J J J A. (3 The first term represents conventional tunnelling and is given in the form J = e h d π Im S G Ɣ N ρ N [f( ev f(] (3 where the elastic rate is defined as Ɣ N = πvn ρn ( and ρ N ( is the bare normal-state DOS at the Fermi energy. The second term describes the branch crossing process (the process with crossing through the Fermi surface in the language of the Blonder Tinkham Klapwijk (BTK [5]. An electron from the normal lead is converted into hole-like excitation in the superconducting lead: J = e h d π Im S G Ɣ N ρ N [f( ev f(]. (33 The next term corresponds to the process in which the electron tunnels into the superconductor picking up the quasi-particle and creating a Cooper pair: J = 4e h d π Im S Re[G G ] Ɣ N ρ N [f( ev f(]. (34 The last term in equation (3 represents Andreev tunnelling in which an electron from the normal lead is reflected back as a hole and a Cooper pair is created in the superconducting electrode: J A = e h d π Im N G Ɣ N ρ N [f( ev f( ev ]. (35 As we can see, at energies ev < S and zero temperature, the only process which contribute to the total current is the Andreev tunnelling. The remaining ones represent single particle processes which are suppressed 6

5 Electron transport through a strongly interacting QD at ev < S due to the lack of the states in the superconductor. Of course, for energies ev > S all these processes give rise to the current. Even Andreev scattering contributes but is strongly suppressed. It is also worth noting that all these processes (except J proceed through virtual states on the dot. 3. Density of states In the following sections we present the numerical results of electron tunnelling in the N QD S system and show how different terms of equation (3 contribute to the total current and differential conductance. First, we turn our attention to the DOS as it gives valuable information about the system. The most pronounced fingerprint of the Kondo effect in the N QD N system is the Abrikosov Suhl or Kondo resonance at the Fermi level and its temperature dependence. Kondo resonance appears for temperatures below the parameterdependent Kondo temperature T K. In the original Kondo effect, there is an odd number of electrons on the dot, so the total spin is a half-integer. In this case electrons from the leads with energy close to the Fermi level screen the spin on the dot producing resonance at the Fermi energy. If electrodes are made superconducting, the situation is more complicated, as there enters another energy scale, the superconducting transition temperature T c (or equivalently the superconducting order parameter. Also, the Kondo effect takes place provided T K > T c, otherwise it is absent due to the lack of low-energy states in the leads to screen the spin on the dot. Naturally there arises a question as to what will happen if one of the leads is superconducting and another is in the normal state. This has been investigated in [3 33, 35] and it has been found that Kondo effect survives in the presence of superconductivity in one of the electrodes, even for T K <T c. The reason for this is simple. The spin of the dot is screened by electrons in the normal lead. We show that this is really the case and this is seen in the DOS of the QD. In figure we show the DOS of the QD for various positions of the dot energy level E d. It is clearly seen that the Kondo effect, which manifests itself in the resonance on the Fermi level, survives the presence of superconductivity in one electrode. The additional structure at =± coming from the superconducting lead is also visible. This is simply a reflection of the superconducting gap. At this point, it is worth noting that, if E d >, there is a bound-like (Andreev state within the superconducting gap, the position of which depends on E d. In the S QD S system this is a true bound state. However, in the present case, due to the finite DOS in the normal lead, this state acquires a finite width (resonance state. It is very interesting to see what the DOS looks like in the non-equilibrium situation (ev = µ N µ S. Let us recall that the Kondo resonance is located at the Fermi level of the lead. In the N QD N system when ev there emerge two resonances pinned to Fermi levels of the left and right leads, respectively. In the present case, there is a gap in the spectrum of the superconducting lead and we expect only one resonance pinned to the normal metal Fermi level (µ N. As we can see from figure this is really the case the Kondo resonance follows the Fermi level of the normal lead. ρ( E d = Figure. The DOS of the QD for various values of the dot energy level E d. Other parameters are: Ɣ N = Ɣ S =., =., ev =,T = 5 in units of the bandwidth W. ρ( ev = Figure. Equilibrium (ev = and non-equilibrium (ev = ±. DOS of the QD. Other parameters have the following values: Ɣ N = Ɣ S =., =.,E d =.8,T = 5 in units of the bandwidth W. ρ( 4 3 = Figure 3. Equilibrium DOS of the QD for various values of the order parameter (. E d =. and the other parameters have values as in figure. In figure 3 the DOS is plotted for a few values of the superconducting order parameter. As we can see, the Kondo resonance is strongly suppressed in comparison to 7

