Proximity Eliashberg theory of electrostatic field-effect-doping in superconducting films
G. A. Ummarino, E. Piatti, D. Daghero, R. S. Gonnelli, Irina Yu. Sklyadneva, E. V. Chulkov, R. Heid
aa r X i v : . [ c ond - m a t . s up r- c on ] A p r Proximity Eliashberg theory of electrostatic field-effect-doping in superconductingfilms
G. A. Ummarino,
1, 2, ∗ E. Piatti, D. Daghero, R. S. Gonnelli, Irina Yu. Sklyadneva, E. V. Chulkov,
3, 4, 5, 6 and R. Heid Istituto di Ingegneria e Fisica dei Materiali, Dipartimento di Scienza Applicata e Tecnologia,Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129 Torino, Italy National Research Nuclear University MEPhI (Moscow EngineeringPhysics Institute), Kashira Hwy 31, Moskva 115409, Russia Donostia International Physics Center (DIPC),20018 San Sebastian/Donostia, Basque Country, Spain St. Petersburg State University, 199034, St. Petersburg, Russian Federation Departamento de Fisica de Materiales, Facultad de Ciencias Quimicas,UPV/EHU, Apdo. 1072, 20080 San Sebastian/Donostia, Basque Country, Spain Centro de Fisica de Materiales CFM - Materials Physics Center MPC,Centro Mixto CSIC-UPV/EHU, 20018 San Sebastian/Donostia, Basque Country, Spain. Karlsruher Institut f¨ur Technologie, Institut f¨ur Festk¨orperphysik, D-76021 Karlsruhe, Germany
We calculate the effect of a static electric field on the critical temperature of a s-wave one bandsuperconductor in the framework of proximity effect Eliashberg theory. In the weak electrostaticfield limit the theory has no free parameters while, in general, the only free parameter is the thicknessof the surface layer where the electric field acts. We conclude that the best situation for increasingthe critical temperature is to have a very thin film of a superconducting material with a strongincrease of electron-phonon (boson) constant upon charging.
PACS numbers: 74.45.+c, 74.62.-c,74.20.FgKeywords: Field effect, Proximity effect, Eliashberg equations
I. INTRODUCTION
In the last decade, electric double layer (EDL) gat-ing has come to the forefront of solid state physics dueto its capability to tune the surface carrier density of awide range of different materials well beyond the lim-its imposed by solid-gate field-effect devices. The order-of-magnitude enhancement in the gate electric field al-lows this technique to reach doping levels comparable tothose of standard chemical doping. This is possible dueto the extremely large specific capacitance of the EDLthat builds up at the interface between the electrolyteand the material under study .EDL gating was first exploited to control the sur-face electronic properties of relatively low-carrier den-sity systems, where the electric field effect is more read-ily observable. Field-induced superconductivity was firstdemonstrated in strontium titanate and zirconium ni-tride chloride , and subsequently on other insulatingsystems such as perovskites and layered transition-metal dichalcogenides . Significant effort was also in-vested in the control of the superconducting propertiesof cuprates , although in this class of materials themechanism behind the carrier density modulation is stilldebated .More recently however, several experimental studieshave been devoted to the exploration of field effect insuperconductors with a large ( & · cm − ) na-tive carrier density. The interplay between two differ-ent ground states, namely superconductivity and chargedensity waves, was explored in titanium and niobium diselenides . The thickness and gate voltage depen-dence of a high-temperature superconducting phase werestudied in iron selenide, both in thin-film and thinflake forms. The effect of the ultrahigh interface elec-tric fields achievable via EDL gating were also probed instandard BCS superconductors, namely niobium andniobium nitride .With the exception of the work of Ref.28 on niobiumdiselenide, all of these studies have been performed on rel-atively thick samples ( &
10 nm) with a thickness largerthan the electrostatic screening length. These systemsare thus expected to develop a strong dependence of theirelectronic properties along the z direction (z being per-pendicular to the sample surface). As a first approxima-tion, this dependence can be conceptualized by schema-tizing the system as the parallel of a surface layer (wherethe carrier density is modified by the electric field) andan unperturbed bulk. The two electronic systems canbe expected to couple via superconducting proximity ef-fect, resulting in a complicated response to the appliedelectric field that goes well beyond a simple modificationof the superconducting properties of the surface layeralone and is strongly dependent on both the electro-static screening length and the total thickness of the film.So far, the only quantitative assessment of thisphenomenon has been reported in the frameworkof the strong-coupling limit of the BCS theory ofsuperconductivity . A proper theoretical treatment forfield effect on more complex materials, which can be de-scribed only by means of the more complete Eliashbergtheory , is lacking. Given the rising interest in thecontrol of the properties of superconductors by means ofsurface electric fields, the development of such a theo-retical approach would be very convenient not only toquantitatively describe the results of future experiments,but also to determine beforehand the experimental condi-tions (e.g., device thickness) most suitable for an optimalcontrol of the superconducting order via electric fields.In this work, we use the Eliashberg theory of proximityeffect to describe a junction composed by the perturbedsurface layer ( T c = T c,s ), where the carrier density ismodulated (with a doping level per unitary cell x ), andthe underlying unperturbed bulk ( T c = T c,b ). Here s and b indicate “surface” and “bulk” respectively (see Fig. 1).Under the application of an electric field, T c,s = T c,b andthe material behaves like a junction between a super-conductor and a normal metal in the temperature rangebounded by T c,s and T c,b . If the application of the elec-tric field increases (decreases) T c,s , then the surface layerwill be the superconductor (normal metal) and the bulkwill be the normal metal (superconductor). We performthe calculation for lead since all the input parametersof the theory are well-known in the literature for thisstrong-coupling superconductor . II. MODEL: PROXIMITY ELIASHBERGEQUATIONS
