Superconductivity suppression in disordered films: Interplay of two-dimensional diffusion and three-dimensional ballistics
SSuperconductivity suppression in disordered films:Interplay of two-dimensional diffusion and three-dimensional ballistics
Daniil S. Antonenko
1, 2, 3 and Mikhail A. Skvortsov
1, 2 Skolkovo Institute of Science and Technology, Moscow 121205, Russia L. D. Landau Institute for Theoretical Physics, Chernogolovka 142432, Russia Moscow Institute of Physics and Technology, Moscow 141700, Russia (Dated: September 11, 2020)Suppression of the critical temperature in homogeneously disordered superconducting films is aconsequence of the disorder-induced enhancement of Coulomb repulsion. We demonstrate that forthe majority of thin films studied now this effect cannot be completely explained in the assumptionof two-dimensional diffusive nature of electrons motion. The main contribution to the T c suppressionarises from the correction to the electron-electron interaction constant coming from small scales ofthe order of the Fermi wavelength that leads to the critical temperature shift δT c /T c ∼ − /k F l ,where k F is the Fermi momentum and l is the mean free path. Thus almost for all superconductingfilms that follow the fermionic scenario of T c suppression with decreasing the film thickness, thiseffect is caused by the proximity to the three-dimensional Anderson localization threshold and iscontrolled by the parameter k F l rather than the sheet resistance of the film.
1. Introduction.
The principal characteristics ofa superconductor is the value of its transition temper-ature, T c . It is usually assumed that T c is a materialproperty and does not depend on the sample size. How-ever there is a strong experimental evidence of the sys-tematic decrease of the critical temperature in disorderedsuperconducting films with decreasing its thickness, d (V[1], NbN [2–9], TiN [10], MoGe [11, 12], MoSi [13, 14],MoC [15], WRe [16], InO [17] etc. [18]). The sup-pression of T c becomes pronounced typically at d ∼ nm, and for the thinnest films T c may eventually vanish,marking the point of a quantum superconductor-metalor superconductor-insulator transition [19–24].Depending on the underlying structure of a material,two scenarios of T c suppression — fermionic and bosonic— have been identified. The bosonic scenario applies togranular and/or strongly inhomogeneous superconduc-tors with localized preformed Cooper pairs (polycrys-talline TiN, amorphous InO) [25–28], where T c signalsproliferation of superconducting coherence from micro-to macro-scales. In the fermionic scenario, relevant forstructureless homogeneously disordered superconductors(NbN, MoGe, etc.), suppression of superconductivity is aconsequence of the disorder-induced enhancement of elec-tron repulsion [29, 30], which leads to the decrease in theeffective Cooper pairing constant. Despite the commonphysical mechanism of disorder-induced T c suppressionin the fermionic scenario, its description for three- andtwo-dimensional systems is rather different. In the three-dimensional (3D) geometry , enhancementof repulsion due to scattering off the impurity potentialis provided by small distances, not exceeding the meanfree path l . As a result, the whole effect can be com-pletely described by the change in the Cooper pairingconstant. The fermionic mechanism for strongly disor-dered 3D superconductors in the vicinity of the Andersonlocalization threshold ( k F l ∼ , where k F is the Fermi momentum) was studied by Anderson, Muttalib and Ra-makrishnan [31]. They also estimated the correction tothe bare electron-electron interaction constant λ in thecase of weak disorder ( k F l (cid:29) ): δλ/λ ∼ / ( k F l ) . Sim-ilar expressions were reported in Refs. [32, 33]. This es-timate can be easily obtained by cutting the 3D diffusivecontribution at the ultraviolet cutoff r ∼ l . However, asshown by Belitz and Kirkpatrick in their study of weak-localization correction to the conductivity [34], diffusivecontributions in the 3D geometry are extended to theballistic region up to the distances of the order of wave-length and have a relative order of / ( k F l ) rather than / ( k F l ) . Similar extension of the interaction-inducedcontribution from the diffusive to the ballistic region isalso known for the tunneling density of states, both in2D [35] and 3D geometries [36, 37].Disorder-induced renormalization of the electron-phonon interaction and its impact on superconductivitywere studied by Keck and Schmid [38]. They showedthat the displacement of impurities by the lattice vibra-tions leads to the suppression of the interaction with lon-gitudinal phonons and the emergence of the interactionwith transverse phonons. An attempt to account for theimpurity corrections both to the Coulomb and electron-phonon interactions and their influence on T c was takenby Belitz with the help of the exact-eigenstates technique[39] and by solving full Gor’kov equations in the strong-coupling regime [40–42]. A part of his results can beinterpreted as a correction to