Superconducting size effect in thin films under electric field: mean-field self-consistent model
SSuperconducting size effect in thin films under electric field: mean-field self-consistentmodel
P. Virtanen, A. Braggio, and F. Giazotto
NEST, Istituto Nanoscienze-CNR and Scuola Normale Superiore, I-56127 Pisa, Italy ∗ We consider effects of an externally applied electrostatic field on superconductivity, self-consistently within a BCS mean field model, for a clean 3D metal thin film. The electrostaticchange in superconducting condensation energy scales as µ − close to subband edges as a functionof the Fermi energy µ , and follows 3D scaling µ − away from them. We discuss nonlinearities beyondgate effect, and contrast results to recent experiments. I. INTRODUCTION
Quantum oscillations in superconducting propertiesdue to size quantization in thin films were predictedearly , and they were later observed in metallicfilms . Modification of superconducting properties bychanging the electron density by electrostatic fields wasalso observed, and is best studied in high-Tc super-conductors where the charge density can be low enoughto enable efficient gating. Generally, modifications ofcritical temperature T c and critical current I c have beenreported. Modification of I c only was also recently re-ported in Refs. 14 and 15 in metallic thin-film samples,but the proper interpretation in the latter is still unclear.Electrostatics of superconductors is an old problem(see e.g. Ref. 16 for a historical review), and the effectof electric fields on superconducting surfaces are theoret-ically discussed in several works. In these, effects onthe amplitude of superconductivity ( T c ) are usually re-lated to modulation of electronic density of states (DOS),which is also what contributes to the quantum size ef-fects. A common approach is to consider “surface dop-ing” and assume the DOS is modified within a Thomas–Fermi screening length from the surface. Self-consistentlyscreened calculations in superconductors have been pre-viously discussed in Refs. 25–27, in a different context.For the normal state, there is a large literature on micro-scopic calculations with surface screening, which are rou-tine today e.g. using density functional theory. Mod-ification of I c on the other hand is often assumed to comefrom changes in the vortex surface pinning potential. In a simple picture, a static electric field appears asa perturbation of the potential that confines electronswithin the thin film. Static fields generally extend upto a screening length from the surface, and so their ef-fect decreases towards high charge density. Although theeffects increase with the applied electric field, achievablefield magnitude is limited by electric breakdown (e.g. viafield emission ).Electrostatic gating of superconductivity in the BCSmean field picture relies on electron-hole asymmetrywithin an energy window determined by the order pa-rameter and Debye frequency centered at the Fermilevel. In a simple clean thin-film model, strong asym-metry naturally exists in the form of the steplike multi-band 2D DOS, which also gives rise to the quantum size effect, and the picture also extends to weakly disor-dered samples. The only question is to what degree theDOS asymmetry is retained, even though sharp featuresin the DOS are smeared by disorder, and when sam-ples cannot be significantly gated (metallic regime 15),since the Fermi level is not necessarily fixed at a sensi-tive point. Regardless, sharp DOS features can increasethe charge density range in which electrostatic effects arelarge enough to be observed. Motivated by the recentexperimental results 15 where large effects were seen, werevisit the problem.In this work, we write down and solve a simple mean-field model for superconductivity in thin films under elec-tric fields, including self-consistent screening. We pointout connections between the dependence of electrostaticenergy on superconductivity and modulation of super-conductivity by fields, and discuss applicability of “sur-face doping” models in this picture. We also discuss towhat degree nonlinear effects beyond linear electrostaticgating could appear in strong fields. We conclude thateffects such as observed in Ref. 15 likely are not presentin the model considered.The manuscript is structured as follows. In Sec. II weintroduce the mean-field model considered and discussresults obtained for the electric fields and modulationof superconducting properties. Sec. III concludes withdiscussion. II. MEAN-FIELD MODEL