6 M Krawiec and K I Wysokiński ρ( T = T A NS ( 4e-4 3e-4 e-4 T A NS ( 6e-4 4e-4 e ev = e Figure 4. Temperature dependence of the DOS for Ɣ N =.5, Ɣ S =., =.5,E d =.,ev =. e e Figure 6. TNS A ( for different values of the bias voltage ev = (solid line,.3 (dashed line and.3 (dotted line. E d =.8,Ɣ N = Ɣ S =. and =.. The inset shows a large-scale view of the equilibrium TNS A (. peak height (a - Kondo e- (b Kondo - low-energy states are available and active in the Kondo screening. Secondly, the additional structure (peaks and dips appears at energies ±. We have found the saturation of the height of these peaks and dips below T K, and increasing at higher temperature. In [35] these structures have been interpreted as a result of the Kondo effect and are claimed to disappear at higher temperatures. Our calculations show the increase of their heights with temperature for T T K but T<T c. e- e T/ T K e- e- e T/T K 4. Andreev reflections and the Kondo effect Figure 5. Temperature dependence of the height of the peaks (a and dips (b near ± (see figure 3. For comparison the temperature dependence of the Kondo resonance is also shown. The parameters are the same as in figure 3. the N QD N DOS (solid line. The reason for this is that due to the lack of low-lying states in the superconductor, the spin on the dot is weakly screened. Similar conclusions have been reached by Clerk and et al [35] within the NCA approach. Finally, we discuss the temperature dependence of the dot DOS. In figure 4 we show the DOS for a number of temperatures. As expected, the Kondo resonance disappears as the temperature is raised. However, it is important to stress that resonances and dips at ± are also temperaturedependent. This is clearly seen in figure 5, where the heights of the peaks (a and dips (b are shown. The spectral weight of the Kondo resonance is also shown for comparison. It is worth noting that below T K the height of both peaks and dips is constant. As soon as the temperature exceeds T K the height of these resonances and the magnitude of dips start to increase. This effect can be thought of as a transfer of the spectral weight between Kondo resonance and Andreev states. Let us note that Andreev processes still take place at T K <T <T c. Our results agree with those obtained in [35] in a few ways. First, as the superconductivity is turned on, the Kondo peak gradually decreases. This is due to the fact that fewer We have shown that Kondo peak in the DOS survives the presence of the superconductivity. However, should we also expect the peak in the current voltage characteristic (differential conductance, G(eV = dj/d(ev? If we consider tunnelling processes, described by equations (3 (35, we might expect the Kondo peak only in the tunnelling mediated by the Andreev reflections (35. The amplitudes of the other processes are equal to zero (at T = for energies less than the superconducting gap. Let us rewrite equation (35 in the form J A = e h d π T A NS ([f( ev f( ev ]. (36 We have introduced transmittance TNS A (, associated with the Andreev tunnelling, defined as T A NS ( = ē h Ɣ N ρn (ρn ( G (. (37 In fact, at zero temperature and at energies less than the superconducting gap, TNS A ( can be regarded as a total transmittance, because Andreev tunnelling is the only process allowed in these circumstances. TNS A ( for different values of ev is plotted in figure 6. The broad resonances at ±.6 are reflections of the dot energy level E d =.8 for electrons and holes [3], shifted from its original position due to renormalization caused by the strong Coulomb interaction. 8

7 Electron transport through a strongly interacting QD.4.3 E d = e-3 8e-4 Γ N = Γ S Γ N > Γ S Γ N < Γ S G(eV SD. T A NS ( 6e-4 4e-4. e ev SD Figure 7. The Andreev differential conductance G A (ev SD = dj A /d(ev SD for different values of the dot energy level E d =.8 (solid line,.4 (dashed line and. (dotted line. Ɣ N = Ɣ S =. and =.. A more important point is that there is no Kondo peak in equilibrium (ev = transmittance. This is in agreement with [3, 35] and arises because the imaginary part of the anomalous GF G ( behaves like for while its real part is proportional to ln ( andbothvanishfor =. This is sufficient to suppress the Kondo effect. However, as soon as we go away from the ev =, we can observe the Kondo peaks at energies =±ev with approximately equal spectral weight. However, there is strong asymmetry between negative (dashed line and positive (dotted line voltages. While in the former case we have very well-resolved resonances, in the latter these resonances are strongly suppressed. This asymmetry is strictly related to the DOS (see figure, where we also observe such asymmetry, which is associated with different conditions for the Kondo effect in both cases (governed by the difference E d ev. The fact that we observe the Kondo peak for both electrons (ev and for holes ( ev is in contradiction to what has been observed in [3], where only a small kink has emerged for = ev. This is certainly due to the different approximation scheme used in calculations. Since equilibrium transmittance TNS A ( does not show the Kondo peak, we cannot expect it in the differential conductance G A (ev SD = dj A /d(ev SD with J A defined by equation (35, since G A is proportional to the equilibrium TNS A. Indeed, this is what we observe in figure 7, where it is also evident that G A (ev SD is very sensitive to the position of the dot energy level. The larger (negative E d, the smaller the conductance. This can be understood as follows. The probability of the Andreev reflections depends on the DOS (for < ofthe normal electrode as well as the dot itself. The latter is strongly E d -dependent (see figure. For E d well below there are no states on the dot participating in tunnelling between normal electrode and the superconductor. Note that, in fact, Andreev reflections take place between the superconducting electrode and the dot. The lack of the peak in the differential conductance confirms that the Kondo effect is suppressed in the N QD S system with strong on-dot Coulomb repulsion. This result is in full agreement with those of [3, 35, 37] Figure 8. The Andreev transmittance TNS A for different values of the couplings to the leads Ɣ N = Ɣ S =. (solid line, Ɣ N =.5, Ɣ S =. (dashed line and Ɣ N =.,Ɣ S =.5 (dotted line. E d =.8 and =.. 