In general, a proximity effect at a superconduc-tor/normal metal junction is observed as the opening of afinite superconducting gap in the normal metal togetherwith its reduction in a thin region of the superconductorclose to the junction. In our model we use the one bands-wave Eliashberg equations with proximity effect tocalculate the critical temperature of the system. In thiscase we have to solve four coupled equations for the gaps∆ s,b ( iω n ) and the renormalization functions Z s,b ( iω n ),where ω n are the Matsubara frequencies. The imaginary-axis equations with proximity effect are: ω n Z b ( iω n ) = ω n + πT X m Λ Zb ( iω n , iω m ) N Zb ( iω m ) ++Γ b N Zs ( iω n ) (1) Z b ( iω n )∆ b ( iω n ) = πT X m (cid:2) Λ ∆ b ( iω n , iω m ) − µ ∗ b ( ω c ) (cid:3) ×× Θ( ω c − | ω m | ) N ∆ b ( iω m ) + Γ b N ∆ s ( iω n ) (2) ω n Z s ( iω n ) = ω n + πT X m Λ Zs ( iω n , iω m ) N Zs ( iω m ) +Γ s N Zb ( iω n ) (3) Z s ( iω n )∆ s ( iω n ) = πT X m (cid:2) Λ ∆ s ( iω n , iω m ) − µ ∗ s ( ω c ) (cid:3) ×× Θ( ω c − | ω m | ) N ∆ s ( iω m ) + Γ s N ∆ b ( iω n ) (4) where µ ∗ s ( b ) are the Coulomb pseudopotentials in the sur-face and in the bulk respectively, Θ is the Heaviside func-tion and ω c is a cutoff energy at least three times largerthan the maximum phonon energy. Thus we haveΛ s ( b ) ( iω n , iω m ) = 2 Z + ∞ d ΩΩ α s ( b ) F (Ω) / [( ω n − ω m ) +Ω ](5)Γ s ( b ) = π | t | Ad b ( s ) N b ( s ) (0) (6)and thus Γ s Γ b = d b N b (0) d s N s (0) , N ∆ s ( b ) ( iω m ) = ∆ s ( b ) ( iω m ) / q ω m + ∆ s ( b ) ( iω m ) (7)and N Zs ( b ) ( iω m ) = ω m / q ω m + ∆ s ( b ) ( iω m ) (8)where α s ( b ) F (Ω) are the electron-phonon spectral func-tions, A is the junction cross-sectional area, | t | the trans-mission matrix equal to one in our case because the ma-terial is the same, d s and d b are the surface and bulk layerthicknesses respectively, such that ( d s + d b = d where d is the total film thickness) and N s ( b ) (0) are the densitiesof states at the Fermi level E F,s ( b ) for the surface andbulk material. The electron-phonon coupling constantsare defined as λ s ( b ) = 2 Z + ∞ d Ω α s ( b ) F (Ω)Ω (9)and the representative energies as ln ( ω ln,s ( b ) ) = 2 λ s ( b ) Z + ∞ d Ω ln Ω α s ( b ) F (Ω)Ω (10)The solution of these equations requires eleven inputparameters: the two electron-phonon spectral fuctions α s ( b ) F (Ω), the two Coulomb pseudopotentials µ ∗ s ( b ) , thevalues of the normal density of states at the Fermi level N s ( b ) (0), the shift of the Fermi energy ∆ E F = E F,s − E F,b that enters in the calculation of the surface Coulombpseudopotential (as shown later), the value of the sur-face layer d s , the film thickness d and the junction cross-sectional area A . The values of d and A are experimentaldata. The exact value of d s , in particular in the case ofvery strong electric fields at the surface of a thin film,is in general difficult to determine a priori . Thus, weleave it as a free parameter of the model, and we performour calculations for different reasonable estimations of itsvalue.Typically, the bulk values of α b F (Ω), µ ∗ b , N b (0) and E F,b are known and can be found in the literature. Thus,we assume that we need to determine only their surfacevalues. In the next Section, we will use density functionaltheory (DFT) to calculate α s F (Ω), ∆ E F and N s (0). (cid:3046)(cid:3029) (cid:3007)(cid:481)(cid:3046)(cid:3046)(cid:3046)(cid:1499) (cid:3007)(cid:481)(cid:3029)(cid:3029)(cid:3029)(cid:1499)(cid:3046)(cid:2870)(cid:3029)(cid:2870) (cid:8) (cid:142) (cid:135) (cid:133) (cid:150) (cid:148) (cid:139) (cid:133) (cid:3) (cid:134) (cid:145) (cid:151) (cid:132) (cid:142) (cid:135) (cid:3) (cid:142) (cid:131) (cid:155) (cid:135) (cid:148) FIG. 1. (Color online) Scheme of an EDL-gated supercon-ducting thin film. The layer of adsorbed ions and the surfacelayer where the carrier density is perturbed (dark green re-gion) compose the EDL. The unperturbed bulk of the filmis indicated in light green color. For both layers, we indicatethe relevant parameters of the proximity Eliashberg equations(see text for details). Parameters in red, black and white in-dicate the free parameters of the model, data obtained fromthe literature, and the output of the DFT calculations re-spectively. Parameters in yellow are obtained from these bysimple calculations.
The value of the Coulomb pseudopotential in the sur-face layer µ ∗ s can be obtained in the following way: inthe Thomas-Fermi theory where the dielectric functionis ε ( q ) = 1 + k TF q and the bare Coulomb pseudopoten-tial µ is the angular average of the screened electrostaticpotential µ = 14 π ~ v F Z k F V ( q ) ε ( q ) qdq (11)Since V ( q ) = πe q it turns out that µ = k T F k F ln (1 + 4 k F k T F ) . (12)Hence we write µ b = a b ln (1 + 1 a b ) . (13)with a b = 2 k T F,b /k F,b . Since a b can be calculated by nu-merically solving Eq. 13, and by remembering that thesquare of Thomas-Fermi wave number k T F,b ( s ) is propor-tional to N b ( s ) (0), we have a s = a b ( N s (0) N b (0) ) / ( 1 + ∆ E F E F,b ) (14)and thus µ s = a s ln (1 + 1 a s ) . (15) The new Coulomb pseudopotential in the surface layeris thus µ ∗ s ( ω c ) = µ s µ s ln (( E F,b + ∆ E F ) /ω c ) (16)We note that, usually, the effect of electrostatic doping on µ ∗ is very small and can be neglected. We can quantifythe effect on T c of this small modulation of µ ∗ by comput-ing it in the case of maximum doping x = 0 .
40 and verythin film ( d = 5 nm), i.e. when the effect is largest. Asdiscussed in the next Section, the unperturbed Coulombpseudopotential is µ ∗ ( x = 0) = 0 . µ ∗ ( x = 0 .