Self-consistent electrostatic screening and the size ef-fect on superconductivity in a clean superconductingmetal in a simple mean-field approximation is convenientto consider starting from a Hartree–Bogoliubov free en-ergy. It can be obtained by decoupling a long-ranged Coulomb and a (retarded) superconducting con-tact interaction via Hubbard–Stratonovich transforma-tions, and considering only the classical saddle point in a r X i v : . [ c ond - m a t . s up r- c on ] M a r the static limit: F [∆ , φ ] = − T Tr ln G − (1)+ (cid:90) d r (cid:16) ρφ − (cid:15) ∇ φ ) + (cid:90) T d τ | ∆( τ ) | λ ( τ ) (cid:17) , G − = − iω + [ ˆ k m − U − µ − eφ ] τ + ∆( ω ) τ . (2)Here, G is the electron equilibrium Green function, U is abackground potential, µ chemical potential, φ is equiva-lent to the static electric potential, ∆ the superconduct-ing order parameter, and ρ ion and external charge den-sity. The electron charge is − e and we use units with (cid:126) = k B = 1. The first term in the free energy is theelectronic contribution, and the second part contains theelectrostatic and superconducting mean-field contribu-tions. In the absence of currents and magnetic field, atsaddle point with suitable gauge ∆ can be chosen real-valued and the values of vector potential and phase arezero. Above, φ has to be taken as the saddle-point so-lution, which as typical for variational Poisson does notminimize F .Variations vs. φ and ∆ give the Poisson and BCS self-consistency equations: − (cid:15) ∇ φ ( r ) = ρ ( r ) − en e ( r )= ρ ( r ) + eT (cid:88) ω n tr τ G ( r , r , ω n ) , (3)∆( r ) = − T (cid:88) | ω n | <ω c λ ( r ) tr τ G ( r , r , ω n ) , (4)where G satisfies the Gor’kov equations G − G = 1 underthe self-consistent potentials. We also here consider aBCS weak-coupling model, with ∆( ω ) = ∆ θ ( ω c − | ω | ),with the coupling λ taken as constant and the cutoff ω c similar to the Debye frequency. In bulk, the BCS gapequation is then directly ∆ = 2 ω c e − / ( N λ ) with N theDOS per spin at Fermi level.For uniform system, expanding G in Eq. (3) to low-est order in φ results to (cid:15) RPA ( q ) q φ ( q ) = δρ ( q ), where (cid:15) RPA ( q ) = (cid:15) − e q Π( q ; ∆) is the self-consistent static di-electric function of a clean superconductor. In thismodel, the static fields are screened, and external chargedensity affects the electronic DOS, but not the Coulombeffect to λ . The latter is due to considering mean-fieldwith the decoupling assumed; corrections appear fromfluctuations of φ (see e.g. Ref. 42 for explicit calcula-tions), or on mean-field level with different decoupling .Aiming to describe effects on a qualitative level, wenow consider a simplified model, similar to those used inseveral previous studies of the quantum size effect in su-perconducting thin films. A confining potential U ( r ) istaken to be an infinite quantum well at | x | < L/
2, whichsupports some number of populated 2D electronic sub-bands (see Fig. 1). In a static problem without currents,the electric field is perpendicular to the metal surface,
FIG. 1. (a) Schematic of superconducting quantum well ofthickness L with infinite size in other directions, supportingseveral populated subbands, with electric fields E ± imposedon the surfaces. (b) Charge density (6) in a superconductoris determined by the density of states and a occupation factorbroadened by the superconducting interaction. and the problem is inhomogeneous only in x -direction.Moreover, we take as a variational Ansatz ∆ spatiallyconstant inside the well; the resulting energies will thenbe upper bounds to the exact solutions.With these assumptions, the problem is elementaryand mostly given by known results, and can be solvedwithout further approximations. First, G = − (cid:90) ∞−∞ d ξ A N ( r , r (cid:48) , ξ ) (cid:32) u iω − (cid:15) + v iω + (cid:15) uviω − (cid:15) − uviω + (cid:15)uviω − (cid:15) − uviω + (cid:15) v iω − (cid:15) + u iω + (cid:15) (cid:33) , (5)where A N is the normal-state spectral function (perspin), u, v = [ (1 ± ξ(cid:15) )] / and (cid:15) = (cid:112) ξ + ∆ . Dueto the spatial symmetry, the problem reduces to one di-mension. The normal-state DOS per volume is ν ( ξ ) = V (cid:82) d r A N ( r , r , ξ ) = (cid:80) ∞ n =1 mπL θ ( ξ − ξ n ) where ξ n are the2D subband edges and V the film volume. The subbandsand the potential φ are obtained from the Schr¨odinger–Poisson problem, Eq. (3) with n e [ φ ] = ∞ (cid:88) n =1 m | ψ n | γ ( ξ n ) , (6)[ − m ∂ x − µ − eφ ( x )] ψ n = ξ n ψ n , ψ n ( ± L , (7)where ψ n ( x ) are the transverse wave functions of the2D subbands. Here, γ describes the contribution to thecharge from each subband: n ( ξ ) = f ( ξ ) + T (cid:88) | ω | <ω c ξ ∆ ( ω + ξ + ∆ )( ω + ξ ) , (8) γ ( ξ ) = (cid:90) ∞ ξ d ξ (cid:48) π n ( ξ (cid:48) ) T =0 ,ω c = ∞ (cid:39) (cid:112) ξ + ∆ − ξ π , (9)where n ( ξ ) → u f ( (cid:15) ) + v (1 − f ( (cid:15) )) for ω c → ∞ and f is the Fermi function. The occupation factor n is broad-ened by the interactions in a window ∆ around the Fermilevel, with the deviation from the Fermi function start-ing to decay more rapidly beyond the interaction rangeat | ξ | (cid:38) ω c . Variations in the DOS within this windowcontribute to the charge response of the amplitude of su-perconductivity (see Fig. 1). To be specific, we assume an external charge densityoutside the sample (e.g. capacitor plates with constantcharge density) such that the amplitudes of the electricfields at the surfaces are fixed, − ∂ x φ ( ± L ) = E ± . Nu-merically, the nonlinear Poisson problem can be solvediteratively, for a fixed value of ∆.The condensation energy f ns (∆) = ( F [∆ , φ ∗ [∆]] − F [0 , φ ∗ [0]]) / V per volume for fixed ∆ now depends onlyon the density of states. Via direct calculation, f ns (∆) = 1 V (cid:90) ∆0 d∆ dd ∆ F [∆ , φ ∗ ] , (10)where we note that dd ∆ F [∆ , φ ∗ ] = ∂ ∆ F [∆ , φ ∗ ] at the sad-dle point φ ∗ . Further, f ns (∆) = ∆ λ − (cid:90) ∆0 d∆ (cid:88) | ω | <ω c (cid:90) ∞−∞ d ξ ν ( ξ ) T ∆ ω + ξ + ∆ (11) ≡ ∆ λ − m πL (cid:90) ∆0 d∆ ∆ ∞ (cid:88) n =1 g ( ξ n ∆ ) , (12) g ( y ) = T ∆ (cid:88) | ω | <ω c (cid:90) ∞ y d x x + 1 + ( ω/ ∆) . (13)Here, g ( y ) → (cid:82) ∞ y d x √ x π arctan ω c / ∆ √ x for T = 0 andfurther g ( y ) → arsinh( ω c / ∆) − arsinh ( y ) for ω c (cid:29) ∆, T = 0. For T = 0 and ω c → ∞ , f ns (∆) → ∆ λ − m ∆ πL (cid:88) ξ n <ω c [ η ( ω c / ∆) − η ( ξ n / ∆)] , (14)where η ( y ) = arsinh y + ( (cid:112) y + 1 − | y | ) y . The self-consistent value ∆ ∗ is attained at a solution of f (cid:48) ns (∆ ∗ ) =0. Separating out an electrostatic contribution by sub-tracting the result for some reference potential φ : δf ns ≡ f ns (∆; φ ∗ ) − f ns (∆; φ ) (15)= − m πL (cid:90) ∆0 d∆ ∆ ∞ (cid:88) n =1 (cid:16) g ( ξ n ∆ ) − g ( ξ (0) n ∆ ) (cid:17) (16) (cid:39) mL ∞ (cid:88) n =1 δγ ( ξ n ) δξ n , T = 0 , ω c → ∞ , (17)where δγ ( ξ ) ≡ γ ( ξ, ∆) − γ ( ξ, ∆ = 0) from Eq. (9),and δξ n ≡ ξ n − ξ (0) n . The result (16) includes bothgating and any nonlinear effects (e.g. energy associ-ated with quantum capacitance) in strong electric fields.Note that the above electrostatic energy contribution de-pends on the electric fields only via ξ n = ξ n [ φ ], an exactstatement in the model here. It is also possible to express the electrostatic energydirectly in terms of the self-consistent electric field, atsmall field strengths. Consider expansion of the elec-tronic energy around a reference electric potential, con-sidering small φ = φ − φ and ρ = ρ − ρ : − T Tr ln G − + T Tr ln G − | φ = φ = (cid:90) d r ( − e ) n e [∆ , φ ]( r ) φ ( r )+ 12 (cid:90) d r d r (cid:48) e Π[∆ , φ ]( r , r (cid:48) ) φ ( r ) φ ( r (cid:48) ) + . . . , (18)where n e is the electron density and Π the density re-sponse function. Solving the resulting saddle-point equa-tion for φ and substituting the solution into F gives,after integration by parts: f [∆] = f [∆ , φ ] + 1 V (cid:90) d r (cid:16) ρ φ + 12 ρ φ , ∗ (cid:17) (19)= f [∆ , φ ] + (cid:88) ± ∓ (cid:15) E ± L [ φ + 12 φ , ∗ ] x = ± L + C , (20) ρ = − (cid:15) ∇ φ , ∗ − (cid:90) d r (cid:48) e Π[∆ , φ ]( r , r (cid:48) ) φ , ∗ ( r (cid:48) ) , (21)where C is independent of ∆. In this order of expan-sion in small φ , the additional electrostatic field en-ergy in (19) coincides with the standard expression. Thelinear