the bare electron-electroncoupling constant δλ/λ ∼ /k F l . However, Belitz’s re-sults were called into question by Finkel’stein [43] bydemonstrating that elastic diagrams, intimately relatedto the correction to the tunneling density of states [44, 45]and claimed to be essential by Belitz, actually do not con-tribute to the leading order of T c shift.The main difference of the two-dimensional (2D) geom-etry compared to the 3D case is that the renormalization a r X i v : . [ c ond - m a t . s up r- c on ] S e p Figure 1. Experimentally relevant hierarchy of length scalesin disordered superconducting films. effect does not boil down to the energy-independent shiftof the coupling constant λ and requires a summation ofthe leading logarithms. Conventional description of T c suppression in thin superconducting films substantiallyrelies on
2D diffusive nature of electron motion, whichis motivated by the experimentally relevant hierarchy oflength scales λ F (cid:28) l (cid:28) d (cid:28) ξ , see Fig. 1. (Here λ F isFermi wavelength, ξ = (cid:112) (cid:126) D/T c is the superconductingcoherence length in the dirty limit, and D is the diffu-sion constant.) In this paradigm, enhancement of disor-der with the decrease of the film thickness d is related tothe increase of the sheet resistance of the film, R (cid:3) .The effect of T c shift due to the interplay of disorderand interaction was studied on a perturbative level inRefs. [44–48], where the 2D diffusive contribution to the T c shift was calculated: δT c T c = − λ πg log (cid:126) T c τ ∗ , (1)where T c is the critical temperature of a bulk super-conductor, g = h/e R (cid:3) = (2 / π )( k F l )( k F d ) (cid:29) is thedimensionless film conductance, and λ is the dimension-less coupling constant of the electron-electron interac-tion (for the screened Coulomb interaction, λ = 1 / ).The parameter τ ∗ is the time when diffusion becomestwo-dimensional: τ ∗ = max { τ, τ d } , where τ is the elasticscattering time and τ d = d / D is the time of diffusionacross the film thickness [43, 46]. In real space, the log-arithm in Eq. (1) is accumulated from the 2D diffusionfrom the length scale max( l, d ) to the coherence length ξ .The correction (1), inversely proportional to the film con-ductance, is conceptually similar to the weak-localization[49, 50] and interaction-related [30] corrections to the 2Dconductivity, while two out of three powers of the loga-rithm are due to the exponential sensitivity of T c to thecoupling constant λ BCS .The first-order perturbative result (1) has later beengeneralised to the case of arbitrarily strong T c suppres-sion by Finkel’stein, who managed to sum the lead-ing logarithms with the help of the renormalization-group technique [43, 51]. The same result can be ob-tained by solving the self-consistency equation with anenergy-dependent Cooper coupling λ E,E (cid:48) = λ BCS − γ g log[1 / max( E, E (cid:48) ) τ ∗ ] [52]. For the screened Coulombinteraction ( λ = 1 / ), the nonperturbative expression for the critical temperature as a function of the dimen-sionless film conductance valid until superconductivity isfully suppressed is given by: log T c T c = 1 γ − γ g log γ + γ g γ − γ g , (2)where γ g = 1 / √ πg and γ = 1 / log( (cid:126) /T c τ ∗ ) . Expression(2), where γ is considered as a fitting parameter , was usedby Finkel’stein [43] to explain the observed dependenceof T c in MoGe films [11] on the film thickness, the latterbeing directly related to the dimensionless conductance g . Since then, such an explanation of experimental dataon superconductivity suppression in disordered films hasbecome generally accepted [14, 15, 53].According to Eqs. (1) and (2), T c suppression in thin( d (cid:28) ξ ) superconducting films is entirely determined bythe dimensionless sheet conductance g . Such a statementperfectly fits the general theoretical framework of scaling[49], justified by the renormalization-group analysis ofthe nonlinear sigma model in the 2D space [54–56].However, interpretation of experimental data on T c ( d ) dependence with the help of Eq. (2) encounters a num-ber of significant difficulties. The first one is the internalinconsistency of the approach that treats γ as a free fit-ting parameter. As follows from Table I, which containsexperimental data on different films, the values of γ − fit ob-tained by fitting T c ( d ) dependence with the help of Eq.