4) = 0 . d s = d T F we find respectively T c = 7 . T c = 7 . T c | ∆ µ ∗ = − . T c | ∆ µ ∗ /T c = 0 . µ ∗ becomes ill-defined as the Thomas-Fermi dielec-tric function is no longer strictly valid for very large elec-tric fields. Nevertheless, the true dielectric function ε ( q )should still be a function of the ratio k T F /k F , whichin the free-electron model is independent on the normaldensity of states at the Fermi level. Thus, Eq. 11 shouldstill be able to describe the behavior of the system as afirst approximation. III. CALCULATION OF α s F (Ω) , ∆ E F AND N s (0) DFT calculations are performed within the mixed-basis pseudopotential method (MBPP) . For leada norm-conserving relativistic pseudopotential includ-ing 5 d semicore states and partial core corrections isconstructed following the prescription of Vanderbilt .This provides both scalar-relativistic and spin-orbit com-ponents of the pseudopotential. Spin-orbit coupling(SOC) is then taken into account within each DFT self-consistency cycle (for more details on the SOC imple-mentation see ). The MBPP approach utilizes a com-bination of local functions and plane waves for the basisset expansion of the valence states, which reduces the sizeof the basis set significantly. One d type local functionis added at each lead atomic site to efficiently treat thedeep 5 d potential. Sufficient convergence is then achievedwith a cutoff energy of 20 Ry for the plane waves. Theexchange-correlation it treated in the local density ap-proximation (LDA) . Brillouin zone (BZ) integrationsare performed on regular k-point meshes in conjuctionwith a Gaussian broadening of 0 . × ×
16 meshes are used, while for the calculationsof density of states and electron-phonon coupling (EPC)even denser 32 × ×
32 meshes are employed.Phonon properties are calculated with the density-functional perturbation theory as implemented inthe MBPP approach , which also provides direct accessto the electron-phonon coupling matrix elements. Theprocedure to extract the Eliashberg function is outlinedin Ref. . Dynamical matrices and corresponding EPCmatrix elements are obtained on a 16 × ×
16 phononmesh. These quantities are then interpolated using stan-dard Fourier techniques to the whole BZ, and the Eliash-berg functions are calculated by integration over the BZusing the tetrahedron method on a 40 × ×
40 mesh.SOC is consistently taken into account in all calculationsincluding lattice dynamical and EPC properties. It iswell known from previous work, that SOC is necessaryfor a correct quantitative description of both the phononanomalies and EPC of undoped bulk lead .Charge induction is simulated by adding an ap-propriate number of electrons during the DFT self-consistency cycle, compensated by a homogeneous back-ground charge to retain overall charge neutrality. Elec-tronic structure properties and the Eliashberg functionare calculated for face centred cubic (fcc) lead with thelattice constant a = 4 .
89 ˚A as obtained by optimizationfor the undoped case. For doping levels considered here,we found that to a good approximation charge inductiondoes not change the band structure but merely results ina shift of E F . In a previous study, the variation of theEPC was studied as function of the averaging energy .The present approach goes beyond this analysis by takinginto account explicitly the effect of charge induction onthe screening properties, which modifies both the phononspectrum and the EPC matrix elements.Finally we point out that, since this DFT approachsimulates the effect of the electric field by adding extracharge carriers to the system together with a uniformcompensating counter-charge (Jellium model ) is unableto describe inhomogeneous distributions caused by thescreening of the electric field itself. A more completeapproach has been developed in Ref. 58, and requires theself-consistent solution of the Poisson equation; however,this method is currently unable to compute the phononspectrum of the gated material, making it unsuitable forthe application of the proximity Eliashberg formalism. IV. RESULTS AND DISCUSSION
We start our calculations by fixing the input parame-ters for bulk lead according to the established literature.We set T c,b to its experimental value T c,b = 7 .
22 K. Theundoped α b F (Ω) gives a corresponding electron-phononcoupling λ b = 1 . ω c = 60meV and a maximum energy ω max = 70 meV in theEliashberg equations, we are thus able to determine thebulk Coulomb pseudopotential to be µ ∗ b = 0 . T c,b .In Fig. 2a we show the calculated electron-phononspectral functions α F (Ω) resulting at the increase of (cid:19)(cid:17)(cid:19) (cid:19)(cid:17)(cid:20) (cid:19)(cid:17)(cid:21) (cid:19)(cid:17)(cid:22) (cid:19)(cid:17)(cid:23) (cid:26)(cid:17)(cid:19)(cid:26)(cid:17)(cid:24)(cid:27)(cid:17)(cid:19)(cid:27)(cid:17)(cid:24)(cid:28)(cid:17)(cid:19) (cid:19) (cid:20) (cid:21) (cid:22) (cid:23) (cid:24) (cid:25) (cid:26) (cid:27) (cid:28) (cid:20)(cid:19)(cid:19)(cid:20)(cid:21)(cid:22)(cid:23)(cid:24)(cid:25)(cid:26) (cid:19)(cid:17)(cid:19) (cid:19)(cid:17)(cid:20) (cid:19)(cid:17)(cid:21) (cid:19)(cid:17)(cid:22) (cid:19)(cid:17)(cid:23)(cid:20)(cid:17)(cid:24)(cid:20)(cid:17)(cid:27)(cid:21)(cid:17)(cid:20)(cid:21)(cid:17)(cid:23)(cid:21)(cid:17)(cid:26)(cid:22)(cid:17)(cid:19) (cid:19)(cid:17)(cid:23)(cid:19)(cid:19)(cid:19)(cid:17)(cid:22)(cid:19)(cid:19)(cid:19)(cid:17)(cid:20)(cid:24)(cid:19)(cid:19)(cid:17)(cid:19)(cid:26)(cid:24) (cid:55) (cid:70) (cid:3)(cid:3) (cid:11) (cid:46) (cid:12) (cid:19)(cid:17)(cid:19)(cid:19)(cid:19)(cid:72)(cid:91)(cid:83) (cid:70)(cid:69) (cid:11)(cid:72) (cid:16) (cid:18)(cid:88)(cid:17)(cid:70)(cid:17)(cid:12) (cid:68) (cid:21) (cid:41) (cid:11) (cid:58) (cid:12) (cid:58) (cid:3)(cid:3)(cid:11)(cid:80)(cid:72)(cid:57)(cid:12) (cid:91) (cid:68) (cid:91)(cid:3)(cid:71)(cid:82)(cid:83)(cid:76)(cid:81)(cid:74)(cid:3)(cid:11)(cid:72) (cid:16) (cid:18)(cid:88)(cid:17)(cid:70)(cid:17)(cid:12) (cid:79) (cid:22)(cid:17)(cid:19)(cid:22)(cid:17)(cid:24)(cid:23)(cid:17)(cid:19)(cid:23)(cid:17)(cid:24)(cid:24)(cid:17)(cid:19) (cid:90) (cid:79) (cid:81) (cid:3) (cid:11) (cid:80) (cid:72) (cid:57) (cid:12) FIG. 2. (Color online) Panel a: calculated surface electron-phonon spectral function for five different value of chargedoping (electrons/unitary cell) 0.00 (violet solid line), 0.075(blue solid line), 0.15 (green solid line), 0.30 (orange solidline) and 0.40 (red solid line). We also show the experimentalelectron-phonon spectral function determined via tunnelingmeasurements (black solid line). All curves are shifted by aconstant offset equal to one. Panel b: calculated values ofelectron-phonon coupling constants λ (green diamonds andrhombus) and representative energies ω ln (brown pentagons)versus charge doping. Panel c: calculated critical temperatureversus charge doping for a system without proximity effect.All dash-dot lines acts as guides to the eye. the doping level x . Specifically, we plot the curves corre-sponding to x = 0 . , . , . , . , .
