Fluctuation-induced noise in out-of-equilibrium disordered superconducting films
We study out-of-equilibrium transport in disordered superconductors close to the superconducting transition. We consider a thin film connected by resistive tunnel interfaces to thermal reservoirs having different chemical potentials and temperatures. The nonequilibrium longitudinal current-current correlation function is calculated within the nonlinear sigma model description and nonlinear dependence on temperatures and chemical potentials is obtained. Different contributions are calculated, originating from the fluctuation-induced suppression of the quasiparticle density of states, Maki-Thompson and Aslamazov-Larkin processes. As a special case of our results, close-to-equilibrium we obtain the longitudinal ac conductivity using the fluctuation-dissipation theorem.
An equilibrium in nature is rare and it is rather an exception than the rule. A vast majority of natural processes, ranging from large-scale flows in the atmosphere to the electric charge transfer found throughout the technological realm, are out-of-equilibrium processes. Yet, the physics of the equilibrium state is much more studied and better understood. A major difficulty impeding the similar progress in the research of the nonequilibrium processes is that while all thermodynamics-based science rests on the law that, in equilibrium, any system assumes the state with the minimal free energy, the equally powerful and fundamental principles that govern the far-from-equilibrium behaviors still wait to be revealed. There has been an important advance both theoretical and experimentalBlanter and Bütikker (2000); Naz (2003); Kogan (1996); de Jong and Beenakker (1996); Kopnin (2001) employing a variety of approaches to out-of-equilibrium problems in electronic systemsKopnin (2001); Khlus (1987); Lesovik (1989); Büttiker (1990); Beenakker and Büttiker (1992); Shulman and Kogan (1969); Nagaev (2000, 1991); Altshuler et al. (1994); Gutman and Gefen (2001); among them, those based on the Keldysh techniqueKeldysh (1965); Kamenev and Levchenko (2009) seem to appear the most promising as paving a way towards general method which would allow treating interacting nonequilibrium systems on a common fundamental ground.
In this work we undertake the study of a disordered superconducting system employing the theory that has been proven to effectively tackle the low-energy excitation physics, the Keldysh nonlinear sigma model Feigel’man et al. (2000). More specifically, we focus on the nonequilibrium phase transition between the normal and the superconducting state. In a vicinity of the transition, when the system is in the normal state, the behavior of the system is governed by fluctuations of the superconducting order parameter. Fluctuation-induced short-living Cooper pairs are formed and contribute critically both into the thermodynamic and transport characteristics of the systemsLarkin and Varlamov (2005). In disordered thin films the temperature range where fluctuations are essential is determined by the sheet resistance (being in any case significantly larger than the one of bulk superconductorsAslamazov and Larkin (1968a, b); Maki (1968); Larkin and Varlamov (2005)) and depends on the particular process involved, extending often to temperatures well above the superconducting transition temperatureLarkin and Varlamov (2005); Baturina et al. (2012).
Here we study the influence of superconducting fluctuations on dynamic properties of a thin film in the fluctuational region of a normal state. The film is driven out of the equilibrium due to contacts with thermal reservoirs having different temperatures and chemical potentials. The Keldysh Ginzburg-Landau-like action under nonequilibrium conditions and several effects of nonequilibrium superconducting fluctuations were addressed in Refs. Chtchelkatchev and Vinokur, 2009; Petković et al., 2010, 2011 for this setting. Out-of-equilibrium fluctuation contributions to the dc electrical conductivity were calculated in Ref. Petković et al., 2011. Importantly, under nonequilibrium conditions the fluctuation-dissipation theorem (FDT) is generally violated and there is no fundamental relation between the current fluctuations (i.e. noise) and conductivity, in contrast to close-to-equilibrium conditions where this relation holds. Therefore, unlike in equilibrium systems, the nonequilibrium noise is not fully tied to the conductivity and can carry additional information, not contained in the conductivity. This poses a problem of the independent calculation of noise in out-of-equilibrium state.
Shot noise in noninteracting diffusive electron system was studied in Refs. Nagaev, 2000; Altshuler et al., 1994. The influence of the Coulomb interaction on shot noise in disordered systems was analyzed in Refs. Beenakker and Büttiker, 1992; Nagaev, 2000; Kozub and Rudin, 1995; Gutman and Gefen, 2001, while the influence of Bardeen-Cooper-Schrieffer (BCS) interaction on shot noise due to electric current flow above the critical temperature was considered only up to the second order in the electric field in Ref. Nagaev, 1991. In the present work, we focus on a film above the nonequilibrium superconducting transition and calculate nonlinear dependence of the Nyquist noise on the temperatures and chemical potentials of the thermal baths that are in contact with the film, see Fig. 1.
