Zero-Field Surface Charge Due to the Gap Suppression in d-Wave Superconductors
Ezekiel Sambo Joshua, Hikaru Ueki, Wataru Kohno, Takafumi Kita
aa r X i v : . [ c ond - m a t . s up r- c on ] A ug Journal of the Physical Society of Japan
FULL PAPERS
Zero-Field Surface Charge Due to the Gap Suppression in d -Wave Superconductors Ezekiel Sambo Joshua , Hikaru Ueki , Wataru Kohno , and Takafumi Kita Department of Physics, Hokkaido University, Sapporo 060-0810, Japan Department of Mathematics and Physics, Hirosaki University, Hirosaki, Aomori 036-8561, Japan
We perform a microscopic study on the redistribution of electric charge near the surface of a model d -wave supercon-ductor cut along the [110] direction, with a Fermi surface appropriate for cuprate superconductors, using the augmentedquasiclassical equations. We identify two possible mechanisms for the redistribution of charged particles di ff erent fromthe well-known magnetic Hall e ff ect, namely; the pair potential gradient (PPG) force due to surface e ff ects on the pairpotential and the pressure di ff erence between the normal and superconducting regions arising from the slope of the den-sity of states (SDOS) in the normal states at the Fermi level. Our present results show that in spite of the absence ofsupercurrents, electric charge is induced around the surface. Moreover, the charging e ff ect due to the SDOS pressuredominates over that due to the PPG force for all the realistic electron-fillings n = .
8, 0 .
9, and 1 .
15, at all temperatures.In addition, for the filling n = .
15, the PPG force and the SDOS pressure contributions have the same negative signs,which gives a larger total surface charge i.e., both the sign and amount of the surface charge depends greatly on theFermi-surface curvature. We have also calculated the local density of states (LDOS) within the augmented quasiclassicaltheory. Spatially varying local particle-hole asymmetry appears in the LDOS, which suggests the presence of electriccharge.
1. Introduction
In general, electrostatic charge redistribution implies theaction of certain forces on charged particles. For instance, innormal metals, semiconductors as well as in superconduc-tors, the magnetic Lorentz force results in the Hall e ff ect.Exclusively in superconductors, in the presence of inhomo-geneities such as surfaces or interfaces, other forces appearnotwithstanding the absence of magnetic fields, namely; thepair-potential gradient (PPG) force which originates fromthe spatial variation of the pair potential, and the pressure dueto the slope of the density of states (SDOS) in the normalstates at the Fermi level. These forces appear in the vor-tex state in type-II superconductors and are expected in thepresence of surfaces and interfaces.Quite recently, the augmented quasiclassical equations in-corporating the three force terms were derived, with the stan-dard Eilenberger equations
12, 13) as the leading-order contri-butions and the force terms as first-order corrections in termsof the quasiclassical parameter δ ≡ ~ / h p F i F ξ ≪
1, where p F is the magnitude of the Fermi momentum, h· · · i F is the Fermisurface average, and ξ is the coherence length at zero temper-ature.
4, 7, 11)
These augmented quasiclassical equations havebeen used in the study of the Hall e ff ect in superconductorsin the Meissner state, and in the vortex state.
5, 7, 8, 11, 15, 16)
However, to the best of our knowledge, they have not yet beenapplied to surface systems.It has been shown that in the isolated vortex core of anisotropic type-II superconductor the PPG force gives up to 10to 10 times larger contribution to the electric charging com-pared to the Lorentz force. Even more recently, Ueki et al. found that the SDOS pressure gives the dominant contributionnear the transition temperature, while the PPG force domi-nates as the temperature approaches zero.
Masaki studiedthe charged and uncharged vortices in a chiral p -wave super-conductor based on the augmented quasiclassical equations.He pointed out that the vortex-core charge is dominated by the contributions from the angular derivative terms in the PPGforce terms. Using a simplified picture of a system consist-ing of a vortex core in the normal state surrounded by a su-perconducting material, Khomskii et al. showed that a finitedi ff erence in chemical potential δµ ,
9, 10)
In the context of surface charging, Fu-rusaki et al. studied spontaneous surface charging in chiral p -wave superconductors based on the Bogoliubov–de Gennes(BdG) equations. They found two contributions, one contribu-tion originates from the Lorentz force due to the spontaneousedge currents, while the other contribution has topologicalorigin and is related to the intrinsic angular momentum of theCooper pairs. They also calculated the surface charging dueto the Lorentz force acting on the spontaneous edge currentsin d + is -wave superconductors based on the BdG equations. Emig et al. also showed using a phenomenological analysis onthe basis of Ginzburg–Landau theory that the presence of sur-faces in d -wave superconductors can induce charge inhomo-geneity due to the suppression of the energy gap. Volovikand Salomaa discussed the appearance of electric charge atthe edges and vortex core of even electrically neutral p -wavesuperfluids.