5. Asymmetric coupling The QD asymmetrically coupled to the normal-state electrodes shows the anomalous Kondo effect [39], which has also been observed experimentally [, 3]. This anomaly shows up as the non-zero position of the Kondo resonance in the differential conductance. In other words, if we increase one of the couplings to the leads (Ɣ L (R, the zero-bias anomaly moves to non-zero voltages [39]. In the N QD S system, there is no Kondo resonance in the differential conductance. Similarly it is absent in the equilibrium transmittance. On the other hand, the Kondo peak emerges when the system is in non-equilibrium. So, in fact, we could expect the non-zero bias Kondo peak in the differential conductance of the dot asymmetrically coupled to the leads. We have calculated Andreev transmittance and differential conductance for a number of couplings to the leads. An example is shown in figure 8. We have not observed any Kondo peak at non-zero voltages regardless of how big the asymmetry (Ɣ N /Ɣ S. The reason for this might be that for the N QD N system the shift of the Kondo peak to non-zero values is very small [39], and in the present case the Kondo resonance cannot develop because there are too few states in the transmittance spectra around the Fermi energy. However, the asymmetry in the couplings leads to another interesting behaviour of the Andreev transmittance as well as differential conductance. Namely, it turns out that Ɣ N has a greater influences on TNS A (G A around energies close to the value of the superconducting gap than Ɣ S does (see dashed line in figure 8. On the other hand, the transmittance (conductance around E d < is affected more by Ɣ S (compare the dotted line in figure 8. We can also note that the positions of the broad resonances, corresponding to the dot energy level E d, depend on asymmetry. This is due to the real part of self-energy. It is more important that Ɣ N and Ɣ S shift the positions of the resonances in opposite directions: Ɣ N towards Fermi energy µ N = µ S = while Ɣ S to higher energies. In the present approach, the Andreev transmittance (and differential conductance vanishes at zero energy. However, it 9

8 M Krawiec and K I Wysokiński e-3 e n N =., n S =..5,.5.5,..,.5 T A NS E d ρ( 5 4 e-7 3 -E d e-9 e- e- e e e3 Γ N /Γ S Figure 9. T A NS at energies: E d (solid line, E d (dashed line, (dotted line and (dot-dashed line as a function of Ɣ N /Ɣ S for E d =.8, =. and fixed Ɣ N Ɣ S = Figure. DOS in various realizations of the e h asymmetry indicated in the figure. The parameters have the following values: E d =.8, =.,Ɣ N = Ɣ S =.. shows interesting properties at the other characteristic energies of the system, such as E d or. The Andreev transmittance TNS A at these energies is shown in figure 9 as a function of the asymmetry in the coupling Ɣ N /Ɣ S. We see that the tunnelling due to the Andreev processes at energies ±E d is likely to take place when the couplings are more or less symmetric, i.e. Ɣ N = Ɣ S. A more surprising result is that the largest probability of these processes at energies ± is for large asymmetry Ɣ N /Ɣ S. As we have already mentioned, such behaviour can be explained by renormalization of the dot energy level due to the real part of self-energy, which depends on Ɣ N and Ɣ S in rather a complicated way. 6. Particle hole asymmetry peak height.6 N,S. N.8 S n N (n S Up to now, we have presented results for the special case, namely the electron hole (e h symmetry in the leads. It is well known [5] that the particle hole asymmetry in the normal metal/superconductor tunnel junctions and metallic contacts suppresses the Andreev reflections due to the fact that the reflection and transmission probabilities are different for incident electrons and holes. In the QD coupled to the normal and superconducting electrodes, we can also expect that this asymmetry will play a role. On the other hand, this asymmetry is already present in the strongly interacting QD (U = in the Kondo regime (E d <, as that studied here. However, the asymmetry in the leads can further modify the Andreev tunnelling. Let us start with the effect of the e h asymmetry on the DOS. As we can see from figure, the asymmetry plays rather a minor role. The most pronounced difference is when the concentrations in both normal and superconducting electrodes are changed radically. Otherwise the effect is small. It seems to be sensitive to average concentration in both electrodes. In figure we show the height of the Kondo resonance as a function of the electron concentration. As we can see in figure, the spectral weight of the Kondo peak strongly