(2) typically lie in the interval ÷ . The issue is thatthese values significantly exceed the theoretical estimate γ − = L d = ln( (cid:126) /T τ d ) (last column in Table I), and inhalf of the cases exceed even the quantity L = ln( (cid:126) /T τ ) (last but one column in Table I). Taking into account thatthe perturbative shift of T c , according to Eq. (1), is pro-portional to the third power of this logarithm, one canconclude that the discrepancy between the microscopictheory and the result of the fit with Eq. (2) appears to bevery large. One can try to save the situation by pointingto the fact that γ − fit should also contain the contributionof 3D diffusion, but that makes the usage of Eqs. (1) and(2) dubious as they were obtained under the assumptionof 2D diffusion.Another problem with interpreting experimental datain terms of Eq. (2) is an implicit assumption that the ef-fect of T c suppression is determined by the dimensionlessfilm conductance only. However in real thin films, theimpurity concentration and hence the mean free path l do vary with the film thickness due to peculiarities of thefabrication process. Large amount of experimental dataon the critical temperature of thin films has been anal-ysed in Ref. [59], where it has been demonstrated that T c is primarily dependent on the 3D bulk conductivity σ ∝ k F l rather than the 2D sheet conductance g ∝ k F ld .Inapplicability of Eq. (2) for the description of T c sup-pression in thin films is actually a consequence of (i) toonarrow interval of 2D diffusion (from d to ξ ), which Table I. Parameters of superconducting films [57]: bulk criti-cal temperature T c , thickness d , mean free path l , the valueof γ obtained from fitting T c ( g ) dependence with Eq. (2)and the values of the two logarithms: L = log( (cid:126) /T c τ ) and L d = log( (cid:126) /T c τ d ) .Mat. Ref. T c , K d , nm l , ˚A γ − fit L L d NbN [4] 15 ÷ ∼ . ÷ . NbN [5] 15 ÷ . ÷ . NbN [8] 17 > < − . ÷ . ÷ . MoGe [11, 51] 7 . ÷ ∼ < . MoSi [13] 7 ÷ < . MoC [15] 8 ÷ < . ÷ . WRe [16] 6 ÷ < . Nb [58] 7 . ÷
18 11.7 5.2 < . appears to be insufficient to explain the observed mag-nitude of the effect and (ii) the smallness of the pref-actor /g ∼ ( k F l ) − ( k F d ) − . Hence for a quantitativedescription of experimental data, one has to specify an-other mechanism of disorder-induced enhancement of theCoulomb interaction that is not related to 2D diffusion.In the present paper, we demonstrate that existing ex-perimental data on T c suppression in thin films can beconvincingly explained assuming that the main contribu-tion stems from the processes of three-dimensional ballis-tic motion of electrons with a typical distance betweenthe interaction point and the point of impurity scatteringof the order of several wavelengths. Our main result isthe amendment of the perturbative expression (1) for T c shift: δT c T c = − αk F l − λ πg log (cid:126) T c τ d , (3)where the added first term accounts for the contributionof the 3D ballistic region. We emphasize that since allscales starting from the Fermi wavelengths contribute to T c suppression, keeping the last term originating fromthe 2D diffusion region on the background of the first onemay be justified only for materials with exceptionally low T c or very small thickness (in particular, for atomicallythin films [60]).The coefficient α in Eq. (3) is nonuniversal and de-pends on the details of the interaction and the structureof the random potential. In the model of weak short-ranged electron repulsion (amplitude λ ) and Gaussianwhite-noise random potential, it is given by α = πλ log ω D /T c λ log E F /ω D ) . (4)For realistic superconducting films with the Coulombinteraction one should expect a material dependent value α ∼ .
2. The model.
We consider a model of s -wave super-conductivity with a phonon-mediated electron attractiondescribed by the potential V ph ( r ) = − ( λ ph /ν ) δ ( r ) effec-tive in the in the energy strip of ω D near the Fermi energy,and a short-range repulsion with the potential V ( r ) =( λ/ν ) δ ( r ) and an energy cutoff at E F . We will workin the weak-coupling approximation, λ ph , λ (cid:28) , andneglect disorder-induced renormalization of the phononvertex beyond the ladder approximation [38]. Disor-der is modeled by a random potential with the Gaus-sian white-noise statistics described by the correlator (cid:104) U ( r ) U ( r (cid:48) ) (cid:105) = δ ( r − r (cid:48) ) / πντ , where ν is the densityof states at the Fermi level (for one spin projection) and τ is the elastic scattering time.In the absence of disorder-induced renormalization ofthe interaction vertices, T c is given by the standard ex-pression of the Bardeen-Cooper-Schrieffer (BCS) theory: T c = ω D exp ( − /λ BCS ) , (5)where the effective coupling constant is λ BCS = λ ph − λ λ log E F /ω D . (6)The second term, known as the Tolmachev logarithmin Russia and as the Coulomb pseudopotential in theWest, describes the effect of the Coulomb repulsion tothe Cooper channel undergoing logarithmic renormaliza-tion in the energy window from E F to ω D [61–64].The critical temperature is determined by the pole ofthe Cooper ladder at zero momentum and frequency inthe Matsubara diagrammatic technique. In the presenceof a random potential, the diagrammatic series shouldbe averaged over disorder in every possible way. In theleading order (no-crossing approximation), this processreduces to independent averaging of the product of thetwo Green functions, G E G − E , connecting the interac-tion vertices ( λ ph or λ ), which is done via insertion of acooperon. According to the Anderson theorem [65–67],the result is disorder independent and leads to the ex-pressions (5) and (6) for the critical temperature.