19, 20)
In a d x − y -wave superconductor with a specularly reflec-tive surface cut along the [110] direction, the order parameteris suppressed near the surface and vanishes at the sur-face, due to a change in its sign along the classical quasipar-ticle trajectories. This sign change also results in the forma-tion of zero energy states (ZES) near (at) the edges of thesematerials. ZES in d -wave superconductors are detectablethrough the observation of zero-bias conductance peaks in thespectra of scanning tunneling spectroscopy at oriented sur-faces of the d -wave crystals. Hayashi et al. discussed theconnection between the Caroli–de Gennes–Matricon states at the vortex core of an s -wave superconductor and the occur-rence of electric charge. They concluded that the particle-
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FULL PAPERS hole asymmetry inside the vortex core observed through thelocal density of states (LDOS) implies the corresponding ex-istence of charge at the vortex core. Recently, Masaki alsodiscussed the connection between particle-hole asymmetryand vortex charging in superconductors.
Surface chargingin d -wave superconductors may also have a similar connec-tion with particle-hole asymmetry in the LDOS, which is ex-pected to appear due to first-order quantum corrections withinthe augmented quasiclassical theory.Even in the absence of magnetic fields, surface e ff ects in d -wave superconductors result in the appearance of the PPGforce due to the suppression of the pair potential near the sur-face and the pressure due to the SDOS at the Fermi level.Hence the presence of oriented surfaces in d -wave supercon-ductors is expected to be accompanied by the redistribution ofcharged particles.In this paper, we report the accumulation of charged parti-cles around the [110] specular surface of a d x − y -wave super-conductor with a Fermi surface used for cuprate superconduc-tors, due to the PPG force and the SDOS pressure, notwith-standing the absence of magnetic field.Although surface charging without time-reversal symmetrybreaking itself has already been suggested in Ref. 18, our mi-croscopic theory clarifies the origin of the charging, whichis missing from the Eilenberger equations. The augmentedquasiclassical theory describes the particle-hole-asymmetricLDOS which also cannot be described at the level of theEilenberger equations. In our calculations, it is shown that theSDOS pressure gives the dominant contribution to the surfacecharge for the realistic electron-fillings n = .
8, 0 .
9, and 1 . d -wave su-perconductor within the augmented quasiclassical theory. Inaddition, we discuss the structure of the LDOS, especially theparticle-hole asymmetry due to the SDOS pressure and thePPG force as an indicator of the existence of surface charge.This asymmetric behaviour in the LDOS caused by surfacecharging is expected to be observed.This paper is organized as follows. In Sect. 2, we givea summary of the quasiclassical formalism relevant to ourpresent study. This is based on the formulation in Ref. 11. Ournumerical procedures and results are summarized in Sect. 3.In Sect. 4, we give a summary of the present study and discussfuture directions.
2. Formalism
The quasiclassical equations augmented with the PPGterms are given in the Matsubara formalism by
8, 11) h i ε n ˆ τ − ˆ ∆ ˆ τ , ˆ g i + i ~ v F · ∂ ˆ g − i ~ n ∂ ˆ ∆ ˆ τ , ∂ p F ˆ g o + i ~ n ∂ p F ˆ ∆ ˆ τ , ∂ ˆ g o = ˆ0 , (1)where ε n = (2 n + π k B T is the fermion Matsubara energy( n = , ± , . . . ), v F is the Fermi velocity while p F is theFermi momentum, and ∂ is the gauge-invariant di ff erential operator.
8, 11)
The commutators are given by [ˆ a , ˆ b ] ≡ ˆ a ˆ b − ˆ b ˆ a and { ˆ a , ˆ b } ≡ ˆ a ˆ b + ˆ b ˆ a . The functions ˆ g = ˆ g ( ε n , p F , r ) andˆ ∆ = ˆ ∆ ( p F , r ) are the quasiclassical Green’s functions and thepair potential, respectively. We have used the static gauge as E ( r ) = − ∇ Φ ( r ) and B ( r ) = ∇ × A ( r ), where E is the electricfield, B is the magnetic field, Φ is the scalar potential, and A is the vector potential.We consider the spin-singlet pairing state without spinparamagnetism. The matrices ˆ g , ˆ ∆ and ˆ τ are written asˆ g = " g − i fi ¯ f − ¯ g , ˆ ∆ = " ∆ φ ∆ ∗ φ , ˆ τ = " − , (2)where the functions ∆ = ∆ ( r ) and φ = φ ( p F ) denote the am-plitude of the energy gap and the basis function of the energygap, respectively. The barred functions in the Matsubara for-malism are defined generally by ¯ X ( ε n , p F , r ) ≡ X ∗ ( ε n , − p F , r ).Following the procedure used in Ref. 14, we carry out a per-turbation expansion of f and g in terms of δ as f = f + f + · · · and g = g + g + · · · . Then the main part in the standard Eilen-berger equations