400 e − /unitcell. We calculate the spectral functions until x = 0 . e − / cell because for larger values of doping an instabil-ity emerges in the calculation processes. We can see thephonon softening evidenced by a reduction of ω ln with in-creasing doping level. The increase of the carrier densitygives rise to two competing effects: the value of ω ln (i.e.the representative phonon energy) decreases while thevalue of electron-phonon coupling costant λ increases (seeFig. 2b). Since the critical temperature is an increasingfunction of both ω ln and λ , in general this could result ineither a net enhancement or suppression of T c , depending x ( e − /cell ) λ ω ln ( meV ) N (0) states/ ( eV spin ) ∆ E F ( meV ) µ ∗ T c ( K )0.000 1.5612 4.8431 0.25866 0.00 0.14164 7.22000.075 1.5582 4.8432 0.25754 108.42 0.14136 7.21970.150 1.6137 4.7176 0.25611 218.77 0.14116 7.31650.300 2.0237 4.2175 0.26770 435.07 0.14074 8.28620.400 2.5392 3.5668 0.27833 571.62 0.14048 8.9406TABLE I. Calculated input parameters with DFT and calculated critical temperature with Eliashberg theory without proximityeffect. on which of the two contributions is dominant. Conse-quently the ideal situation for obtaining largest criticaltemperature in an electric field doped material is to havea strong increase of λ and ω ln concurrently. In the case oflead the contribution from the increase of λ is dominantover that from the reduction of ω ln , giving rise to a netincrease of the superconducting critical temperature (aswe report in Fig. 2c). In addition, in Table I we summa-rize all the input parameters of the proximity Eliashbergequations as obtained from the DFT calculations.Having determined the response of the superconduct-ing properties of a homogeneous lead film to a modula-tion of its carrier density, we can now consider the behav-ior of the more realistic junction between the perturbedsurface layer and the unperturbed bulk. In order to doso, however, it is now mandatory to select a value forthe thickness of the perturbed surface layer. Close to T c , the superfluid density is small and the screeningis dominated by unpaired electrons. Thus, a very roughapproximation would be to set d s to the Thomas-Fermiscreening length d T F , which for lead can be estimated tobe 0.15 nm . However, we have recently shown thatthis assumption might not be satisfactory in the pres-ence of the very large electric fields that build up in theelectric double layer. Indeed, our experimental findingson niobium nitride indicated that the screening lengthincreases for very large doping values . However, it isreasonable to assume the exact entity of this increase tobe specific to each material. Thus, while the qualitativebehavior can be expected to be general, the exact valuesof d s determined for niobium nitride cannot be directlyapplied to lead.In order not to lose the generality of our approach, wecalculate the behavior of our system for three differentchoices of the behavior of d s . We start by expressing d s = d T F [1 + m Θ( x − x )], where m is a dimensionlessparameter indicating how much d s expands beyond theThomas-Fermi value for large doping levels, and x is thespecific doping value upon which this increase in d s takesplace. By selecting x = 0 .