term ∼ ρ φ describes a gate effect on super-conductivity, which in this approach we see to be re-lated to the ∆-dependence of the equilibrium potential φ . Using Eq. (21) the quadratic part can be expressedas ∼ φ (cid:15) RPA φ . It corresponds to a (quantum) capac-itance modulation by superconductivity. The re-sult (19) can be directly used for computing δf ns (∆) (if δφ ≡ φ [∆] − φ [0] is known) and is equivalent with (16)in the small-field limit. However, due the ∆-dependenceof φ it is not necessarily very practical to compute, assolving the nonlinear Poisson problem is still required.However, the above expressions can be used as a consis-tency check.As noted above, we consider charge density ρ = ρ + ρ where ρ outside the sample fixes the electric field atthe surface. Finally, we need to specify the background(“ion”) charge density ρ . The electric potential due to ρ together with U gives the pseudopotential for the electronsystem. For simplicity, unless otherwise mentioned, be-low we assume ρ = en e [∆ = 0 , φ = 0 , µ ], which resultsto φ = 0 as the solution in the normal state, and µ becoming the parameter that fixes the charge density inthe normal state. This is of course a crude toy model ofthe surface electron behavior, even within Hartree-typemodels , but likely modifies mainly the precise positionsof the subbands but not the main qualitative features ofthe effect of the screening of external charges on super-conductivity. A. Size effect in electric field
In the same way as the variation in thickness, gatingby a surface electric field can in principle make a singlesubband edge ξ n to cross the Fermi level, which resultsto a jump in superconducting properties. Such responsecan be larger than in bulk material, and is not capturedby “surface doping” models often used for the electricfield effect, where the LDOS ν ( x, ξ ) is assumed tobe modified in a surface layer of thickness of a screeninglength λ T F according to bulk relations. In addition, thefield screening is not exactly Thomas-Fermi type, butthis causes less relevant changes than the difference inthe DOS.The order of magnitude of δf ns can be estimated ina Thomas–Fermi approximation. Taking φ ( L/ x (cid:48) ) (cid:39)− Eλ T F e x (cid:48) /λ TF for x (cid:48) < λ T F = (cid:112) (cid:15) / ( e ν F ), δξ n (cid:39) (cid:104) n | ( − e ) φ | n (cid:105) = λ T F
L eE + q (2 λ T F k n ) , (22)where k n = πn/L and q ( z ) = z / (1+ z ). From Eq. (17),and keeping only the smallest | ξ n | < ω c , δf ns (cid:39) | f ns, D | eEa (cid:112) ξ n + ∆ + | ξ n | π k F L ) q (2 λ T F k n ) , (23)where f ns, D = − mk F π ∆ is the bulk 3D condensationenergy and a = 4 π(cid:15) / ( me ) the Bohr radius. The aboveresult is valid in the leading order in ∆, as φ is assumedto be independent of it. The factor q (2 λ T F k n ) in real-ity depends on details of the screening, and below weconsider it as a constant of order of magitude 1.Including the next-order eigenvalue perturbation δξ (2) n in (16) and considering terms of order E gives thesecond-order correction, δf (2) ns (cid:39) mL ∞ (cid:88) k,n =1 δγ ( ξ n ) − δγ ( ξ k ) ξ n − ξ k |(cid:104) k | ( − e ) φ | n (cid:105)| , (24)where n = k means the limit ξ k → ξ n . This energy con-tribution is associated with the change Π[∆ , − Π[0 , .However, it is of the same order in ∆ as the changeΠ[0 , φ [∆]] − Π[0 ,
0] due to the ∆-dependent shift in theself-consistent equilibrium potential, which we have ne-glected here. As a consequence, Eq. (24) is not theonly contribution to the E term, and solving the self-consistent electrostatic problem is in general required. Conversely, calculation of the effect of superconductiv-ity on the dielectric function requires taking the self-consistency of ∆ = ∆ ∗ [ φ ] into account. The overlap factor q above depends on how accuratethe Thomas-Fermi screening assumption is close to thesurface. For the simple problem here, we can solve thePoisson equation numerically. Such a solution is illus-trated in Fig. 2a for λ T F (cid:28) L . Since λ T F ∼ k F , x / L e / (a) 10
25 0 25 E + L /0.20.00.2 f n s / | f n s , D | (b) FIG. 2. (a) Self-consistent electric potential φ and its mod-ulation δφ = φ (∆) − φ (∆ = 0) for E − = 0, E + = 3 . / nm, L = 10 nm, µ = ξ (0)18 + 0 . / ( mL ) = 1 .