12, 13) are obtained from the leading-ordercontribution in terms of the quasiclassical parameter as ε n f + ~ v F · ∇ − i e A ~ ! f = ∆ φ g , (3)where e < g = ¯ g = sgn( ε n ) p − f ¯ f . Moreover, the self-consistency equationfor the pair potential is given by ∆ = π Γ k B T n c X n = h f φ i F , (4)where the cuto ff n c is determined from (2 n c + π k B T = ε c with ε c denoting the cuto ff energy, and Γ denotes the cou-pling constant responsible for the Cooper pairing, defined by Γ ≡ − N (0) V e ff with V e ff and N (0) denoting the constant ef-fective potential and the normal-state density-of-states (DOS)per spin and unit volume at the Fermi level, respectively. Weobtain the expression for the first-order Green’s function g interms of quasiclassical parameter δ from Eq. (1) as v F · ∇ g = − i ∇ + i e A ~ ! ∆ ∗ φ · ∂ f ∂ p F − i ∇ − i e A ~ ! ∆ φ · ∂ ¯ f ∂ p F + i ∆ ∗ ∂φ∂ p F · ∇ − i e A ~ ! f + i ∆ ∂φ∂ p F · ∇ + i e A ~ ! ¯ f . (5)We note that the momentum derivative of φ terms come in the g equation for anisotropic superconductors. The LDOS is obtained as N s ( ε, r ) = N (0) h Re g R0 + Re g R1 i F + N ′ (0) ε h Re g R0 i F + N ′ (0)2 h Im( f R0 ∆ ∗ φ ) + Im( ¯ f R0 ∆ φ ) i F , (6)where the functions g R0 , and f R0 are the quasiclassical re-tarded Green’s functions which are obtained by solving Eqs.(3) and (5) with the following transformation: g R0 , ( ε, p F , r ) =
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FULL PAPERS g , ( ε n → − i ε + η, p F , r ) and f R0 ( ε, p F , r ) = f ( ε n →− i ε + η, p F , r ), η is a positive infinitesimal constant, and thebarred functions in the real energy formalism are defined gen-erally by ¯ X ( ε, p F , r ) ≡ X ∗ ( − ε, − p F , r ). The electric field is expressed as − λ ∇ E ( r ) + E ( r ) = − π k B Te ∞ X n = h ∇ Im g i F − e N ′ (0) N (0) Z ˜ ε c + ˜ ε c − d ε ¯ n ( ε ) ε D ∇ Re g R0 E F − ce N ′ (0) N (0) ∇ | ∆ | , (7)where λ TF ≡ p d ǫ / e N (0) denotes the Thomas–Fermiscreening length with thickness d
15, 38) and the vacuum per-mittivity ǫ , and the function ¯ n ( ε ) = / (e ε/ k B T +
1) is theFermi distribution function for quasiparticles. The first termon the RHS of Eq. (7) is the PPG term, while the secondand third terms are the contributions from the SDOS pressure.Furthermore, it can be seen that the third term depends on thegradient of the amplitude of the pair potential. The parameter c first introduced by Khomskii et al. is given by c ≡ Z ˜ ε c + ˜ ε c − d ε ε tanh ε k B T c , (8)where T c denotes the superconducting transition temperatureat zero magnetic field. The cuto ff energies ˜ ε c ± are determinedby Z ˜ ε c + ˜ ε c − N s ( ε, r ) d ε = Z ˜ ε c + ˜ ε c − N ( ε ) d ε, N s ( ˜ ε c ± , r ) = N ( ˜ ε c ± ) . (9) ff erencein the homogeneous system We introduce the normal DOS, N ( ε ), expressed as N ( ε ) ≡ Z st BZ d p x d p y (2 π ~ ) δ ( ε − ǫ p + µ ) , (10)where ǫ p denotes the single particle energy. p x and p y are the x and y -components of the quasiparticle momentum, respec-tively, while µ is the chemical potential. We should also re-strict momentum integration in Eq. (10) to the first Brillouinzone. The superconducting DOS in the homogeneous systemis given by N s ( ε ) = N (0) * | ε | q ε − ∆ φ θ ( | ε | − ∆ bulk | φ | ) + F + N ′ (0) (cid:28) sgn( ε ) q ε − ∆ φ θ ( | ε | − ∆ bulk | φ | ) (cid:29) F , (11)where ∆ bulk denotes the gap amplitude in the bulk.The chemical potential di ff erence between the normal andsuperconducting states of the homogeneous system is givenby δµ = − N ′ (0) N (0) Z ˜ ε c + ˜ ε c − d εε ¯ n ( ε ) N bulks0 ( ε ) N (0) − − c N ′ (0) N (0) ∆ , (12)where N bulks0 ( ε ) is the LDOS in the bulk obtained from the Fig. 1.
Schematic representation of quasiparticle momentum axes ˜ p x and˜ p y transformed to p x and p y , respectively by a π/ p y direction. standard Eilenberger equations with the zeroth-order in δ as N bulks0 ( ε ) = N (0) * | ε | q ε − ∆ φ θ ( | ε | − ∆ bulk | φ | ) + F . (13)The details of the derivation of Eqs. (6), (7), (11), and (12)are available in Ref. 11.
3. Numerical Results
We here perform numerical calculations for a quasi-two-dimensional semi-infinite system with a single specular sur-face at x =
0. As a starting point, we introduce the single-particle energy on a two-dimensional square lattice used forhigh- T c superconductors,
14, 39, 40) ǫ p = − t cos ˜ p x a ~ + cos ˜ p y a ~ ! + t cos ˜ p x a ~ cos ˜ p y a ~ − ! + t cos 2 ˜ p x a ~ + cos 2 ˜ p y a ~ − ! , (14)with the lattice constant a , the dimensionless hopping param-eters t / t = / t / t = − /
5, and the momenta ˜ p x and˜ p y given by ˜ p x = ( p x + p y ) / √ p y = ( p y − p x ) / √ d -wave pair-ing as φ = C (cos ˜ p x − cos ˜ p y ), and use v s = in the absenceof external magnetic field and without spontaneous edge cur-rents. This is the same as using ∆ = ∆ ∗ and A = . Herethe real constant C is determined via the normalization con-dition h φ i F =
1, and v s is the superfluid density defined by v s ≡ ( ~ / m )( ∇ ϕ − e A / ~ ), with m and ϕ denoting the electronmass and the phase of the pair potential, respectively.First, we obtain the self-consistent solutions to the stan-dard Eilenberger equations in Eqs. (3) and (4) using the Ric-cati method.