3. Diffusive contribution.
In order to find the shiftof T c , one has to take into account processes describingan interplay of interaction and disorder in the next orderwith respect to no-crossing diagrams [43–45, 47, 48, 51].The leading diagrammatic contributions in the diffusiveregion are shown in Fig. 2, where the interaction (zigzagline) is crossed by the impurity ladders — diffusons andcooperons — depicted as gray blocks. The diagram (a)has a mirror counterpart, while the diagram (b) containstwo additional contributions with an impurity line con-necting the Green functions with the energy of the samesign (Hikami box) [68]. Analytical expression for T c shiftcontains a summation over two Matsubara energies E (a) (b) Figure 2. Inelastic diagrams for the diffusive contribution( q (cid:28) /l and E, E (cid:48) (cid:28) /τ , where q is the momentum car-ried by the interaction line) to the Cooper susceptibility thatdetermine T c shift. The shaded blocks in the center of the dia-grams are cooperons and diffusons connecting the Green func-tions with the opposite Matsubara energy signs. The shadedtriangles in the corners of the diagrams designate renormaliza-tion of the phonon vertex by the impurity ladders and laddersof electron interaction with the constant λ . and E (cid:48) : δT c T c = − πλν (cid:18) λ ph λ BCS (cid:19) T E F (cid:88) E,E (cid:48) > u ( E ) u ( E (cid:48) ) I E,E (cid:48) EE (cid:48) , (7)where the factor λ ph /λ BCS and the logarithmic function u ( E ) = θ ( ω D − E ) − ( λ log ω D /T ) / (1+ λ log E F /T ) repre-sent renormalization effects, which can be introduced byadding λ -interaction ladders to the left and right vertexof the diagram [64]. In the diffusive region, the quan-tity I E,E (cid:48) in the film geometry can be expressed via anintegral over the 2D in-plane momentum q (cid:107) and a sumover the transverse modes of the Laplace operator withthe Neumann boundary conditions ( q z = 2 πm/d , with m = 0 , 1, . . . ) carried by the interaction line [64]: I E,E (cid:48) = τd (cid:88) q z (cid:90) d q (cid:107) (2 π ) f q ( E + E (cid:48) ) [3 − f q ( E + E (cid:48) )]1 − f q ( E + E (cid:48) ) . (8)To trace the crossover to the ballistic region, we writecooperons and diffusons beyond the diffusive approx-imation and express them via the function f q ( ω ) =( ql ) − arctan[ ql/ (1 + | ω | τ )] , which corresponds to the onestep of the impurity ladder at arbitrary values of ql and ωτ , but under the conditions q (cid:28) k F , ω (cid:28) E F . Ananalogous approach was used in Ref. [69] to calculate thefluctuation conductivity at arbitrary disorder strength.The leading 2D diffusive contribution stems from themode with q z = 0 . Cutting the integral over q at themomentum /d and the energy summation at ω D , andtaking into account that for realistic films studied in ex-periments the Debye frequency ω D is comparable to (cid:126) /τ d [9], we arrive at the well-known result (1) with τ ∗ ∼ τ d .Note however that the extraction of the 2D diffusive con-tribution out of expressions (7) and (8) is complicatedby the fact that the contributions of other regions are infact larger. Indeed, at the scale q ∼ /d the 2D loga-rithmic behavior is changed to a linearly divergent one Figure 3. Sketch of the dependence of integrand in Eq. (8)on q (at not too large E + E (cid:48) ). In the region q > /d it hasa weak q dependence, changing by a factor of π / at thecrossover from the diffusive to ballistic motion (at q ∼ /l ). due to excitation of higher transverse modes, making themomentum integral three-dimensional. One can estimatethe contribution of the 3D diffusive region by introducingan artificial cutoff at q ∼ /l , which gives δT (diff, 3D) c T c ∼ − λ ( k F l ) log ω D T c . (9)This contribution has only two out of three logarithmicfactors but nevertheless it exceeds Eq. (1) by the parame-ter d/l (cid:29) . However, nothing prevents considering evengreater momenta in Eq. (8) and study the ballistic region q (cid:29) /l . Remarkably, in this region the integrand ofEq. (8) still obeys the /q behavior, but with a differentnumerical prefactor. This means that the main contribu-tion to the integral originates from momenta of the orderof Fermi momentum, q ∼ k F . This region requires a spe-cial treatment, which will be done below. Schematicallythe role of different momentum regions is illustrated inFig. 3. Up to logarithmic factors coming from the energysummations, the integral of the shown curve determinesthe contribution of the corresponding regions to T c shift.
4. Ballistic contribution.
In this Section, we studythe ballistic contribution to T c shift originating from pro-cesses with momentum transfer q > /l . Due to theassumption l (cid:28) d , electron motion can be assumed tobe three-dimensional. This contribution is described bythe diagrams shown in Fig. 2, where we left only oneimpurity line out of the diffusive ladder, correspondingto scattering on one impurity. For an accurate calcula-tion, one should reconsider expression (8), relaxing theassumption q (cid:28) k F .The ballistic contribution can be described as a cor-rection to the bare (unrenormalized) repulsive electron-electron coupling constant in the Cooper channel, λ c ,which in the leading order coincides with λ (Fig. 4(a)).The leading corrections are given by the diagramsFig. 