22 eV, ∆ = 760 µ eV, ω c = 34 meV, T = 0. (b) Change δf ns in the condensationenergy at fixed ∆, in units of the 3D bulk condensation en-ergy f ns, D = − ν D ( µ ) | ∆ | . Results from Eq. (16) (solid),the small-field expression (20) (dashed), and Eq. (23) with q ( z ) = 1 (dotted) are shown. screening is not fully exponential, but the electric po-tential exhibits 1 /k F oscillations. The correction δφ = φ (∆) − φ (∆ = 0) to the equilibrium electrostatic poten-tial from superconductivity is small in the high chargedensity regime considered here. The chemical potentialis chosen to be close to a subband edge in the figure.The corresponding dependence of δf ns on the electricfield magnitude is shown in Fig. 2b, together with thecorresponding result from Eq. (23). The electrostatic en-ergy expression (20) is also shown, and coincides withthe exact result in the small-field regime. Generally, theelectric field effect is appreciable only for E + L (cid:38) µ . Inthe estimate from Eq. (23), we here set q ( z ) = 1, to ac-count for the expectation that likely for the true screen-ing length λ T F k F (cid:38)
1. The second-order correction (24)is neglibile for these parameters, being higher order in λ T F eE/ ( Lω c ), and the nonlinearity visible in the resultoriginates from g ( ξ ).The linear gating effect can be suppressed with acharge-neutral field configuration E + = E − = E , whichcorresponds to an experiment using floating gate elec-trodes (i.e. placing the system inside a plate capacitor).The result from Poisson equation for this case is shownin Fig. 3, together with the result for δf ns . Imposing thefield on both sides produces a larger δφ . However, as thelinear contribution to the free energy cancels, the mod-ulation δf ns arises from the next-order effect and is anorder of magnitude smaller than with the gate effect. Al-though the energy can still be expressed also via Eq. (20),the eigenvalue perturbation result (24) does not agree aswell, as expected.Whether electrostatic effects are significant dependson how large the modulation δf ns is compared to f ns .The dependence of their ratio on the chemical poten- x / L e / (a) 10
25 0 25 E + L /0.000.020.04 f n s / | f n s , D | (b) FIG. 3. Same as Fig. 2, for the symmetric field configuration E + = E − . Dotted line indicates Eq. (24). [ /( mL )]10 f n s / | f n s , D | c
34 meV
FIG. 4. Electrostatic condensation energy increase δf ns vs µ for E + = 0 .
76 V / nm, other parameters as in Fig. 2. Dashedline indicates Eq. (23), taking q ( z ) = 1. tial, and hence charge density, is shown in Fig. 4, at arelatively large external field. The magnitude of δf ns depends strongly on whether the chemical potential islocated near a band bottom, where the effect is ampli-fied (see Fig. 1), which produces the oscillations visible inFig. 4. When µ is close to a subband bottom, the magni-tude appears to be captured well by Eq. 23 (dashed line).When the chemical potential is not close to a band bot-tom, depending on the ratio between the subband spac-ing and the cutoff ω c , the electric field effect can vary byorder of magnitude. Note that as long as | ξ n | (cid:28) ω c forsome n , the result is dominated by the smallest ξ and thecutoff ω c < ∞ is of limited importance. The sum (16) isconvergent also for ω c → ∞ . However, these results arebased on the simple weak-coupling model for supercon-ductivity, and the precise shape of the modulation maybe sensitive to details of the interaction. Regardless, fromthe above results one can see that the relative magnitudeat resonance scales as ∝ ∆ /µ , and not as (∆ /µ ) as onewould expect for the amplitude response in 3D bulk .Away from the subband edge resonances, δf ns ∝ (∆ /µ ) .The self-consistent value of ∆ ∗ , f (cid:48) ns (∆ ∗ ) = 0, is shown
13 14 15 16 17 k F L /0.60.81.0 * / (a) 1550 1600 1650 1700 [ /( mL )]0.60.81.0 (b) E ( Vnm )0.00.763.8
FIG. 5. Self-consistent ∆ ∗ ( T = 0) vs (a) L and (b) µ , with g = N , D λ = 0 .