13, 41–43)
The relevant boundary condition used inthe bulk region is obtained by carrying out a gradient ex-pansion up to the first-order, as shown in Appendix A.We also assume mirror reflection at the surface such that Q ( ε n , p F , ≡ Q ( ε n , p ′ F , Q , where p F and p ′ F are the Fermi momenta before and after reflection atthe surface, respectively, and are related by p ′ F = p F − n ( n · p F ),with n = − ˆ x . We note that we need to solve the Riccati-typeequation (see Eqs. (A ·
1) and (A ·
5) in Appendix A) by numer-ical integration towards the − ˆ x direction for v F x <
3. Phys. Soc. Jpn.
FULL PAPERSFig. 2. (Color online) Temperature dependences of the self-consistent gapamplitude for the d x − y -wave state with a smooth [110] surface at x =
0. Attemperatures T = . T c (green solid line), 0 . T c (blue long dashed line), and0 . T c (red short dashed line), for the filling n = . bulk at x = x c ≫ ξ to the surface at x = x direction for v F x > x = x = x c . We also use the solutions obtained by the gradient ex-pansion of the Riccati-type equation up to the first-order (seeAppendix A) in the region of | v F x | ≪ h v F i F , where v F is themagnitude of the Fermi velocity.Next, we solve Eq. (7) to obtain the surface electric fieldwith the boundary conditions where the electric field van-ishes at the surface and the first term on the LHS of Eq. (7)is neglected near the bulk, using Eq. (5) and substituting theGreen’s functions f and g R0 into Eq. (7) accordingly. We ob-tain the retarded Green’s functions by performing the transfor-mation ε n → − i ε + η and using the same procedures as in thecalculation of the Matsubara Green’s functions. The deriva-tives ∂ f /∂ x and ∂ f /∂ p F x in Eq. (5) are also shown in Ap-pendix A. We then calculate the corresponding charge densityusing Gauss’ law, ∇ · E ( x ) = ρ ( x ) / d ǫ . Furthermore, we calcu-late the LDOS, by substituting the retarded Green’s functions g R0 , and f R0 into Eq. (6). We also use g R1 = δ = . t = ∆ , and λ TF = . ξ , where ∆ denotes the gap ampli-tude at zero temperature. We discuss our numerical results in the following. Fig-ure 2 shows the self-consistent gap amplitude for the d -wavepaired superconductor with a [110] oriented surface at thefilling n = .
9. The pair potential shows spatial variationin the surface region. The suppression of the pair-potentialaround the surface is caused by the symmetry of the pair po-tential given by φ ( p F ) = − φ ( p ′ F ) and mirror reflection impliesthat f ( ε n , p F , = f ( ε n , p ′ F , The pair potential is ap-proximately described by ∆ ( x ) = ∆ bulk tanh( x /ξ ), where ξ is the healing length defined as ξ = lim x → x [ ∆ bulk / ∆ ( x )].Note that ξ is well described by the coherence length ξ c ≡ [ h ~ v x φ i F / h φ i F ] ∆ − .In Fig. 3, the superconducting DOS and the normal DOSin the homogeneous system at the filling n = . ε = ˜ ε c + and ˜ ε c − . They connect more smoothlytaking into account higher order derivatives of the DOS at Fig. 3. (Color online) Superconducting DOS N s ( ε ) (green solid line) andthe normal DOS N ( ε ) (red dashed lines) in the homogeneous system attemperature T = . T c , for the filling n = .
9, in units of N (0) over − ∆ ≤ ε ≤ ∆ . Fig. 4. (Color online) Temperature dependence of the chemical potentialdi ff erence δµ between the normal and the superconducting states of the ho-mogeneous system at the filling n = . the Fermi level, although the higher-order derivatives con-tribute little to quantities. Thus, we can perform the fol-lowing calculations using these cuto ff energies ˜ ε c ± . We alsoobtain N ′ (0) ∆ / N (0) = − . − . − . n = .
8, 0 .
9, and 1 .
15, respectively, fromthe normal DOS calculations for each filling. Here we notethat the sign of N ′ (0) ∆ / N (0) for all the fillings is negative,since the Van Hove singularity in the DOS obtained from Eq.(14) is at about n = .
5, but not at a half filling. In Fig. 4,we show the temperature dependence of the chemical poten-tial di ff erence between the normal and superconducting statesof the homogeneous system for the filling n = . ff ective) chemicalpotential di ff erence between the surface and the bulk is whatbrings about the redistribution of charged particles.Furthermore, Fig. 5 shows the total surface charge withcontributions from the PPG force and the SDOS pressure forthe filling n = .
9. The SDOS pressure gives the dominantcontribution to the surface charge within our present model at
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FULL PAPERSFig. 5. (Color online) Total surface charge density (red solid line) due tothe PPG force (blue long dashed line) and the SDOS pressure (green shortdashed line), in units of ρ ≡ d ǫ ∆ / | e | ξ , at temperature T = . T c , for thefilling n = .