The paper is organized as follows. In Sec. II we introduce the nonlinear sigma model for superconductors within the Keldysh technique. In Sec. III we find the current correlation function for the case of noninteracting electrons in a nonequilibrium disordered thin film. We further consider the BCS interaction. In Sec. IV we introduce nonequilibrium fluctuation propagators. Then we proceed with calculations of different contributions to the current-current correlation function caused by superconducting fluctuations: the density of states contribution is calculated in Sec. V, the Maki-Thompson in Sec. VI and the Aslamazov-Larkin in Sec. VII. In Sec. VIII we summarize our results. Some calculation details are given in Appendices.
Ii Keldysh formalism: disordered superconductors
In this section we introduce the notation and provide the basic equations needed for the calculation of the current-current correlation function. We present the model, discuss its applicability and explain the procedure that allows us to analyze the fluctuations of the superconducting order parameter in the metallic state. In order to treat the nonequilibrium physics, we employ the Keldysh technique Keldysh (1965). We start with the nonlinear sigma model and then further discuss the calculation of the current-current correlations.
ii.1 Nonlinear sigma model
The nonlinear sigma model can be used to describe the low-energy physics for superconductors with BCS interaction in the presence of short-range quenched disorder (see Appendix A). The partition function takes the form , where the action consists of three partsFeigel’man et al. (2000); Kamenev and Andreev (1999)
Here and in the following we set . The fields and are Hubbard-Stratonovich fields introduced to decouple the four-fermion terms originating from the Hamiltonian describing the BCS interaction and disorder, respectivelyFeigel’man et al. (2000); Kamenev and Andreev (1999). The contributions to the action (1) are given by
Here is the diffusion coefficient, and it carries information about the disorder. The bare single particle density of states at the Fermi level per one spin projection is denoted by . The superconductive coupling constant is positive. The matrix field satisfies the nonlinear relation . The check symbol denotes matrices that are defined in the tensor product of the Keldysh and Nambu spaces. The former and the latter are spanned by the Pauli matrices and , , respectively, and we define , . One uses different notation for the same matrices for convenience, and . We assume implicitly the multiplication in the time-space, and “Tr” includes the integration over the real space. The subscript denotes the gauge transformed fields , and . The fields and are defined in same way as , where and are the vector and the scalar potential, respectively. The field is given by , and . is defined in the same way. We have also defined . The quantum (q) and classical (cl) components of the fields are respectively defined as the half-sum and the half-difference of the field values at the lower and the upper branches of the Keldysh time-contour. The field becomes the superconducting order parameter at the mean-field (saddle-point) level, while the saddle point equation for produces the Usadel quasiclassical equations, where plays the role of the quasiclassical Greens function. The covariant spatial derivative is given by . We stress that the nonlinear sigma model action captures the low-energy physics at energy scales much smaller that the elastic scattering rate. It is valid in the limit when the lifetime of fluctuating Cooper pairs is much greater than the elastic scattering time.
We further explain the strategy to treat the superconducting fluctuations. First we find the saddle point equation for of the action (1), in the absence of the BCS interaction (i.e. ). It reads as Larkin and Ovchinnikov (1975, 1977); Kamenev and Andreev (1999); Feigel’man et al. (2000)
After Wigner transforming we obtain that can be related to the quasiparticle electron/hole distribution functions via . One then considers massless fluctuations around the normal-metal saddle point solution, since massive modes can be integrated out in the Gaussian approximation and lead to unimportant renormalization of the parameters in the action. The massless fluctuations satisfy and are conveniently parameterized as Feigel’man et al. (2000)
such that . Here we introduced four real fields with representing diffuson degrees of freedom and the two complex fields for Cooperon degrees of freedom. One now substitutes matrix given by Eq.(7) into the action (1) and requires that the terms linear in vanish. This leads to a kinetic equation that electron and hole distribution function have to satisfy. Next we switch on the BCS interaction assuming the system is in the normal state. In that case the average value of the superconducting order parameter is zero, and therefore one again obtains the normal-metal saddle point solution (4). In order to study the influence of the BCS interaction, one has to consider the fluctuations. Assuming that the system is not too close to the transition, we take into account quadratic fluctuations around the metallic saddle point solution. Integrating them out, one obtains the effective action depending only on the superconducting order parameter and electromagnetic fields. This action allows us to treat the superconducting fluctuations in the normal metallic state. Levchenko and Kamenev (2007); Petković et al. (2011)
ii.2 Current-current correlation function
Our aim in the following is to calculate the symmetrized two-operator current correlation function
under nonequilibrium conditions. It can be obtained differentiating the nonlinear sigma model partition function
In the rest of the paper we focus on the particular system shown in Fig. 1. We consider a thin superconducting film connected by resistive interfaces to thermal reservoirs having different chemical potentials and temperatures. We assume that the system is in zero magnetic field and in the normal state but close to the transition into the superconducting state. We study a stationary situation (). Therefore, the noise depends only on the time difference . In the following we use the notation for the energy dependence of the distribution function.