14 fixed and other parameters as in Fig. 2.Here, ∆ = 2 ω c e − /g . in Fig. 5a as a function of the film thickness, showingthe well-known quantum size effect. The correspondingdependence on the chemical potential is shown in Fig. 5b,for several values of the external electric field. In thisfigure, it is obvious that the electrostatic field simplygates the system: the size effect physics is dominatedby the ξ n closest to the chemical potential, so that thegate-induced shift δξ n is identical to a chemical potentialshift − δµ .The above discussion can be compared to a surfacedoping model, where the DOS is assumed to change ina surface layer of thickness λ T F , and the system is con-sidered a superconducting bilayer in the Cooper limit.In such a model, δF ns /F ns ∼ − λ TF L δ ν F λ ∼ δν F λ TF λν F L .The general form of Eq. (23) can then be recoveredby including (ad-hoc) the main features of the multi-band DOS in δν F . This can be done by writing δν F ν F = πk F L ∂ ξ n θ ∆ ( ξ n ) λ T F eE , where θ ∆ ( ξ ) is a broadened unitstep function with width ∆. For the problem here, al-though the actual form of δν ( x, E ) is different, this sim-pler model captures the main effects. Surface dopingmodels have indeed been successful in understanding pre-vious experimental results. B. Superfluid weight
The effect of the electrostatic field on the phase fluc-tuations can be studied via the superfluid weight D sij , which describes the free energy cost of superflow ∆( r ) ∝ e i A · r : F [∆ ∗ , φ ∗ , A ] = F [∆ ∗ , φ ∗ ,
0] + (cid:126) V D sij A i A j + . . . . (25)The “vector potential” A describing the superflow canbe introduced in Eq. (2) by replacement ˆ k (cid:55)→ ˆ k + A .The calculation of D s is standard for multiband BCSsuperconductor. Since the current operators in y/z di-rections do not here couple different bands, the result for i, j ∈ { y, z } is D sij = δ ij (cid:80) n n s ( ξ n ) / ( mL ) where n s ( ξ n )is the BCS superfluid density: n s ( ξ ) = 2 m (cid:90) ∞ ξ d ξ (cid:48) π [ n ( ξ (cid:48) ) + ( ξ (cid:48) − ξ ) f (cid:48) ( (cid:15) (cid:48) )] , (26)where n ( ξ ) is given by Eq. (8) and (cid:15) (cid:48) = (cid:112) ( ξ (cid:48) ) + ∆ .As well known, n s ( ξ ) → n e ( ξ ) = 2 mγ ( ξ ) at T = 0. Theelectrostatic modulation of the superfluid stiffness is thensimilar to that of the charge density, i.e., small in themetallic regime. Similar conclusion then applies to thephase stiffness, and quite likely also to the phase-slip en-ergy barrier . These results however apply in the cleanlimit. III. DISCUSSION AND CONCLUSIONS
We discussed an elementary BCS/Hartree–Bogoliubovmean field model for the size effect under self-consistentelectrostatic fields in superconducting thin films, andstudied it based on numerically exact solutions. As thesize modulation in superconducting properties decays rel-atively slowly with increasing charge density, it increasesthe response to applied electric fields, effectively changingthe small parameter from (∆ /µ ) to ∆ /µ for fine-tunedvalues of µ , also in films thick compared to the screeninglength.The mean-field approach likely is not useful in de-scribing atomically thin, or strongly disordered and re-sistive samples, where fluctuation effects matter. Phase–plasmon fluctuation effects in principle can be includedin the approach above in a standard way by expanding inRe ∆, Im ∆ and V = − iφ around the mean-field solution.A priori, in view of some existing results, however,it’s not clear why such corrections would depend stronglyon the external electric field.Large electrostatic size effects in thin-film systems areexpected to be visible mainly in relatively low chargedensities, e.g. semiconducting materials. As noted in previous works, it appears likely this is a main effect inhigh-Tc superconductors. The modulation of screeningby superconductivity will also appear in proximity sys-tems, e.g. in semiconductor/superconductor hybrids recently considered as Majorana fermion platforms.With regard to the large modification of superconduct-ing critical current by electric fields reported in Refs. 15,it then appears somewhat less likely these results can beunderstood