9, with η = . n = .
9. We also confirmed that the SDOS pressure was dom-inant at not only n = .
9, but also at n = . n = . Therefore, the contributionfrom the PPG force to the surface charge becomes small, andthe SDOS pressure is dominant in a wide parameter range ofthe semi-finite system, compared to the vortex system. Figure6 shows the total surface charge with contributions from thePPG force and the SDOS pressure for the filling n = . n = . ff erent from the one at n = .
15. To ex-plain this, we expand the quasiclassical Green’s functions interms of the pair potential up to third order, assume that thepair potential has the form ∆ ≃ ∆ bulk tanh( x /ξ ), and substi-tute them into Eq. (7). We then obtain the charge density dueto the PPG force at x = ρ PPG (0) ∼ − a (3) ~ d ǫ ∆ ξ e * φ ∂ v F x ∂ p F x + F , (15)where ρ PPG represents the charge density induced by the PPGforce and a (3) ≡ π k B T P ∞ n = ε − n . As described in Appendix B,this approximation is valid near the critical temperature. Ac-cording to Eq. (15), we see that the filling dependence is de-termined only by h φ ∂ v F x /∂ p F x i ≃ h ∂ v F x /∂ p F x i ∝ ( R (n)H ) xx = ( R (n)H ) yy , where R (n)H is the Hall coe ffi cient. Therefore, thecharge density induced by the PPG force also changes its signaround n = ff erent from that of the VanHove singularity.In Fig. 7, the temperature dependence of the total surfacecharge for the filling n = . Fig. 6. (Color online) Total surface charge density (red solid line) due tothe PPG force (blue long dashed line) and the SDOS pressure (green shortdashed line), in units of ρ ≡ d ǫ ∆ / | e | ξ , at temperature T = . T c , for thefilling n = .
15, with η = . Fig. 7. (Colour online) Temperature dependence of the total surface chargeinduced by the PPG force and SDOS pressure, for the filling n = .
9, in unitsof ρ ≡ d ǫ ∆ / | e | ξ , with η = .
01, at temperatures T = . T c (green shortdashed line), 0 . T c (red solid line), and 0 . T c (blue long dashed line). potential shown in Fig. 2. One may notice that the temperaturedependence of the second order derivative of ρ ( x & x is notmonotonic. The second order derivative of ρ ( x =
0) is givenby ( d ρ ( x ) / dx ) x = ∝ − [ ρ SDOS (0) /ξ + ρ PPG (0) /ξ ],where ξ SDOS ( ξ PPG ) is defined by the value of x at the first peakof the charge density due to the SDOS pressure (PPG force).Thus, not only ρ i (0) but also ξ i is necessary when we consider( d ρ ( x ) / dx ) x = . In the present case, although ρ (0) decreasesmonotonically as the temperature decreases, ( d ρ ( x ) / dx ) x = behaves nonmonotonically because of the competition be-tween ρ SDOS (0) /ξ < ρ PPG (0) /ξ > et al. considered the surface charging in d -wave su-perconductors phenomenologically. In fact, the right-handside in Eq. (7) reproduces the gradient of Eq. (3.1) in Ref.18 with the following approximations: (i) neglect the contri-butions from the PPG force, (ii) use the approximation for ∇ g R0 as ∇ g R0 ≃ ∇ g (2)0 ( ε n → − i ε + η ) (see Appendix B) (iii) c ∼ ln( ε c / k B T c ), where ε c represents the energy cuto ff of theorder of the Debye temperature. With these approximations,
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FULL PAPERSFig. 8. (Color online) LDOS N s ( ε, x ) at x = ξ (blue solid line), and 2 ξ (red long dashes), with η = .
04, at temperature T = . T c , for the filling n = . Fig. 9. (Color online) Deviation δ N PPGs ( ε, x ) − δ N PPGs ( ε, ∞ ), where δ N PPGs ( ε, x ) is the deviation of the LDOS from the Eilenberger solution. Weuse η = .
04, temperature T = . T c , and at electron filling n = . we obtain the electric-field equation equivalent to the gradientof Eq. (3.2) of Ref. 18 as follows:( − λ ∇ + E SDOS ≃ − ˜ c e N ′ (0) N (0) ∇ ∆ , (16)where ˜ c ≡ ln( ε c / k B T c ). From this relation, we find the SDOSpressure is essentially equivalent to the mechanism studied inRef. 18. If we substitute ∆ ( x ) ≃ ∆ bulk tanh( x /ξ ) into Eq. (16)with approximation ( − λ ∇ + E SDOS ≃ E SDOS , we obtaincharge density as ρ SDOS ≃ − ˜ ce N ′ (0) d ǫ ∆ N (0) ξ " − x ξ ! + x ξ ! , (17)which naturally satisfies the charge neutrality. On the otherhand, however, the charge density described in Eq. (3.4) ofRef. 18 is derived by using the asymptotic approximationfrom the bulk to the surface for the source term of the electric-field equation, i.e., the behaviour of ρ SDOS is quite di ff erentfrom the charge density by Ref. 18 around the surface. Specif- Fig. 10. (Color online) Deviation δ N SDOSs ( ε, x ) − δ N SDOSs ( ε, ∞ ), where δ N SDOSs ( ε, x ) is the deviation of the LDOS from the Eilenberger solution.We use η = .