There are four different contributions to the current correlation function
where denotes the noise in the noninteracting case (), and the other terms are contributions induced by superconducting fluctuations. The main processes are: i)coherent Andreev reflections of quasiparticles on the local fluctuations of the superconducting order parameter resulting into the so-called Maki-Thompson (MT) contributionsMaki (1968); Thompson (1970); ii) formation of Cooper pairs and their involvement into charge transfer is described by the Aslamazov-Larkin (AL) correctionsAslamazov and Larkin (1968a), and iii) the suppression of the single-particle density of states (DOS) due to quasiparticle participation in Cooper-pairing Larkin and Varlamov (2005). Randomness in the production and dissociation of fluctuating Cooper pairs, and in other closely related processes discussed above, gives the corresponding contributions to the current noise. In the following sections we evaluate and analyze the individual contributions. We focus on the regime where the superconducting fluctuations can be treated perturbatively, i.e., the fluctuation contribution are small compared to the noise in the noninteracting case. This implies that the film is not too close to the superconducting transition and that the physics is dominated by Gaussian fluctuations. Then one finds . Here is the Ginzburg number, is the film thickness and denotes the sum of the contributions induced by fluctuations of the superconducting order parameter.
Iii Noninteracting electrons
We consider noninteracting electrons first and evaluate Eq. (10) taking into account nonequilibrium conditions. After differentiating the partition function with respect to the vector potential field in Eq. (10), we substitute the field with its saddle-point solution given by Eq. (4). We do not take here into account the fluctuations around , since they are responsible for weak-localization effects. Since the main aim of this paper is to find the noise close to the superconducting transition, and since the weak-localization contribution is a non-singular function in the vicinity of the transition, we do not consider it here. Then, we obtain the longitudinal current-current correlation function to beNagaev (2000)
Here the Drude conductivity is and the integration is always from to if not specified differently. In equilibrium, the distribution function is and the expression (12) simplifies to
This universal relation between the conductivity and the noise is known as the FDT. The microscopic details about the disorder strength are hidden in the diffusion constant, i.e., in . We stress that the range of applicability of the nonlinear sigma model is , where is the elastic scattering time. This explains why in Eq. (14) appears the frequency independent Drude conductivity .
Next we analyze Eq. (12) in out-of-equilibrium conditions. Then, the FDT is generally violated. We assume that the film is thin such that the Thouless energy corresponding to diffusion across the film , well exceeds all the relevant energy scales. Here denotes the film thickness. The current across the interface separating the substrate and the film is , where is the tunneling resistance per unit area characterizing the interface and the subscript denotes the substrate. A similar equation holds for the interface between the film and the gate. From the continuity equation for the current follows , where . Here and denote the distributions in the substrate and in the gate, respectively. The gauge invariant distribution in the film is defined as and takes the form
The upper (lower) signs correspond to electrons (holes). We assumed very resistive interfaces, such that the resistance of the film can be neglected with respect to the resistance of the interfaces. Also, we assumed .
Now we can proceed with the evaluation of the expression (12). In the case one can calculate it exactly and obtains
where is the equilibrium noise, see Eq. (14). When one of the tunneling resistances is infinitely large, then effectively the system is in contact only with one reservoir and therefore in an equilibrium. In that case or , and as expected Eq. (16) reduces to the equilibrium noise . Moreover, notice that by increasing the temperature of the system, thermal fluctuations increase and the current noise increases. Similarly, in Eq. (16), the gate voltage also leads to an increase of the noise with respect to the equilibrium one (). In the case of zero frequency, the noise was found in Ref. Petković et al., 2010. Eq. (16) also describes a diffusive bridge placed between two reservoirs having different chemical potentialsAltshuler et al. (1994). There, plays the role of the coordinate along the bridge, and after performing the integration over in Eq. (16) one finds the shot noise of that system.