with electrostatic effects of the type discussedabove. At metallic densities ∆ /µ ∼ − , electrostaticeffects in the model here, even at a sharp DOS feature,likely can only give | δf ns /f ns, D | (cid:46) − , which is toosmall to cause large measurable effects. It appears un-likely this is easily rectified by lifting some of the approx-imations we made. This is simply a manifestation of the“Anderson theorem”: the amplitude of conventional su-perconductivity is insensitive to time-reversal symmetricperturbations, and suppressing it requires perturbationslarge compared to µ , which are usually not achievable inthe metallic regime below the electrical breakdown field.Also, as the linear gate effect generally should dominatenonlinearities, whether superconductivity is suppressedor enhanced depends on the sign of the electric field, quiteunlike in Refs. 15. Previously, reduction in the criticalcurrent by an applied field was attributed to modifica-tion of vortex pinning. In Ref. 15 effects appear alsoin aluminum strips with lateral size (cid:46) ξ , making this ex-planation less favorable. In the clean-limit model here,it also appears unlikely the phase slip rates would be sig-nificantly affected.In summary, we considered effects of electrostaticfields on superconductivity self-consistently within a BCSmodel, connected them to questions about electrostaticenergy, and commented on their relation to recent ex-periments. We obtain results for the size and externalelectric field modulation of superconductivity, and con-trast results to a surface doping model. Expanding aboutthis mean field solution, considering electric field effectson phase slips and phase fluctuations is possible. Experi-mentally, the effects are best visible in low charge densitysystems, e.g. semiconductor hybrid structures. ∗ [email protected] J. M. Blatt and C. J. Thompson, Phys. Rev. Lett. , 332(1963). D. S. Falk, Phys. Rev. , 1576 (1963). A. A. Shanenko, M. D. Croitoru, and F. M. Peeters, Phys.Rev. B , 014519 (2007). A. A. Shanenko, J. A. Aguiar, A. Vagov, M. D. Croitoru,and M. V. Miloˇsevi´c, Supercond. Sci. Tech. , 054001(2015). B. G. Orr, H. M. Jaeger, and A. M. Goldman, Phys. Rev.Lett. , 2046 (1984). Y. Guo, Y.-F. Zhang, X.-Y. Bao, T.-Z. Han, Z. Tang, L.-X.Zhang, W.-G. Zhu, E. G. Wang, Q. Niu, Z. Q. Qiu, J.-F.Jia, Z.-X. Zhao, and Q.-K. Xue, Science , 1915 (2004). Y.-F. Zhang, J.-F. Jia, T.-Z. Han, Z. Tang, Q.-T. Shen, Y. Guo, Z. Q. Qiu, and Q.-K. Xue, Phys. Rev. Lett. ,096802 (2005). X.-Y. Bao, Y.-F. Zhang, Y. Wang, J.-F. Jia, Q.-K. Xue,X. C. Xie, and Z.-X. Zhao, Phys. Rev. Lett. , 247005(2005). J. Mannhart, Mod. Phys. Lett. B , 555 (1992). J. Mannhart, J. G. Bednorz, K. A. M¨uller, D. G. Schlom,and J. Str¨obel, J. Alloys and Compounds , 519 (1993). C. H. Ahn, J.-M. Triscone, and J. Mannhart, Nature ,1015 (2003). D. Matthey, N. Reyren, J. M. Triscone, and T. Schneider,Phys. Rev. Lett. , 057002 (2007). C. H. Ahn, A. Bhattacharya, M. Di Ventra, J. N. Eck-stein, C. D. Frisbie, M. E. Gershenson, A. M. Goldman,I. H. Inoue, J. Mannhart, A. J. Millis, A. F. Morpurgo,
D. Natelson, and J.-M. Triscone, Rev. Mod. Phys. ,1185 (2006). D. Shvarts, M. Hazani, B. Y. Shapiro, G. Leitus,V. Sidorov, and R. Naaman, EPL , 465 (2005). G. De Simoni, F. Paolucci, P. Solinas, E. Strambini, andF. Giazotto, Nat. Nanotech. , 802 (2018); F. Paolucci,G. De Simoni, E. Strambini, P. Solinas, and F. Gia-zotto, Nano Lett. , 4195 (2018); F. Paolucci, G. DeSimoni, P. Solinas, E. Strambini, N. Ligato, P. Virtanen,A. Braggio, and F. Giazotto, Phys. Rev. Applied (2019,accepted). P. Lipavsk´y, J. Kol´aˇcek, K. Morawetz, and E. H. Brandt,Phys. Rev. B , 144511 (2002). B. Y. Shapiro, Phys. Lett. A , 374 (1984). B. Y. Shapiro, Solid State Commun. , 673 (1985). W. D. Lee, J. L. Chen, T. J. Yang, and B.