04, temperature T = . T c , and at electron filling n = . Fig. 11. (Color online) Im f R0 , ξ ∂ Im f R0 /∂ x , and h p F x i F ∂ Im f R0 /∂ p F x at thesurface for the momentum direction ϕ k F = π/ η = . ically, the charge density by Emig et al. ( ρ E ) is described by ρ E ( x ) = − ce N ′ (0) d ǫ ∆ N (0) ξ " e − x /λ TF λ TF − e − x /ξ ξ , (18)Using Eqs. (17) and (18), we make a comparison of the accu-mulated charge per unit area within a surface layer of thick-ness x c estimated from the phenomenological analysis byEmig et al. Q p ( x c ) and the charge from our microscopic cal-culation Q m ( x c ). Defining their ratio as γ ( x c ) ≡ Q p / Q m , weobtain γ ( x c ) = R x c ρ E ( x ) dx R x c ρ SDOS ( x ) dx ∼ − ( x c /ξ )( e − x c /λ TF − e − x c /ξ )tanh( x c /ξ ) , (19)where we use ρ SDOS in Eq. (17) to derive the last expression.From this quantity with λ TF ≪ ξ ( e − ξ /λ TF ∼ γ ( x c = ξ ) ∼ . ∼
1, while γ ( x c = ∼ ξ /λ TF ≫
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FULL PAPERS with thickness x c = λ TF ln( ξ / λ TF ) ∼ λ TF was calculated.Therefore, we conclude from this comparison that (1) ourtheory with approximations (i), (ii), (iii) reproduces the for-malism used in Ref. 18, (2) our estimate of the accumulatedcharge within a surface layer of thickness x c & ξ is con-sistent with the estimate in Ref. 18. In addition, we empha-sise the importance of the PPG-force contribution neglected inRef. 18, since it competes with the SDOS pressure and yieldsnontrivial temperature dependence for the surface charge asshown in Fig. 7.Figure 8 plots the normalized LDOS for the filling n = . x = ξ , and 2 ξ . The zero energy peak structure ap-pears as we move from the bulk towards the surface, and theparticle-hole asymmetry exists in the LDOS even around theZES although the peak of the bound states is at the zero energywhen we choose η to be small enough. This may seeminglycontradict the symmetry consideration on the ZES at the [110]surface.
31, 32)
However this particle-hole asymmetry aroundthe zero energy exists even in the LDOS obtained from thecalculation based on the BdG equations as Fig. 10(c) in Ref.28 and based on the T -matrix method without the quasiclas-sical approximation as Figs. 6 and 8 in Ref. 31, similar to ourresult on the surface LDOS. Therefore, it is possible that onlynon-zero energy states near the zero energy may contribute tothe particle-hole asymmetry, but a detailed study based on theBdG equation may be needed to clarify that.Figures 9 and 10 plot the di ff erence in deviations δ N PPGs ( ε, x ) − δ N PPGs ( ε, ∞ ) and δ N SDOSs ( ε, x ) − δ N SDOSs ( ε, ∞ ).Where δ N PPG(SDOS)s ( ε, x ) is the local deviation from theEilenberger solution due to quantum corrections from thePPG force (SDOS pressure) at x = , ξ , ξ . While δ N PPG(SDOS)s ( ε, x = ∞ ) is the local deviation from the Eilen-berger solution due to quantum corrections from the PPGforce (SDOS pressure) in the bulk region. The deviations δ N PPGs ( ε, x ) and δ N SDOSs ( ε, x ) are defined by δ N PPGs ( ε, x ) = N (0) h Re g R1 i F , (20a) δ N SDOSs ( ε, x ) = N ′ (0) ε h Re g R0 i F + N ′ (0)2 ∆ h (Im f R0 + Im ¯ f R0 ) φ i F . (20b)We observe spatial variation in the local particle-hole asym-metry in the LDOS deviations. The di ff erence in LDOS de-viations between the surface and the bulk due to the PPGforce and due to the SDOS pressure show a change in theirlocal peaks around ε = ε = ± . ∆ . Fur-thermore, the behavior of ∂ Im f R0 /∂ x results in the multipleturning points in the LDOS deviation due to the PPG forcearound ε =
0, since the PPG force and the LDOS terms aret he space-momentum Moyal product for ∆ φ and Im f R0 . Fig-ure 11 plots Im f R0 , ∂ Im f R0 /∂ x , and ∂ Im f R0 /∂ p F x at the surfacefor the momentum direction ϕ k F = π/ η = .
04. Here ϕ k F is defined by ϕ k F ≡ arctan k F y / k F x , k F is the Fermi mo-mentum with the W point as the origin, and k F x and k F y havethe same direction as p F x and p F y . The spatial variation in thelocal electron-hole asymmetry suggests the presence of elec-tric charging at the surface. More precisely, to obtain the vari-ation of the charge density near the surface, the integrationof the LDOS over negative energy range must be non-zero,since the expression for the charge density contains the distri- bution function. Although the LDOS deviation due to thePPG force is greater than that due to the SDOS pressure, itsintegral for the negative energy region is greater for the SDOSpressure than for the PPG force. Therefore, the surface chargedue to the SDOS pressure is larger than that due to the PPGforce. The connection between charging in superconductorsand particle-hole asymmetry in the LDOS had already beendiscussed in the mixed state in type-II superconductors.