4(b) and Fig. 4(c). In the considered model of point-like interaction and delta-correlated disorder, the calcu-lation of these diagrams can be performed analytically (b) (c)(a) Figure 4. (a) Electron-electron interaction vertex λ in theCooper channel with the first impurity line of the surroundingcooperons. (b), (c) Diagrams describing the leading vertexcorrection λ c from the ballistic region. Both diagrams havemirrored counterparts. and leads, generally speaking, to an energy-dependentcorrection δλ cEE (cid:48) to the Cooper-channel coupling: δλ cE,E (cid:48) λ = 2 ( b ) + ( c )( a ) = 2[ P ( E, E (cid:48) ) + P ( E, − E (cid:48) )](2 πντ ) f (2 E ) f (2 E (cid:48) ) λ , (10)where the terms in the brackets correspond to the dia-grams (b) and (c), respectively, and the numerical co-efficient is due to mirrored diagrams. The factors f ( ω ) = 1 / (1 + | ω | τ ) in the denominator originate fromthe momentum integration of a pair of the Green func-tions in Fig. 4(a) (one step of the diffusive ladder).It is convenient to calculate block P ( E, E (cid:48) ) in the co-ordinate representation [37]. Since the electron-electroninteraction as well as the disorder correlator are assumedto be point-like, analytical expression contains only oneintegral over the distance r between the impurity and theinteraction point, so we get: P ( E, E (cid:48) ) = λ πντ (cid:90) d r G + G (cid:48)− [ G + G − ][ G (cid:48) + G (cid:48)− ] , (11)where G ± = G ± E ( r ) are disorder-averaged Green func-tions and the prime refers to the energy argument E (cid:48) .The square brackets denote the real-space convolution: [ G + G − ] = (cid:82) G + ( ρ ) G − ( r − ρ ) d ρ . As will be demon-strated below, the integral over r in Eq. (11) convergeson the scale /k F that allows to replace the Green func-tions by their values in the absence of disorder: G ± = − πν e ± ik F r k F r , [ G + G − ] = 2 πντ | E | τ sin k F rk F r , (12)where the convolution was calculated under the assump-tion E, E (cid:48) (cid:28) E F .One can easily show that the integral in Eq. (11) van-ishes for different signs of the energies E and E (cid:48) , andthus P ( E, E (cid:48) ) ∝ θ ( EE (cid:48) ) . Thereby in the consideredmodel, the ballistic diagrams in Figs. 4(b) and 4(c) arenonzero for the same relation between the energy signs asfor the diffusive diagrams in Figs.2(a) and 2(b), respec-tively. This conclusion is a priori not obvious because asingle impurity line can connect two Green functions of the same energy sign. However, we see that in the caseof the point-like interaction and delta-correlated disorder,these diagrams vanish in the ballistic limit as well.Substituting Eq. (12) to Eq. (11) and then to Eq. (10),we observe that the factors (1 + 2 | E | τ ) and (1 + 2 | E (cid:48) | τ ) in the denominators of [ G + G − ] and [ G (cid:48) + G (cid:48)− ] cancel thesame factors f ( E ) and f ( E (cid:48) ) in Eq. (10). The onlyenergy dependence of δλ cE,E (cid:48) is thus due to the factor θ ( EE (cid:48) ) contained in the block P ( E, E (cid:48) ) . However, italso disappears because of the structure of Eq. (10). Asa result, the correction δλ cE,E (cid:48) appears to be energy-independent: δλ c = πνλ τ (cid:90) d r ( k F r ) (cid:18) sin k F rk F r (cid:19) = πλ k F l . (13)As expected, the integral stems from the scales of theorder of the electron wavelength, which is typical for 3Dmesoscopic effects [34, 70, 71].The obtained correction may be interpreted in thespirit of Ref. [35] as the renormalization of the contribu-tion of electron-electron interaction to the Cooper chan-nel due to scattering on Friedel oscillations caused byimpurities. This correction describes the enhancement ofthe electron-electron repulsion, leading to the increase ofthe Coulomb pseudopotential and, consequently, to thesuppression of the effective coupling constant λ BCS . Sup-pression of T c can be found by substituting λ by λ + δλ c and expanding Eq. (6) in δλ c : δT ( ball, 3D ) c T c = − π λk F l (cid:18) log ω D /T c λ log E F /ω D (cid:19) . (14)
5. Role of elastic diagrams.
Besides inelastic dia-grams shown in Figs. 2 and 4, where the interaction lineconnects the upper and lower Green functions, there is aset of elastic diagrams related to the interaction correc-tion to the one-particle Green function. As demonstratedby Finkel’stein [43] for 2D diffusion, the contribution ofthis set of diagrams is always small: at Dq > ω theycontain a smaller power of a large logarithm, while at Dq < ω their contribution is canceled by contributionsof inelastic diagrams and of an additional set of diagramsrestoring the gauge invariance of the theory. The latterdiagrams become subleading already in the diffusive re-gion at Dq > ω and therefore are not considered in thepresent paper.In the case of an instantaneous electron-electron inter-action, there is an exact relation [39, 44, 45] between thecontribution of elastic diagrams to T c shift and correc-tion to the tunneling density of states δν ( ε ) , which canbe represented [64] in the form analogous to Eq. (7): δT (elast) c T c = (cid:18) λ ph λ BCS (cid:19) T (cid:88) E (cid:90) dε u ( E ) E + ε δν ( ε ) ν . (15)We will use known results for δν ( (cid:15) ) in order to estimatethe contribution (15) of elastic diagrams. ●●●●●●●● ●●●●●●● ● ● ● ● ●●● ● ● ( а ) NbN ●●●●● ) MoC(b ● ●●●●●●●●● ) V(c Figure 5. Experimental data on the dependence of T c on k F l (dots) and their fitting with the help of Eq. (17) (solid line) forsuperconducting films of different thickness and composition: (a) NbN [8], (b) MoC [15], (c) V [1]. The correction to the tunneling density of states of a3D metal in the diffusive region ( | ε | < /τ ) has the form δν diff ( ε ) /ν ∼ λ (cid:112) | ε | τ / ( k F l ) [72]. A simple algebra re-veals that the contribution to T c shift from this region isproportional to / ( k F l ) , which is parametrically smallerthan the contribution of the ballistic region discussed be-low.The correction to the tunneling density of states inthe 3D ballistic region ( | ε | > /τ ) was studied in Refs.[36, 37] and appeared to be linear in energy and generallyasymmetric with respect to the Fermi level. In the caseof a point-like interaction, delta-correlated disorder, andparabolic electron spectrum, it is finite only for energiesbelow the Fermi energy and has the form δν ball ( ε ) /ν ∼ λ | ε | θ ( − ε ) / ( k F l ) [37]. Then Eq. (15) yields δT ( ball, 3D, elast ) c T c ∼ λ k F l (cid:18) log ω D /T c λ log E F /ω D (cid:19) , (16)which is parametrically smaller than the leading contri-bution (14) under the model assumption λ (cid:28) . The ab-sence of a linear-in- λ contribution from elastic diagramsis related to the fact that, contrary to Eq. (7) with twologarithmic summations over E and E (cid:48) , the integral (15)in the 3D ballistic region is not logarithmic. The con-clusion that elastic diagrams do not contribute to theleading T c shift is presumably quite general and relatedto the fact that the tunneling density of states is not athermodynamic quantity.