depends on the electron concentration in both electrodes. Moreover, there is a strong asymmetry with respect to the n = point, i.e. the peak is higher when the concentration of electrons is higher. This result is expected, as in the original Figure. The height of the Kondo peak as a function of the concentrations of the electrons n N (n S in the normal (superconducting lead. The curve marked N, S corresponds to situation when electron concentration in both electrodes is changed, N (S when it is changed in the normal (superconducting electrode only. The parameters are the same as in figure. Kondo effect the resonance at zero energy emerges due to the screening of the conduction electrons. So we can expect that it should depend on their concentration, as it does. We observe a similar effect when the electron concentration is changed in one lead only. It almost does not depend in which lead n is changed. However, for large e h asymmetry, the concentration of electrons in the normal lead seems to play a more important role. Now let us turn to the Andreev reflections and their modifications due to the e h asymmetry. The Andreev transmittance TNS A ( (equation (37, shown in figure is also affected by the concentration of the electrons in the leads. However, the quantitative behaviour of TNS A ( seems not to depend on the e h asymmetry. The most pronounced qualitative differences occur for the energies = E d. For a decreasing number of electrons in the normal lead, TNS A ( also decreases around these energies. On the other hand, the effect is the opposite if we decrease the electron concentration in the superconducting lead. We can easily explain the dependence

9 Electron transport through a strongly interacting QD T A NS ( 3e-4 e-4 e-4 n N =., n S =..5,.5.5,.., Figure. Andreev transmittance for various electron concentrations in the leads. The model parameters are the same as in figure. of the TNS A ( on the number of normal electrons, taking note of the fact that the probability of Andreev reflections is larger when the number of electrons in the normal lead is large and the number of holes small. However, TNS A ( also depends on the concentration of electrons and holes in the superconducting lead. This can be understood as follows. The number of electron-like (hole-like quasi-particles in the superconductor is proportional to the concentration of the electrons (holes in this lead in the normal state. In the Andreev process, if two electrons enter the superconductor, the electron-like quasiparticle is created. This means that the probability of Andreev reflections depends on the number of electron-like and holelike quasi-particles in the superconducting lead. So, with an increase of the number of holes in the superconductor, the probability of Andreev reflections of the impinging electrons is larger. This is exactly what we can see from figure. The modifications of TNS A ( due to e h asymmetry are not so large as the modifications due to the asymmetry in the couplings (see figure 8, nevertheless e h asymmetry influences the Andreev tunnelling. 7. Conclusions In conclusion, we have studied a strongly interacting QD connected to normal and superconducting leads. Using the EOM technique for the non-equilibrium GFs, we derived the formula for the current in terms of various tunnelling processes. This technique allowed us to calculate at once all the GFs emerging in the problem and to perform a consistent decoupling procedure for the higher-order GFs. We have discussed the problem of the interplay between the Kondo effect and Andreev reflections. While the Kondo resonance is present in the DOS, there is no zero-bias anomaly in the differential conductance. As a matter of fact, the Andreev conductance is strongly suppressed for zero-bias voltages. At the level of our approximation the superconducting electrons do not participate in the Kondo effect. Finally, we have discussed the problem of asymmetry in the couplings to the leads and found large modifications of the Andreev conductance due to this effect, mainly for energies around the dot level and superconducting gap. We have also studied the properties of the system when the concentration of electrons in the leads can be changed. However, the modifications of the Andreev tunnelling due to this effect are small. We hope the effects we have discussed here can be observed experimentally. The difficulty is to make one of the electrodes superconducting and the other normal. The only experiment in which interplay between the Kondo effect and the superconductivity is observed has been performed in a geometry in which a QD is coupled to two superconducting leads [5]. These authors have used a carbon nanotube QD and have found that the superconducting correlation does not destroy Kondo coherence provided T K > T c. This agrees with the calculations presented here and, in particular, with the gradual changes of the Kondo peak with an increasing superconducting gap (. References [] Ferry D K and Goodnick S M 997 Transport in Nanostructures (Cambridge: Cambridge University Press [] Glazman L I and Raikh M E 988 JETP Lett [3] Ng T K and Lee P A 988 Phys. Rev. Lett [4] Kawabata A 99 J. Phys. Soc. Japan 6 3 [5] Herschfield S, Davies J H and Wilkins J W 99 Phys. Rev. Lett Herschfield S, Davies J H and Wilkins J W 99 Phys. Rev. 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