15, 37)
On the other hand, unlike the vortex case, the contributionfrom the LDOS near the gap edge, rather than near zero en-ergy, is dominant in the surface charge redistribution.One may wonder why we can use the cuto ff energies ˜ ε ± obtained from the DOS in the homogeneous system for thecalculation of the surface charge, despite the fact that the de-viation in Fig. 10 is large even at ε = ± ∆ . We again empha-size the following. Even if the cuto ff energies are increasedbecause the connection is not smooth, the di ff erence betweenthe LDOS only taking into account the first derivative of theDOS at the Fermi level and the normal DOS will only be-come larger. This does not satisfy Eq. (9). To connect themmore smoothly, we need to consider the higher-order deriva-tive of the DOS at the Fermi level, but the higher-order deriva-tives contribute little to quantities. Therefore, we have usedthe same cuto ff energies as those of the homogeneous systemas an approximation. Surface charge measurement in superconductors may bepossible using atomic force microscopy (AFM) in the non-contact mode. It is known that as the atomically sharp tip ofthe AFM cantilever approaches the surface of the supercon-ductive material, the cloud of charged particles around the tipforms an electric dipole. The sample on the other hand, pilesup charged particles in response to this dipole, so as to screenitself.
The observation of surface charge in this scenario re-duces to observing the electrostatic interaction between thedipole at the tip of the cantilever and the charge at the surfaceof the sample.
4. Summary
In summary, we have performed a microscopic calculationon surface charging at a single [110] specularly reflective sur-face of a d -wave superconductor with a Fermi surface usedfor cuprate superconductors, using the augmented quasiclas-sical theory. We have shown that the SDOS pressure gives thedominant contribution to the charging compared to the PPGforce, for all the realistic electron-fillings n = .
8, 0 . .
15 at all temperatures. In addition, since the charge due toPPG force and that due to the SDOS pressure at n = .
15 havethe same signs, the PPG force and the SDOS pressure may in-duce a larger surface charge in electron-doped d -wave super-conductors compared to hole-doped superconductors. Boththe sign and amount of the surface charge depends greatlyon the Fermi-surface curvature. We have also calculated theLDOS within the augmented quasiclassical theory, taking intoaccount the contributions due to the PPG force and the SDOSpressure. At the surface, the LDOS shows a peak structurewhich signifies the presence of ZES. The bulk region showsa (nodal) gap-like structure which is a characteristic of thesuperconducting state. We have also shown the existence ofparticle-hole asymmetry (the SDOS pressure gives a locally
7. Phys. Soc. Jpn.
FULL PAPERS larger particle-hole asymmetry at the filling n = .
9) in theLDOS. This spatially varying local asymmetry suggests thepresence of electric charge.Although our present study is restricted to a smooth sur-face without edge currents, the presence of surface rough-ness is expected to a ff ect the surface states and may conse-quently alter the surface charge. In addition, surface imper-fections appear in the process of fabricating real samples, it istherefore important to consider the e ff ects of surface imper-fections, theoretically. It is relatively easier to consider sur-face roughness within the quasiclassical theory using the ran-dom S-matrix theory or by adding a disorder-induced self-energy.
46, 47)
Furthermore, in the presence of edge currents,the PPG force contribution to the charging e ff ect may be en-hanced due to the appearance of the phase terms of the pairpotential.
15, 17)
Other possible models for the d -wave surfacestate include the presence of a subdominant pairing near thesurface and are characterized by spontaneously broken time-reversal symmetry.
27, 48–50)
Although the experimental identi-fication of the evidence of these admixed states is still con-troversial, due to contradictory experimental data on cupratesuperconductors,
51, 52) they are very interesting. The e ff ects ofbroken time-reversal symmetry on the charging properties atthe surfaces of d -wave superconductors are not consideredwithin our present model. A combination of surface rough-ness and the presence of spontaneous edge currents may re-veal very interesting physics in relation to the surface charg-ing in d -wave superconductors and chiral superconductors.We are grateful to Marie Ohuchi for very insightful discus-sions. E. S. J. is supported by the Ministry of Education, Cul-ture, Sports, Science, and Technology (MEXT) of Japan, H.U. and W. K. are supported in part by JSPS KAKENHI GrantNo. 15H05885 (J-Physics) and JSPS KAKENHI Grant Num-ber 18J13241, respectively. The computation in this work wascarried out using the facilities of the Supercomputer Center,the Institute for Solid State Physics, the University of Tokyo. Appendix A: Boundary conditions based on gradient ex-pansion
We start from the Riccati form of Eq. (3).