2, we allow the upper half ofour doping values to go beyond the Thomas-Fermi ap-proximation. We then perform proximity-coupled Eliash-berg calculations for m = 0 , , d = 5 , , , ,
40 nm, always assuming thejunction area to be A = 10 − m . Note that the case m = 0 of course corresponds to the case where the ma- terial satisfies the Thomas-Fermi model for any value ofdoping: in this case, the model has no free parameters.In Fig. 3 we plot the evolution of T c upon increasingelectron doping and assuming that the Thomas-Fermi (cid:19)(cid:17)(cid:19)(cid:19) (cid:19)(cid:17)(cid:19)(cid:24) (cid:19)(cid:17)(cid:20)(cid:19) (cid:19)(cid:17)(cid:20)(cid:24) (cid:19)(cid:17)(cid:21)(cid:19) (cid:19)(cid:17)(cid:21)(cid:24) (cid:19)(cid:17)(cid:22)(cid:19) (cid:19)(cid:17)(cid:22)(cid:24) (cid:19)(cid:17)(cid:23)(cid:19)(cid:26)(cid:17)(cid:21)(cid:21)(cid:26)(cid:17)(cid:21)(cid:22)(cid:26)(cid:17)(cid:21)(cid:24)(cid:26)(cid:17)(cid:22)(cid:19)(cid:26)(cid:17)(cid:23)(cid:19)(cid:26)(cid:17)(cid:25)(cid:19)(cid:27)(cid:17)(cid:19)(cid:19) (cid:19) (cid:24) (cid:20)(cid:19) (cid:20)(cid:24) (cid:21)(cid:19) (cid:21)(cid:24) (cid:22)(cid:19) (cid:22)(cid:24) (cid:23)(cid:19)(cid:26)(cid:17)(cid:21)(cid:21)(cid:26)(cid:17)(cid:21)(cid:22)(cid:26)(cid:17)(cid:21)(cid:24)(cid:26)(cid:17)(cid:22)(cid:19)(cid:26)(cid:17)(cid:23)(cid:19)(cid:26)(cid:17)(cid:25)(cid:19)(cid:27)(cid:17)(cid:19)(cid:19) (cid:69) (cid:55) (cid:70) (cid:3)(cid:3) (cid:11) (cid:46) (cid:12) (cid:91)(cid:3)(cid:71)(cid:82)(cid:83)(cid:76)(cid:81)(cid:74)(cid:3)(cid:11)(cid:72) (cid:16) (cid:18)(cid:88)(cid:17)(cid:70)(cid:17)(cid:12) (cid:71) (cid:86) (cid:32)(cid:71) (cid:55)(cid:41) (cid:32)(cid:19)(cid:17)(cid:20)(cid:24)(cid:3)(cid:81)(cid:80)(cid:3)(cid:71)(cid:32)(cid:3)(cid:24)(cid:3)(cid:81)(cid:80)(cid:3)(cid:3)(cid:71)(cid:32)(cid:20)(cid:19)(cid:3)(cid:81)(cid:80)(cid:3)(cid:3)(cid:71)(cid:32)(cid:21)(cid:19)(cid:3)(cid:81)(cid:80)(cid:3)(cid:3)(cid:71)(cid:32)(cid:22)(cid:19)(cid:3)(cid:81)(cid:80)(cid:3)(cid:3)(cid:71)(cid:32)(cid:23)(cid:19)(cid:3)(cid:81)(cid:80) (cid:68) (cid:3) (cid:55) (cid:70) (cid:3)(cid:3) (cid:11) (cid:46) (cid:12) (cid:71)(cid:3)(cid:73)(cid:76)(cid:79)(cid:80)(cid:3)(cid:87)(cid:76)(cid:70)(cid:78)(cid:81)(cid:72)(cid:86)(cid:86)(cid:3)(cid:11)(cid:81)(cid:80)(cid:12) (cid:3)(cid:19)(cid:17)(cid:19)(cid:26)(cid:24)(cid:3)(cid:72) (cid:16) (cid:18)(cid:88)(cid:17)(cid:70)(cid:17)(cid:15)(cid:3)(cid:71) (cid:86) (cid:32)(cid:71) (cid:55)(cid:41) (cid:32)(cid:19)(cid:17)(cid:20)(cid:24)(cid:3)(cid:81)(cid:80)(cid:3)(cid:19)(cid:17)(cid:20)(cid:24)(cid:19)(cid:3)(cid:72) (cid:16) (cid:18)(cid:88)(cid:17)(cid:70)(cid:17)(cid:15)(cid:3)(cid:71) (cid:86) (cid:32)(cid:71) (cid:55)(cid:41) (cid:32)(cid:19)(cid:17)(cid:20)(cid:24)(cid:3)(cid:81)(cid:80)(cid:3)(cid:19)(cid:17)(cid:22)(cid:19)(cid:19)(cid:3)(cid:72) (cid:16) (cid:18)(cid:88)(cid:17)(cid:70)(cid:17)(cid:15)(cid:3)(cid:71) (cid:86) (cid:32)(cid:71) (cid:55)(cid:41) (cid:32)(cid:19)(cid:17)(cid:20)(cid:24)(cid:3)(cid:81)(cid:80)(cid:3)(cid:19)(cid:17)(cid:23)(cid:19)(cid:19)(cid:3)(cid:72) (cid:16) (cid:18)(cid:88)(cid:17)(cid:70)(cid:17)(cid:15)(cid:3)(cid:71) (cid:86) (cid:32)(cid:71) (cid:55)(cid:41) (cid:32)(cid:19)(cid:17)(cid:20)(cid:24)(cid:3)(cid:81)(cid:80) FIG. 3. (Color online) Panel a: calculated critical temper-ature versus charge doping for five different values of filmthickness d = 5 nm (orange stars), d = 10 (blue down tri-angles), d = 20 nm (red circles), d = 30 nm (green up tri-angles) and d = 40 nm (black squares) with surface layerthickness d s = d TF . Panel b: calculated critical temper-ature versus film thickness for four different charge doping(electrons/unitary cell): 0 .
075 (black squares), 0 .
150 (red cir-cles), 0 .
300 (green up triangles), 0 .
400 (blue down triangles)with d s = d TF . The two graphs are in semi-logarithmic scale( log ( T c − . model always holds ( m = 0 and d s = d T F ), for differ-ent values of film thickness. The calculations show thatthe qualitative increase in T c with increasing doping levelthat we observed in the homogeneous case is retained alsoin proximized films of any thinckess (see Fig. 3a). How-ever, the presence of a coupling between surface and bulkinduced by the proximity effect gives rise to a key differ-ence with respect to the homogeneous case, namely, astrong dependence of T c on film thickness in the dopedfilms. Indeed, the magnitude of the T c shift with respectto the homogeneous case is heavily suppressed already infilms as thin as 5 nm. This behavior is best seen in Fig.3b, where we plot the same data as a function of the to-tal film thickness for all doping levels. As we can see theincrease of critical temperature drops dramatically withincreasing film thickness. We have not calculated thecritical temperature for monolayer films since the approx-imations of the model would no longer apply in this case:in particular the unperturbed electron-phonon spectralfunction would have been different from the bulk-like onewe employed in our calculations .We now consider the effect of the different degrees ofconfinement for the induced charge carriers at the surfaceof the films. We do so by allowing the perturbed surfacelayer to spread further in the depth of the film for largeelectron doping, i.e. by increasing the m parameter inthe definition of d s . In Fig. 4 we plot the evolution of T c with increasing electron doping and for different filmthicknesses, in the two cases m = 1 ( d s is allowed to ex-pand up to 2 d T F = 0 . m = 4 ( d s is allowed toexpand up to 5 d T F = 0 .