Next we discuss the influence of the electron-electron interaction on the distribution function (15). In general, the interaction leads to smearing of a noninteracting distribution function. However, when the inelastic length is large in comparison to the system dimensions, inelastic processes can be neglected. The inelastic length is expected to increase as the temperature and/or applied voltage decrease, allowing one to tune the ratio of the characteristic system length and the inelastic relaxation length. Coulomb-interaction induced corrections to the noise in the equilibrium have been studied by Altshuler and AronovAltshuler and Aronov (1985). Out-of-equilibrium, the influence of the the Coulomb interaction on the shot noise in diffusive contacts has been studied in the limit of large inelastic length in Refs. Beenakker and Büttiker, 1992; Nagaev, 2000; Gutman and Gefen, 2001, while contributions due to inelastic collisions have been studied in Ref. Kozub and Rudin, 1995. In the present paper, we assume that time-of-flight of the quasiparticles through the film is much smaller than the typical energy-relaxation time. We consider a thin film characterized by the large inelastic scattering length and focus on the influence of the BCS interaction on the longitudinal transport in the normal state.
Iv Superconducting fluctuations
Having discussed the noninteracting case in the previous section, we start the analysis of the influence of the BCS interaction on the current noise. The system is assumed to be in the normal state, but in the vicinity of the nonequilibrium superconducting transition. To be more precise, we do not discuss the Berezinskii-Kosterlitz-Thouless transition, but the crossover that in the equilibrium happens at the BCS critical temperature . In the following we analyze this crossover in nonequilibrium conditions, studying influence of the gate voltage and temperatures of the reservoirs on the Cooper pair lifetime and the processes discussed in Sec. II.2. In this and in the following sections we consider the case where the Ginzburg-Landau rate is much smaller than the effective temperature , defined below by Eq. (22).
The saddle point equation of the action (1) for the superconducting order parameter has the trivial solution with the average value of the superconducting order parameter being zero. Then, one recovers the normal-metal saddle point (4) for the field . Therefore, now we include the massless fluctuations around it. We substitute Eq. (7) into the action (1) and expand it to the second order in , Eq. (8). Since we are neither interested in the weak localization correction nor in the Altshuler-Aronov type corrections, but in fluctuations caused by the fluctuating superconducting order parameter , in the following we consider only the Cooperon degrees of freedom. After integrating out the Cooperon degrees of freedom, we obtain a Ginzburg-Landau-like action which depends only on the order parameter . This calculation was performed in Ref. Petković et al., 2011 and the correlation functions of Cooperon fields have been obtained. They read as 111Note that in Ref. Petković et al., 2011 it was used the notation and and similarly for the Fourier transform in the time-domain. This notation is confusing since then the convention is that complex conjugation includes the change of momentum and energy, and . In the following, we will use and . ,Petković et al. (2011)
The average is with respect to the action given by Eq. (1) and it includes averaging over the fluctuations of , and . Here denotes retarded/advanced fluctuation propagators and is the Keldysh propagator. Their low frequency and low momentum behavior is given by the following expressionsPetković et al. (2011):
Here and in Eqs. (17-19) we set . The parameters appearing in the retarded and advanced propagators are functionals of . They are given by the following expressions , , and . The symbol denotes the principal value of the integral. The nonequilibrium Ginzburg-Landau (GL) rate is defined as , where is the Debye energy. The GL time denotes the lifetime of the fluctuation induced Cooper pairs. It carries the information how far the system is from the transition and at the transition 222More precisely, in two dimensions this condition does not correspond to a real superconducting transition, but to a crossover. it becomes infinitely large, signaling that the Cooper pairs become long living. We point out that the fluctuation propagators given above are valid when the system is in normal state but in the vicinity of the transition such that . Substituting the distribution function (15) in the previous equations, one evaluates all these parameters and finds that they are functions of temperatures of the reservoirs and , the ratio of tunneling resistances and the gate voltage Petković et al. (2011):
Here is the digamma function, defined as , where is the gamma function. We expressed the critical temperature as , where is the Euler constant. Now one easily finds the parameter .