13, 41–43) v F x ∂ a ∂ x = − ε n a − ∆ φ a + ∆ φ, (A · a = a ( ε n , p F , x ) is the Riccati parameter and is related to f and g as f = a + a ¯ a , g = − a ¯ a + a ¯ a . (A · of Eq. (3) using the ex-pansion a ≈ a (0) + a (1) , which gives a (0) = ∆ φε n + p ε n + ∆ φ , a (1) = − v F x p ε n + ∆ φ ∂ a (0) ∂ x ,∂ a (0) ∂ x = − a (0)2 p ε n + ∆ φ d ∆ dx φ + a (0) ∆ d ∆ dx . (A · ∂ f /∂ x and ∂ f /∂ p F x in Eq. (5) are ex-pressed as ∂ f ∂ x = + a ¯ a ) ∂ a ∂ x − a ∂ ¯ a ∂ x ! ,∂ f ∂ p F x = + a ¯ a ) ∂ a ∂ p F x − a ∂ ¯ a ∂ p F x ! . (A · ∂ a /∂ x is obtained from Eq. (A · ∂ a /∂ p F x is given bysolving the following equation: v F x ∂∂ x ∂ a ∂ p F x = − ε n ∂ a ∂ p F x − ∆ ∂φ∂ p F x a − ∆ φ a ∂ a ∂ p F x + ∆ ∂φ∂ p F x − ∂ v F x ∂ p F x ∂ a ∂ x , (A · ·
5) used near the bulkis given by ∂ a ∂ p F x ≈ ∂ a (0) ∂ p F x + ∂ a (1) ∂ p F x ,∂ a (0) ∂ p F x = − a (0)2 p ε n + ∆ φ ∆ ∂φ∂ p F x + a (0) φ ∂φ∂ p F x ,∂ a (1) ∂ p F x = v F x ∆ φ ε n + ∆ φ ) ∂φ∂ p F x ∂ a (0) ∂ x − p ε n + ∆ φ ∂ v F x ∂ p F x ∂ a (0) ∂ x + v F x ∂ a (0) ∂ x ∂ p F x ! ,∂ a (0) ∂ x ∂ p F x = a (0)2 ( ε n + ∆ φ ) / ∆ φ d ∆ dx ∂φ∂ p F x − a (0) p ε n + ∆ φ ∂ a (0) ∂ x ∆ ∂φ∂ p F x − a (0)2 p ε n + ∆ φ d ∆ dx ∂φ∂ p F x + ∂ a (0) ∂ x φ ∂φ∂ p F x . (A · Appendix B: Derivation of Eq. (15)
Here, we derive Eq. (15) using the following steps: (i) ex-pand the quasiclassical Green’s functions in terms of pair po-tential up to third order based on the assumptions | ∆ ( x ) | ≪ ∆ and ~ v F x ( d n ∆ / dx n ) = O ( ∆ n + ), which are valid near T c , (ii)substitute the expanded Green’s functions into the first term ofEq. (7), and (iii) neglect the Thomas–Fermi term of Eq. (7).Here, we consider the solutions only in ε n > f = ∞ X ν = f ( ν )0 , g = + ∞ X ν = g ( ν )0 . (B · ·
1) into Eq. (3) with the initial conditions f (0)0 = g (0)0 =
1, and g (1)0 =
0, we obtain the followingrecursive relation: f ( ν )0 = " ∆ φ g ( ν − ε n − ~ v F x ε n ∂ f ( ν − ∂ x , (B · ~ v F x d ∆ / dx = O ( ∆ ). Using Eq. (B ·
2) withinitial conditions and normalization condition g = − f ¯ f , f ( ν )0 and g ( ν )0 up to third order are derived as follows f (1)0 = ∆ φε n , f (2)0 = − ~ φ v F x ε n d ∆ dx ,
8. Phys. Soc. Jpn.
FULL PAPERS f (3)0 = − " ( ∆ φ ) ε n − ( ~ v F x ) ε n d dx ∆ φ, (B · g (2)0 = − ( ∆ φ ) ε n , g (3)0 = . (B · ∆ = ∂ Im g /∂ x , which is included in the elec-tric field equation in Eq. (7), is given by ∂ Im g ∂ x = − ~ ε n " φ ∂ v F x ∂ p F x + v F x φ ∂φ∂ p F x ! d ∆ dx d ∆ dx − v F x φ ∂φ∂ p F x ∆ d ∆ dx . (B · (cid:16) − λ d dx + (cid:17) E PPG x ( x ) ≃ ~ a (3) e " D φ ∂ v F x ∂ p F x E F + D v F x φ ∂φ∂ p F x E F ! d ∆ dx d ∆ dx − D v F x φ ∂φ∂ p F x E F ∆ d ∆ dx , (B · E PPG x represents the x -component of electric field in-duced by the PPG force a (3) ≡ π k B T P ≤ n ≤ n c ε − n . Moreover,neglecting the term related to the Thomas–Fermi screeninglength, and using the assumption λ TF ≪ ξ , we obtain theapproximated charge density as ρ PPG ( x ) ≃ ~ a (3) ǫ e " D φ ∂ v F x ∂ p F x E F + D v F x φ ∂φ∂ p F x E F ! d ∆ dx ! + D φ ∂ v F x ∂ p F x E F d ∆ dx d ∆ dx − D v F x φ ∂φ∂ p F x E F ∆ d ∆ dx . (B · ρ PPG ( x ) ≡ d ǫ ( dE PPG x / dx ). Therefore, ρ PPG ( x ) is ex-pressible only in terms of d n ∆ ( x ) / dx n in high-temperature re-gion where the present approximation is valid. Substitutingthe pair potential assumed as ∆ ( x ) ≃ ∆ bulk tanh( x /ξ ) into Eq.(B ·
6) and taking the limit x →
0, we arrive at Eq. (15).
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