6. Conclusion.
In the present paper we studied theinfluence of the 3D ballistic region of electrons motionon the critical temperature degradation of moderatelydisordered superconducting films ( k F l (cid:29) ). Assumingthe model of a point-like repulsion and delta-correlateddisorder, we calculated the perturbative contribution ofthis region to T c suppression given by the first term inEq. (3). When comparing our theory with experimentaldata, one should take into account that in real samples λ ∼ / due to the Coulomb interaction and that thenumerical factor in Eq. (4) is model-specific. In general,one might expect that the ballistic contribution to T c shift has form δT c /T c = − α/k F l with α ∼ . The second term in Eq. (3) describes the standard con-tribution to T c suppression originating from the region oftwo-dimensional electron diffusion, where the logarithmstems from the spatial scales between the film thickness d and the coherence length ξ . The smallness of this in-terval for realistic films and a relatively large value ofthe dimensionless conductance g ∼ ( k F l )( k F d ) makes itpractically negligible compared to the three-dimensionalballistic contribution.Fig. 5 presents the fits of experimental data ( T c , k F l )for superconducting films of different thicknesses made ofthree different materials following the fermionic scenarioof T c suppression by the formula T c = (1 − α/k F l ) T c , (17)where α and T c are treated as fitting parameters. Arather good agreement is observed, with the material-dependent value of α being of the order of one, as ex-pected. We emphasize that the data for NbN presentedin Fig. 5(a) refer to thick films [8], for which there is notwo-dimensional diffusive region at all (see Table I).Based on (i) the observed agreement between experi-mental data and Eq. (17), (ii) intrinsic inconsistencies ofthe theory behind Eq. (2) mentioned above, and (iii) thefindings of Ref. [59], which indicate that T c is primar-ily dependent on the 3D conductivity rather than the2D sheet conductance, we make the following practicallyrelevant conclusion: For a substantial fraction of not too thin moderatelydisordered superconducting films that follow the fermionicscenario of superconductivity suppression, the latter isgoverned by the proximity to the threshold of three-dimensional Anderson localization and controlled by theparameter k F l . Two-dimensional diffusion effects, con-trolled by dimensionless conductance g are also present,but they typically constitute only a small correction ontop of three-dimensional ballistic effects. The authors are grateful to I. S. Burmistrov,M. V. Feigel’man, A. M. Finkel’stein, P. Samuely,P. Szab´o, K. S. Tikhonov, and P. M. Ostrovsky for usefuldiscussions. The research was supported by a grant fromthe Russian Science Foundation No. 20-12-00361. [1] A. A. Teplov, “Critical magnetic fields in layered thin-film vanadium-carbon structures,” Zh. Eksp. Teor. Fiz. , 802 (1976), [Sov. Phys. JETP , 422 (1976)].[2] Z. Wang, A. Kawakami, Y. Uzawa, and B. Komiyama,“Superconducting properties and crystal structures ofsingle-crystal niobium nitride thin films deposited at am-bient substrate temperature,” J. Appl. Phys. , 7837(1996).[3] A. Semenov, B. G¨unther, U. B¨ottger, H.-W. H¨ubers,H. Bartolf, A. Engel, A. Schilling, K. Ilin, M. Siegel,R. Schneider, D. Gerthsen, and N. A. Gippius, “Op-tical and transport properties of ultrathin NbN films andnanostructures,” Phys. Rev. B , 054510 (2009).[4] Y. Noat, V. Cherkez, C. Brun, T. Cren, C. Carbillet,F. Debontridder, K. Ilin, M. Siegel, A. Semenov, H.-W.H¨ubers, and D. Roditchev, “Unconventional supercon-ductivity in ultrathin superconducting NbN films studiedby scanning tunneling spectroscopy,” Phys. Rev. B ,014503 (2013).[5] K. Makise, T. Odou, S. Ezaki, T. Asano, and B. Shi-nozaki, “Superconductor-insulator transition in two-dimensional NbN / MgO and