-S. Chiou,Physica C , 167 (1996). L. Burlachkov, I. B. Khalfin, and B. Y. Shapiro, Phys.Rev. B , 1156 (1993). P. Lipavsk´y, K. Morawetz, J. Kol´aˇcek, and T. J. Yang,Phys. Rev. B , 052505 (2006). K. Morawetz, P. Lipavsk´y, J. Kol´aˇcek, and E. H. Brandt,Phys. Rev. B , 054525 (2008). K. Morawetz, P. Lipavsk´y, and J. J. Mareˇs, New J. Phys. , 023032 (2009). G. A. Ummarino, E. Piatti, D. Daghero, R. S. Gonnelli,I. Y. Sklyadneva, E. V. Chulkov, and R. Heid, Phys. Rev.B , 064509 (2017). T. Koyama, J. Phys. Soc. Jpn. , 2102 (2001). M. Machida and T. Koyama, Phys. Rev. Lett. , 077003(2003). B. D. Woods, T. D. Stanescu, and S. Das Sarma, Phys.Rev. B , 035428 (2018). N. D. Lang and W. Kohn, Phys. Rev. B , 4555 (1970). J. Mannhart, D. G. Schlom, J. G. Bednorz, and K. A.M¨uller, Phys. Rev. Lett. , 2099 (1991). J. W. Gadzuk and E. W. Plummer, Rev. Mod. Phys. ,487 (1973). C. J. Adkins and J. R. Waldram, Phys. Rev. Lett. , 76(1968). D. I. Khomskii and F. V. Kusmartsev, Phys. Rev. B ,14245 (1992). D. I. Khomskii and A. Freimuth, Phys. Rev. Lett. , 1384(1995). A. van Otterlo, M. Feigel’man, V. Geshkenbein, andG. Blatter, Phys. Rev. Lett. , 3736 (1995). D. Belitz and T. R. Kirkpatrick, Rev. Mod. Phys. , 261(1994). T. M. Rice, J. Math. Phys. , 1581 (1967). V. Ambegaokar, U. Eckern, and G. Sch¨on, Phys. Rev.Lett. , 1745 (1982). A. van Otterlo, D. S. Golubev, A. D. Zaikin, and G. Blat-ter, Eur. Phys. J. B , 131 (1999). R. E. Prange, Phys. Rev. , 2495 (1963). J. Seiden, J. Phys. France , 561 (1966). P. Morel and P. W. Anderson, Phys. Rev. , 1263 (1962). S. Fischer, M. Hecker, M. Hoyer, and J. Schmalian, Phys.Rev. B , 054510 (2018). H. J. Schulz, Phys. Rev. Lett. , 2462 (1990). P. W. Anderson, J. Phys. Chem. Solids , 26 (1959). M. Tinkham,
Introduction to Superconductivity , 2nd ed.(McGraw-Hill, New York, 1996). K. M. Hong, Phys. Rev. B , 1766 (1975). V. Eyert, J. Comp. Phys. , 271 (1996). The confining potential U is not necessary for the formu-lation, except for enabling the use of the constant-∆ ap-proximation, see Appendix B. Alternatively, one could base the expansion (18) aroundthe normal-state potential φ = 0 and include a third-order term (cid:82) Γ ( x, x (cid:48) , x (cid:48)(cid:48) ) φ ( x ) φ ( x (cid:48) ) φ ( x (cid:48)(cid:48) ). D. J. Scalapino, S. R. White, and S. Zhang, Phys. Rev. B , 7995 (1993). J. S. Langer and V. Ambegaokar, Phys. Rev. , 498(1967). A. Vuik, D. Eeltink, A. R. Akhmerov, and M. Wimmer,New J. Phys. , 033013 (2016). Appendix A: Density response function in thin film
The static density response in a superconducting infi-nite potential well can be found, in a situation transla-tionally invariant vs. y and z (i.e. response to a chargesheet). First, we haveΠ( x, x (cid:48) ) = T (cid:88) ω n tr G ( x, x (cid:48) ) τ G ( x (cid:48) , x ) τ (A1)= 2 (cid:90) ∞−∞ d ξ d ξ A N ( r , r (cid:48) ; ξ ) A N ( r , r (cid:48) ; ξ ) ∗ (A2) × n ( ξ , ∆) − n ( ξ , ∆) ξ − ξ , where the trace and the Matsubara sum has been evalu-ated, and n ( ξ ) = u ξ f ( (cid:15) ξ ) + v ξ (1 − f ( (cid:15) ξ )). The normal-state spectral function for a potential well is A N ( x, x (cid:48) ; ξ ) = ∞ (cid:88) p =1 L sin[ k p ( x + L k p ( x (cid:48) + L δ ( ξ − E p ) , (A3)where k p = πpL , E p = k p m . Then we have,Π( x, x (cid:48) ) = 8 mL ∞ (cid:88) pq =1 sin[ k p ( x + L k p ( x (cid:48) + L × sin[ k q ( x + L k q ( x (cid:48) + L × γ ( E p − µ, ∆) − γ ( E q − µ, ∆) E p − E q , which can be evaluated. Here, the terms p = q imply thelimit E p → E q . Appendix B: Confining potential
In a more realistic model than in the main text, wewould set U = 0 and the electrons would be confined inthe metal film due to the attractive potential from theionic charge density ρ >
0. However, in such calcula-tions the simplifying assumption of a spatially constant∆ is not permissible, as discussed below.The charge density in uniform 3D metal for µ → −∞ (i.e. deep in the vacuum), with constant ∆, is ρ e ( µ, T = 0 , ∆) = (2 m ) / π (cid:90) ∞− µ d ξ (cid:112) µ + ξn ( ξ ) (B1) (cid:39) (2 m ) / π ∆ √− µ . The corresponding Poisson equation in a Thomas-Fermiapproximation becomes ∂ x φ ( x ) (cid:39) e(cid:15) − ρ e ( µ − φ ( x ) , T = 0 , ∆) (cid:39) a (cid:112) φ ( x ) , (B2) ⇒ φ ( x ) = (cid:16) √ ax (cid:17) / , ρ e ( x ) ∝ x − / . (B3) From the solution, we find the electrostatic field fails toconfine the “superconducting” electrons, and an infiniteamount of total charge (cid:82) ∞ x d x ρ e ( x ) leaks to the vacuum,which is unphysical. In the exact solution, the mean field | ∆( r ) ||