75 nm). We can first observe howa different value of d s does not change the qualitative be-havior of the films. The evolution of T c with increasingelectron doping is still comparable to both the homoge-neous case and the proximized films in the Thomas-Fermilimit. The suppression of the T c increase with increasingfilm thickness is also similar to the latter case. How-ever, the magnitude of the T c shift for the same values offilm thickness and doping level per unit cell is clearly themore enhanced the larger the value of d s . This is to beexpected, as larger values of d s increase the fraction ofthe film that is perturbed by the application of the elec-tric field and reduce the T c shift dampening operated bythe proximity effect. In principle, for values of m largeenough (or film thickness d small enough) one could reachthe limit value d s ≃ d and recover the homogeneous casewhere the T c shift is maximum.All the calculations we performed so far assumed thatone could directly control the induced carrier density per unit volume , x , in the surface layer, without an ex-plicit upper limit. However, this is not an experimentallyachievable goal in a field-effect device architecture. Inthis class of devices, the polarization of the gate electrodeallows one to tune the electric field at the interface andthus the induced carrier density per unit surface , ∆ n D ,required to screen it, i.e. ∆ n D = R d s ∆ n D dz withinour model is distributed within a layer of thickness d s .In general, the determination of the exact depth profile (cid:19)(cid:17)(cid:19)(cid:19) (cid:19)(cid:17)(cid:19)(cid:24) (cid:19)(cid:17)(cid:20)(cid:19) (cid:19)(cid:17)(cid:20)(cid:24) (cid:19)(cid:17)(cid:21)(cid:19) (cid:19)(cid:17)(cid:21)(cid:24) (cid:19)(cid:17)(cid:22)(cid:19) (cid:19)(cid:17)(cid:22)(cid:24) (cid:19)(cid:17)(cid:23)(cid:19)(cid:26)(cid:17)(cid:21)(cid:21)(cid:26)(cid:17)(cid:21)(cid:22)(cid:26)(cid:17)(cid:21)(cid:24)(cid:26)(cid:17)(cid:22)(cid:19)(cid:26)(cid:17)(cid:23)(cid:19)(cid:26)(cid:17)(cid:25)(cid:19)(cid:27)(cid:17)(cid:19)(cid:19) (cid:19)(cid:17)(cid:19)(cid:19) (cid:19)(cid:17)(cid:19)(cid:24) (cid:19)(cid:17)(cid:20)(cid:19) (cid:19)(cid:17)(cid:20)(cid:24) (cid:19)(cid:17)(cid:21)(cid:19) (cid:19)(cid:17)(cid:21)(cid:24) (cid:19)(cid:17)(cid:22)(cid:19) (cid:19)(cid:17)(cid:22)(cid:24) (cid:19)(cid:17)(cid:23)(cid:19)(cid:26)(cid:17)(cid:21)(cid:21)(cid:26)(cid:17)(cid:21)(cid:22)(cid:26)(cid:17)(cid:21)(cid:24)(cid:26)(cid:17)(cid:22)(cid:19)(cid:26)(cid:17)(cid:23)(cid:19)(cid:26)(cid:17)(cid:25)(cid:19)(cid:27)(cid:17)(cid:19)(cid:19) (cid:69) (cid:91)(cid:3)(cid:71)(cid:82)(cid:83)(cid:76)(cid:81)(cid:74)(cid:3)(cid:11)(cid:72) (cid:16) (cid:18)(cid:88)(cid:17)(cid:70)(cid:17)(cid:12) (cid:55) (cid:70) (cid:3)(cid:3) (cid:11) (cid:46) (cid:12) (cid:91)(cid:3)(cid:71)(cid:82)(cid:83)(cid:76)(cid:81)(cid:74)(cid:3)(cid:11)(cid:72) (cid:16) (cid:18)(cid:88)(cid:17)(cid:70)(cid:17)(cid:12) (cid:71) (cid:86) (cid:32)(cid:19)(cid:17)(cid:20)(cid:24)(cid:62)(cid:20)(cid:14) (cid:84) (cid:11)(cid:91)(cid:16)(cid:19)(cid:17)(cid:21)(cid:12)(cid:64)(cid:3)(cid:81)(cid:80)(cid:3)(cid:71)(cid:32)(cid:3)(cid:24)(cid:3)(cid:81)(cid:80)(cid:3)(cid:71)(cid:32)(cid:20)(cid:19)(cid:3)(cid:81)(cid:80)(cid:3)(cid:71)(cid:32)(cid:21)(cid:19)(cid:3)(cid:81)(cid:80)(cid:3)(cid:71)(cid:32)(cid:22)(cid:19)(cid:3)(cid:81)(cid:80)(cid:3)(cid:71)(cid:32)(cid:23)(cid:19)(cid:3)(cid:81)(cid:80) (cid:68) (cid:3) (cid:55) (cid:70) (cid:3)(cid:3) (cid:11) (cid:46) (cid:12) (cid:71) (cid:86) (cid:32)(cid:19)(cid:17)(cid:20)(cid:24)(cid:62)(cid:20)(cid:14)(cid:23) (cid:84) (cid:11)(cid:91)(cid:16)(cid:19)(cid:17)(cid:21)(cid:12)(cid:64)(cid:3)(cid:81)(cid:80)(cid:3)(cid:71)(cid:32)(cid:3)(cid:24)(cid:3)(cid:81)(cid:80)(cid:15)(cid:3)(cid:3)(cid:71)(cid:32)(cid:20)(cid:19)(cid:3)(cid:81)(cid:80)(cid:3)(cid:71)(cid:32)(cid:21)(cid:19)(cid:3)(cid:81)(cid:80)(cid:3)(cid:71)(cid:32)(cid:22)(cid:19)(cid:3)(cid:81)(cid:80)(cid:3)(cid:71)(cid:32)(cid:23)(cid:19)(cid:3)(cid:81)(cid:80) FIG. 4. (Color online) Calculated critical temperature versuscharge doping for five different values of film thickness d = 5nm (orange stars), d = 10 (blue down triangles), d = 20 nm(red circles), d = 30 nm (green up triangles) and d = 40nm (black squares) with surface layer thickness d s = d TF [1 + m Θ( x − . m = 1 and panel b m = 4). The twographs are in semi-logarithmic scale ( log ( T c − . of this distribution requires the self-consistent solution ofthe Poisson equation ; however, as a first approximationwe can consider this distribution to be constant, obtain-ing an effective doping level per unit volume simply as x = ∆ n D /d s . This procedure allows one to employ thesame DFT-Eliashberg formalism we developed before inorder to simulate a field effect experiment on a supercon-ducting thin film.In addition, in the previous calculations we supposedthat d s can only take on two values as a function of x ,depending on the threshold value x . When we considerthe field-effect architecture, however, the parameters m and x in the expression d s = d T F [1 + m Θ( x − x )] areno longer independent as in the previous case. More-over, according to our recent experimental findings onniobium nitride , d s is a monotonically increasing func-tion of ∆ n D . We include this behavior in our calcula-tions in the following way: Once the maximum dopinglevel x is selected, m = m (∆ n D ) is automatically de-termined by the requirement d s (∆ n D ) = ∆ n D · x forany x > x . Fig. 5a shows the resulting dependenceof the doping per unit volume x and surface layer thick- (cid:24) (cid:20)(cid:19) (cid:21)(cid:19) (cid:22)(cid:19) (cid:23)(cid:19) (cid:20)(cid:19)(cid:20)(cid:19)(cid:19)(cid:21)(cid:24)(cid:19)(cid:19)(cid:20)(cid:19) (cid:20)(cid:23) (cid:20)(cid:19) (cid:20)(cid:24) (cid:21)(cid:91)(cid:20)(cid:19) (cid:20)(cid:22) (cid:24)(cid:91)(cid:20)(cid:19) (cid:20)(cid:24) (cid:26)(cid:17)(cid:21)(cid:21)(cid:26)(cid:17)(cid:22)(cid:26)(cid:17)(cid:24)(cid:27)(cid:17)(cid:19)(cid:28)(cid:17)(cid:19) (cid:20)(cid:19) (cid:20)(cid:23) (cid:20)(cid:19) (cid:20)(cid:24) (cid:21)(cid:91)(cid:20)(cid:19) (cid:20)(cid:22) (cid:24)(cid:91)(cid:20)(cid:19) (cid:20)(cid:24) (cid:26)(cid:17)(cid:21)(cid:21)(cid:26)(cid:17)(cid:22)(cid:26)(cid:17)(cid:24)(cid:27)(cid:17)(cid:19)(cid:28)(cid:17)(cid:19)(cid:20)(cid:19) (cid:20)(cid:23) (cid:20)(cid:19) (cid:20)(cid:24) (cid:21)(cid:91)(cid:20)(cid:19) (cid:20)(cid:22) (cid:24)(cid:91)(cid:20)(cid:19) (cid:20)(cid:24) (cid:19)(cid:17)(cid:20)(cid:19)(cid:17)(cid:21)(cid:19)(cid:17)(cid:22)(cid:19)(cid:17)(cid:23)(cid:19)(cid:17)(cid:19)(cid:24)(cid:19)(cid:17)(cid:24) (cid:15) (cid:3) x (cid:3)(cid:32)(cid:3)(cid:19)(cid:17)(cid:23)(cid:3)(cid:72) (cid:16) (cid:18)(cid:88)(cid:17)(cid:70)(cid:17)(cid:15) (cid:3) x (cid:3)(cid:32)(cid:3)(cid:19)(cid:17)(cid:22)(cid:3)(cid:72) (cid:16) (cid:18)(cid:88)(cid:17)(cid:70)(cid:17) (cid:39) (cid:81) (cid:21)(cid:39) (cid:3)(cid:11)(cid:70)(cid:80) (cid:16)(cid:21) (cid:12) (cid:91) (cid:3) (cid:11) (cid:72) (cid:16) (cid:18) (cid:88) (cid:17) (cid:70) (cid:17) (cid:12) (cid:19)(cid:17)(cid:20)(cid:20)(cid:20)(cid:19) (cid:3) (cid:71) (cid:86) (cid:3) (cid:11) (cid:81) (cid:80) (cid:12) (cid:71)(cid:69) (cid:55) (cid:70) (cid:3) (cid:11) (cid:19) (cid:17) (cid:23) (cid:12) (cid:3) (cid:16) (cid:3) (cid:55) (cid:70) (cid:3) (cid:11) (cid:19) (cid:17) (cid:22) (cid:12) (cid:3) (cid:11) (cid:80) (cid:46) (cid:12) (cid:71)(cid:3)(cid:73)(cid:76)(cid:79)(cid:80)(cid:3)(cid:87)(cid:75)(cid:76)(cid:70)(cid:78)(cid:81)(cid:72)(cid:86)(cid:86)(cid:3)(cid:11)(cid:81)(cid:80)(cid:12) (cid:3)(cid:20)(cid:17)(cid:27) (cid:151) (cid:20)(cid:19) (cid:20)(cid:23) (cid:3)(cid:70)(cid:80) (cid:16)(cid:21) (cid:3)(cid:23)(cid:17)(cid:19) (cid:151) (cid:20)(cid:19) (cid:20)(cid:23) (cid:3)(cid:70)(cid:80) (cid:16)(cid:21) (cid:3)(cid:27)(cid:17)(cid:19) (cid:151) (cid:20)(cid:19) (cid:20)(cid:23) (cid:3)(cid:70)(cid:80) (cid:16)(cid:21) (cid:3)(cid:3)(cid:20)(cid:25) (cid:151) (cid:20)(cid:19) (cid:20)(cid:23) (cid:3)(cid:70)(cid:80) (cid:16)(cid:21) (cid:3)(cid:3)(cid:22)(cid:21) (cid:151) (cid:20)(cid:19) (cid:20)(cid:23) (cid:3)(cid:70)(cid:80) (cid:16)(cid:21) (cid:68) (cid:70) (cid:55) (cid:70) (cid:3) (cid:11) (cid:46) (cid:12) (cid:39) (cid:81) (cid:21)(cid:39) (cid:3)(cid:11)(cid:70)(cid:80) (cid:16)(cid:21) (cid:12) x (cid:3)(cid:32)(cid:3)(cid:19)(cid:17)(cid:23)(cid:3)(cid:72) (cid:16) (cid:18)(cid:88)(cid:17)(cid:70)(cid:17)(cid:3)(cid:24)(cid:3)(cid:81)(cid:80)(cid:3)(cid:20)(cid:19)(cid:3)(cid:81)(cid:80)(cid:3)(cid:21)(cid:19)(cid:3)(cid:81)(cid:80)(cid:3)(cid:22)(cid:19)(cid:3)(cid:81)(cid:80)(cid:3)(cid:23)(cid:19)(cid:3)(cid:81)(cid:80) (cid:55) (cid:70) (cid:3) (cid:11) (cid:46) (cid:12) (cid:39) (cid:81) (cid:21)(cid:39) (cid:3)(cid:11)(cid:70)(cid:80) (cid:16)(cid:21) (cid:12) x (cid:3)(cid:32)(cid:3)(cid:19)(cid:17)(cid:22)(cid:3)(cid:72) (cid:16) (cid:18)(cid:88)(cid:17)(cid:70)(cid:17)(cid:3)(cid:24)(cid:3)(cid:81)(cid:80)(cid:3)(cid:20)(cid:19)(cid:3)(cid:81)(cid:80)(cid:3)(cid:21)(cid:19)(cid:3)(cid:81)(cid:80)(cid:3)(cid:22)(cid:19)(cid:3)(cid:81)(cid:80)(cid:3)(cid:23)(cid:19)(cid:3)(cid:81)(cid:80) FIG. 5. (Color online) Panel a: dependence of the doping per unit volume x (red up triangles and diamonds) and surface layerthickness d s (blue up triangles and diamonds) on the induced carrier density per unit surface ∆ n D , for two different valuesof the maximum doping level x = 0 . x = 0 . − /unit cell. Panel b and panel c: T c versus induced carrier density perunit surface ∆ n D for five different film thicknesses ( d = 5 nm (orange stars), d = 10 (blue down triangles), d = 20 nm (redcircles), d = 30 nm (green up triangles) and d = 40 nm (black squares) in the cases x = 0 . . − /unit cell respectively.Panels b and c are in semi-logarithmic scale ( log ( T c − . T c ( x = 0 . − T c ( x = 0 .