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The starting point of our analysis is the zero-momentum Cooper susceptibility: L = (cid:90) d r (cid:90) /T dτ (cid:10) ψ + ↓ ( r , τ ) ψ + ↑ ( r , τ ) ψ ↑ (0 , ψ ↓ (0 , (cid:11) . (S1)The divergence of L as a function of temperature T marks the transition to the superconducting state.The basic element of the theory is the disorder-averaged Matsubara Green function G ± E ( k ) = 1 ± iE − ξ k ± i/ τ . (S2)For calculations in the momentum representation, we use the approximation ξ k = υ F ( | k | − k F ) , which breaks downin the vicinity of the Fermi momentum. When working in the real space, we assume a parabolic dispersion of theelectron spectrum: ξ k = k / m − E F .In order to calculate L , we need to draw all possible diagrams with the interaction vertices λ ph and λ , and averagethem over disorder. It is convenient to calculate ladders of repulsive interaction lines λ first and then insert thecorresponding block (denoted as Π ) between the attractive phonon lines λ ph . Summing the corresponding ladder, weobtain L = Π1 − λ ph Π /ν (S3)As the block Π is inserted between the phonon lines, energy cutoff at the Debye frequency ω D is implied at its edges.Equation (S3) allows to express the critical temperature in terms of Π through the relation νλ − ph = Π( T c ) . (S4) Ballistic disorder ladders
In the following calculation we will need the expression for the “ballistic” cooperon and diffuson C ( q , ω ) derived atarbitrary values of ql , ωτ (but we still assume that q (cid:28) k F and ω (cid:28) E F ). Taking E > and E − ω < , we get forone step of the ladder [69]: f q ( ω ) = ν πντ (cid:90) d Ω4 π (cid:90) dξ iE − ξ + i/ τ i ( E − ω ) − ξ − υq − i/ τ = 1 ql arctan ql ωτ . (S5)Summing the geometric series of the diffusive ladder, we obtain C ( q , ω ) = 12 πντ − f q ( ω ) ; C (0 , ω ) = 12 πντ ωτωτ . (S6) Figure S1. Diagrammatic equation for the renormalized Cooper vertex υ ( E ) . Zigzag lines stand for the repulsive interaction λ . Blocks with no impurity lines between the interaction lines are also included. The outer Green functions are not includedinto the expression for υ ( E ) . Renormalization of the phonon vertex
In order to deal with logarithmic contributions originating from various energy intervals, it is convenient to introducethe renormalized phonon vertex υ ( E ) defined as the sum of the sequence of diagrams shown in Fig. S1. In brief, υ ( E ) takes into account ladders of the interaction lines (repulsion constant λ ), which are known to be responsible for the“Tolmachev logarithm” (Morel-Anderson pseudopotential) renormalisation [61–63]. In the quasiballistic region it isimportant to account for the diagrams, where the diffusive ladder may be absent (no impurity lines). Since the vertexcontains the photon interaction, the energy arguments in the pair of Green functions adjacent to the vertex shouldbe smaller than ω D . This property is taken into account by introducing the step function θ ( ω D − | E | ) to the firstterm of the series and restricting integrations over internal energies in the other terms (see below). The renormalisedphonon vertex then takes the form υ ( E ) = (cid:20) πντ | E | τ C (0 , | E | ) (cid:21) u ( E ) = 1 + 2 Eτ Eτ u ( E ) , (S7)where u ( E ) = θ ( ω D − | E | ) − λν T ω D (cid:88) E (cid:48) πνE (cid:48) + (cid:18) − λν (cid:19) T ω D (cid:88) E (cid:48) πνE (cid:48) E F (cid:88) E (cid:48)(cid:48) πνE (cid:48)(cid:48) + · · · = θ ( ω D − | E | ) − λ log ω D /T λ log E F /T . (S8) Anderson theorem
In the leading no-crossing approximation, Π is given by the diagram depicted in the Fig. S2 and is given by Π ( T ) = 2 πντ ω D (cid:88) E f (2 | E | ) υ ( E ) = ν λ log E F /ω D λ log E F /T log ω D T . (S9)This expression appears to be disorder-independent, which leads to the insensitivity of the critical temperature topotential disorder in the leading order (Anderson theorem) [65–67]. Solving Eq. (S4) with
Π = Π , we get the standardBardeen-Cooper-Schrieffer (BCS) expression (5) with the renormalised coupling constant λ BCS given by Eq. (6).