We are now equipped to start the calculation of different fluctuation contributions to the current-current correlation function. However, before doing it, we make a brief digression in order to stress the importance of the above given fluctuation propagators. In close-to-equilibrium conditions, the Aslamazov-Larkin contribution to the conductivity can be obtained by employing together the linear response theory and the phenomenological time-dependent GL (TDGL) equation, while the Maki-Thompson and the DOS, should be derived starting from the microscopic theory. The phenomenological TDGL equation reads as
where the thermal noise satisfies
However, if one is interested in a nonlinear dependence on the drive, then the previous equation is not a good starting point. Notice that even for the simple setup here considered (see Fig. 1), the TDGL equation takes different formPetković et al. (2011) than the usual phenomenological TDGL equation (25). It is given by
where the nonequilibrium noise satisfies the following condition
In Eq. (27), the retarded fluctuation propagator is given by Eq. (21) with many drive-dependent parameters that we discussed above. Only in the equilibrium, and , Eqs. (27) and (28) coincide with Eqs. (25) and (26), while otherwise they are different. Therefore, we point out that in order to study a nonlinear dependence of some observable on the drive, one should derive a corresponding TDGL equation starting from the microscopic theory and not to use the phenomenological one (25), as is usually the case in literature.
V Density of states contribution
In this section we start the calculation of the current correlation function (10) in the presence of the BCS interaction. We consider linear response to the in-plane electric field, while the system is driven out-of-equilibrium by thermal baths having different temperatures, between which it is sandwiched, or by an electric field perpendicular to the film, see Fig. 1.
First we consider the density of state contribution to the noise. Taking into account the massless fluctuations around the normal-metal saddle point Eq. (4) to the second order in the Cooperon fields, we collect all the terms of the type and in Eq. (10). They constitute the DOS contribution and together give
The leading contribution in Eq. (V) close to the transition into superconducting state is given by
We use the notation . Notice that Eq. (V) is the only singular part of Eq. (V) for and therefore the most dominant close to the transition. The holes and electrons give the same contribution. This is expected, since the system is in the normal state and there is no long-living Cooper-pair condensate that could allow for a charge imbalance between the holes and electrons. We further evaluate Eq. (V) and find up to logarithmic accuracy:
Here and in the following denotes . In order to discuss the assumptions made in Eq. (V), we introduce
We assume that the system is close to the transition such that and are much smaller than relevant energy scales of the distribution function, i.e., . For the special case and , is exactly zero. Eq. (V) is also valid for and then the condition becomes . When obtaining Eq. (V), we used the fact that the most important contribution in Eq. (V) originates from small momenta and that therefore we can safely cut the momentum integration at the upper limit .
In the case of equal temperatures of the reservoirs , substituting the distribution function (15) in Eq. (V), we obtain the nonlinear dependence of the noise on the gate voltage , temperature and frequency . Integrating out spatial coordinates, one obtains the final result
The function is defined as
where the frequency satisfies . However, for but arbitrary with respect to , one obtains the result by setting to zero in Eq. (V). The expressions for , , and are provided in Sec. IV. In Appendix B we discuss the case of different temperatures of the reservoirs, .
For we also get the same expression from Eq. (V), since it also corresponds to the equilibrium situation. In the case of noninteracting equilibrium electrons, studied in Sec. III, the source of the current fluctuations is thermal fluctuations. Here we obtained the additional contribution (V) to the current noise caused by superconducting fluctuations. The Cooper pair density fluctuates due to randomness in the formation and dissociation of Cooper pairs, and these fluctuations affect the single-particle density of states leading to the noise .
In equilibrium the FDT holds and relates the current fluctuations (noise) to the real part of the ac conductivity that provides information about absorbed energy in the sample:
where . Using it we obtain the density of states contribution to the in-plane ac conductivity in equilibrium:
This result is in the agreement with findings of Ref. Petković and Vinokur, 2013, where it is obtained in a different manner, i.e. directly considering the ac conductivity in equilibrium within the nonlinear sigma model approach. In Eqs. (35,37,38) the GL rate takes its equilibrium value . One obtains that suppression of the single-particle density of states due to superconducting fluctuations gives the negative contribution to the Drude conductivity, see Fig. 2 where as the function of frequency is shown by the red dashed line. We remind the reader that our calculations apply when the frequency, temperatures and the gate voltage are much smaller than the elastic scattering rate, since we use the nonlinear sigma model that captures low-energy physics.