3) versus induced carrier density per unit surface ∆ n D for the five different film thicknesses. ness d s on the induced carrier density per unit surface ∆ n D , for two different values of the maximum dopinglevel x = 0 . x = 0 . − /unit cell. When ∆ n D issmall enough so that x < x , the Thomas-Fermi screen-ing holds, d s = d T F is constant and x linearly increaseswith ∆ n D . As soon as ∆ n D becomes large enoughthat x = x is constant (∆ n D ( x )), the Thomas-Fermiscreening is no longer valid and d s increases linearly with∆ n D .In Fig. 5b and 5c we plot the resulting modulation of T c for five different film thicknesses in the cases x = 0 . . − /unit cell respectively. In both cases we canreadily distinguish between two regimes of ∆ n D . When∆ n D . ∆ n D ( x ), Thomas-Fermi screening holds andwe reproduce the behavior we observed in Fig. 3a. Inthis regime, the induced carrier density directly modu-lates x and thus the electron-phonon spectral function α F (Ω). The T c modulation is thus a result of a directmodification of the material properties at the surface,with proximity effect simply operating a “smoothing”the larger the value of the film thickness. On the otherhand, when ∆ n D > ∆ n D ( x ), the surface properties ( α F (Ω)) are no longer modified by the extra charge car-riers, and the further modulation of T c originates entirelyfrom the proximity effect as determined by the increasein d s .We can also compare the T c shifts for different maxi-mum doping levels x . Fig. 5d shows the difference be-tween the T c corresponding to x = 0 . . − /unitcell as a function of the total film thickness, for differ-ent values of ∆ n D . We can clearly see how T c is alwayslarger for the films with larger x , for any value of filmthickness, even if the associated values of d s are alwayssmaller. This indicates that the maximum achievablevalue of x is dominant with respect to the increase of d s to determine the final value of T c , also in the dopingregime ∆ n D > ∆ n D ( x ).Of course, in a real sample we don’t expect the tran-sition between the two regimes to be so clear-cut, as thesaturation of x to x would occur over a finite range of∆ n D . In this intermediate region, the modulation of α F (Ω) and d s would both contribute in a comparableway to the final value of T c in the film. We stress, how-ever, that in both regimes the proximity effect is funda-mental in determining the T c of the gated film. We alsonote that the proximized Eliashberg equations are ableto account for a non-uniform scaling of the T c shift fordifferent values of film thickness, unlike the models thatuse approximated analytical equations for T c . V. CONCLUSIONS
In this work, we have developed a general method forthe theoretical simulation of field-effect-doping in super-conducting thin films of arbitrary thickness, and we havebenchmarked it on lead as a standard strong-couplingsuperconductor. Our method relies on ab-initio
DFTcalculations to compute how the increasing doping level x per unit volume modifies the structural and electronicproperties of the material (shift of Fermi level ∆ E F , den-sity of states N (0), and electron-phonon spectral func-tion α F (Ω)). The Coulomb pseudopotential µ ∗ is de-termined by simple calculations from some of these pa-rameters. The properties of the pristine thin film (criticaltemperature T c , device area A and total film thickness d )can be obtained either from the literature or experimen-tally from standard transport measurements. For dop-ing values where the Thomas-Fermi theory of screeningis satisfied, the perturbed surface layer thickness is con-stant ( d s = d T F ) and the theory has no free parameters.Once all the input parameters are known, our methodallows to compute the transition temperature T c for ar-bitrary values of film thickness d and electron dopingin the surface layer x by solving the proximity-coupledEliashberg equations in the surface layer and unper-turbed bulk. On the other hand, if no reliable estima- tions of the surface layer thickness d s are available, ourmethod allows one to determine d s ( x ) by reproducingthe experimentally-measured T c ( x ). This allows to probedeviations from the standard Thomas-Fermi theory ofscreening in the presence of very large interface electricfields.We also show how, even in the case where the Thomas-Fermi approximation breaks down and the doping level x can no longer be increased, the transition temperature T c of a thin film can still be indirectly modulated by theelectric field by changing the surface layer thickness d s .For what concerns artificial enhancements of T c in super-conducting thin films, we conclude that very thin films( d . d s , in order to minimize the smoothing operatedby the proximity effect) of a superconductive materialcharacterized by a strong increase of the electron-phonon(boson) coupling upon changing its carrier density arerequired to optimize the effectiveness of the field-effect-device architecture.Finally, our calculations indicate that sizable T c en-hancements of the order of ∼ . ∼ ACKNOWLEDGMENTS
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