Crossing corrections to Π( T ) Contributions to Π beyond the non-crossing approximation are responsible for the shift of T c . Assuming that δ Π issmall and linearizing, we get the following equation for δT c in the first order: νλ − ph = Π ( T c + δT c ) + δ Π( T c ) . (S10)Hence we get for the perturbative shift of T c : δT c T c = δ Π ν (cid:18) λ log E F /T c λ log E F /ω D (cid:19) = δ Π ν (cid:18) λ ph λ BCS (cid:19) . (S11) Figure S2. Cooper bubble Π in the no-crossing approximation. R RAA
Figure S3. Hikami box H ( q, E, E (cid:48) ) made of four Green functions. In general, account for the renormalisation effects can be done with the help of Eq. (S11) and insertion of renormalisedCooper vertices υ ( E ) into the ends of the diagrams, which describe the correction to the Cooper bubble, δ Π . Thisprocedure leads to Eqs. (7) and (15).Applying this technique to the ballistic vertex correction reproduces the result obtained in the Letter by interpretingthis correction as a shift of the bare Cooper-channel constant δλ c and expanding Eqs. (5) and (6). On the other hand,applying the same technique to corrections originating at energies E < ω D (as the main part of the 2D diffusiveFinkel’stein-Ovchinnikov correction) leads to the cancellation of the renormalisation factors. Momentum-space calculation of the critical temperature shift
Below we sketch the derivation of Eqs. (7) and (8), which represent the contribution of inelastic diagrams (depictedin Fig. 2) to the T c shift. The calculation is done in the momentum representation in terms of ballistic diffusons andcooperons [see Eq. (S6)] to assess the crossover to the ballistic region.The first diagram in Fig. 2 represents a correction δ Π a to be inserted between the phonon lines in the Cooperladder. When substituted to Eq. (S11) it results in Eq. (7) with I ( a ) E,E (cid:48) = τd (cid:88) q z (cid:90) d q (cid:107) (2 π ) f q ( E + E (cid:48) ) − f q ( E + E (cid:48) ) , (S12)where Eqs. (S5) and (S6) were used and summation over diffusive modes in the film geometry is implied [to be replacedby usual 3D integration (cid:82) ( d q ) when studying crossover from 3D diffusion to 3D ballistics]. The numerator in Eq.(S12) represents two triangular Hikami boxes in the diagram, while the denominator corresponds to the Cooperonladder.Calculation of the second diagram in Fig. 2 involves computation of the ballistic Hikami box made of of four Greenfunctions (see Fig. S3), which is given by H ( q, E, E (cid:48) ) = 4 πντ f q ( E + E (cid:48) )[1 − f q ( E + E (cid:48) )](1 + 2 Eτ )(1 + 2 E (cid:48) τ ) . (S13)Then the contribution of the second diagram is given by Eq. (7) with I ( b ) E,E (cid:48) = τd (cid:88) q z (cid:90) d q (cid:107) (2 π ) f q ( E + E (cid:48) ) [1 − f q ( E + E (cid:48) )][1 − f q ( E + E (cid:48) )] , (S14)where the denominator originates from two diffusons in the central part of the diagram and the numerator is theHikami box (S13) multiplied by two additional f q ( E + E (cid:48) ) factors, stemming from the integrals of the “bubbles” G E ( p ) G − E (cid:48) ( p + q ) and G − E ( p (cid:48) ) G E (cid:48) ( p (cid:48) − q ) .Finally, in order to be able to trace a crossover to the ballistic region, one should also include the diagram obtainedfrom the second diagram in Fig. 2 by leaving only one out of the two diffusons encircling the interaction line. Thatleads to Eq. (7) with I ( b (cid:48) ) E,E (cid:48) = 2 τd (cid:88) q z (cid:90) d q (cid:107) (2 π ) f q ( E + E (cid:48) ) [1 − f q ( E + E (cid:48) )]1 − f q ( E + E (cid:48) ) . (S15)Finally, summing Eqs. (S12), (S14), and (S15), one arrives at Eq. (8).3 Figure S4. The central part of elastic diagrams to the Cooper susceptibility in the exact-eigenstates representation.
Elastic diagrams
The central part δP elastic of elastic diagrams is depicted in Fig. S4, where we work in terms of the exact eigenstates(labeled by a, a (cid:48) ) of the Hamiltonian in the presence of disorder. The corresponding analytical expression is: δP elastic = T (cid:88) E,E (cid:48) (cid:88) a,a (cid:48) V aa (cid:48) G a ( E ) G a ( − E ) [ G a ( E ) + G a ( − E )] G a (cid:48) ( E (cid:48) ) , (S16)where the matrix element of the interaction V aa (cid:48) = − (1 − s )( λ/ν ) (cid:82) d r | φ a ( r ) | | φ a (cid:48) ( r ) | includes both Fock (exchange)and Hartree terms (with spin degeneracy factor s = 2 in the latter). Here φ a ( r ) are the wavefunctions correspondingto the energies ξ a . The Matsubara Green function in this representation is G a ( E ) = 1 / ( ξ a − iE ) . After some algebra,Eq. (S16) can be rewritten [44, 45] in the form δP elastic = T (cid:88) E iE (cid:88) E (cid:48) (cid:88) aa (cid:48) V aa (cid:48) (cid:2) G a ( E ) − G a ( − E ) (cid:3) G a (cid:48) ( E (cid:48) ) , (S17)where one recognises corrections to the Green functions at coincident points ( δG E and δG − E ) summed over Matsubaraenergies with a factor T / (2 iE ) . Adding renormalised vertices υ ( E ) [Eq. (S7)] and substituting to Eq. (S11), one arrivesat the expression δT elastic c T c = (cid:18) λ ph λ BCS (cid:19) iT (cid:88) E = E n u ( E ) E δG E − δG − E ν . (S18)Now using the analyticity property, which relates the Matsubara Green function with the real-time retarded Greenfunction G R (at coinciding points in our case), G E = 1 π (cid:90) dε Im G R ( ε ) ε − iE = − (cid:90) dε ν ( ε ) ε − iE . (S19)One can finally express [39] the result for the contribution of elastic diagrams to the T cc