Charging in the Abrikosov lattice of type-II superconductors
aa r X i v : . [ c ond - m a t . s up r- c on ] N ov Anomalous Hall effect in the Abrikosov lattice of type-II superconductors
Marie Ohuchi, Hikaru Ueki, , and Takafumi Kita Department of Physics, Hokkaido University, Sapporo, Hokkaido 060-0810, Japan Department of Mathematics and Physics, Hirosaki University, Hirosaki, Aomori 036-8561, Japan (Dated: November 11, 2020)We study the temperature and magnetic field dependence of the vortex-core charge in theAbrikosov lattice of an s -wave superconductor based on the augmented quasiclassical equationswith the Lorentz force and the pair-potential gradient (PPG) force. It is shown that the chargingis dominated by the spatial derivative terms of the pair potential in the PPG force, where the PPGforce terms are divided into the two parts, namely, (i) the spatial derivative of the pair potentialand (ii) the product of the vector and pair potential. This originates from the fact that the Lorentzforce and vector potential terms almost cancel each other out. We also clarify that the vortex-corecharging due to the PPG force is caused by the anomalous Hall effect on circulating supercurrents,since the charge is large even in an isolated vortex and in the limit of high- κ , which is also thezero field limit, and originates primarily from the phase of the pair potential, i.e. supercurrents.Moreover, we see that the vortex-core charge peaks near zero temperature and about half the uppercritical field, which makes it the most suitable region for a measurement of the charge. I. INTRODUCTION
It is known that vortices in type-II superconductorshave not only a single magnetic flux quantum but alsoaccumulated charge. This vortex-core charging has at-tracted the interest of many researchers, and this hasled to numerous theoretical studies. The earliest studieson the vortex-core charging were carried out based onthe phenomenological theory [1–3] and the Bogoliubov–de Gennes (BdG) equations [4–6]. London included theLorentz force acting on supercurrents in his phenomeno-logical equation, which provides a transparent way toshow the existence of the vortex-core charge due to theLorentz force acting on circulating supercurrents [1, 7].Khomskii and Freimuth also calculated the vortex-corecharge due to the chemical potential difference betweenthe vortex-core region in the normal state and its sur-rounding region in the superconducting state by adopt-ing the pair potential in the form of the step function[2, 8]. Matsumoto and Heeb first performed a self-consistent calculation based on the BdG equations, in-cluding Maxwell’s equations, in order to calculate thevortex-core charging in a chiral p -wave superconductor[5]. The vortex-core charging in a chiral p -wave super-conductor was also calculated by using the Ginzburg–Landau (GL) Lagrangian with the Chern–Simons term[9]. Moreover, the dynamical dipole charge in the vor-tex core in type-II superconductors under an AC electro-magnetic field was calculated by Eschrig et al . [10, 11].It was also shown that the electric charge accumulateseven at the vortex core of electrically neutral p -wave su-perfluids, although it is small compared with the coreof superconductors [12]. Experimentally, the vortex-corecharge in cuprate superconductors was estimated usingthe NMR/NQR measurements [13].Recently, the augmented quasiclassical (AQC) equa-tions of superconductivity with the three force termswere derived by incorporating the next-to-leading-ordercontributions in the expansion of the Gor’kov equa- tions [14, 15] in terms of the quasiclassical parameter δ ≡ /k F ξ to study the vortex-core charging in type-IIsuperconductors [16], where k F is the amplitude of theFermi wavenumber and ξ ≡ ~ v F / ∆ is the coherencelength with the energy gap ∆ at zero magnetic fieldand zero temperature, and the amplitude of the Fermivelocity v F . It has been elucidated that the vortex-corecharging is caused by the three forces: (i) the Lorentzforce that acts on supercurrents [1, 17, 18], (ii) pair-potential gradient (PPG) force [16, 19–21], and (iii) thepressure difference arising from the slope in the densityof states (SDOS) [2, 8, 16]. In the isolated vortex sys-tem near the lower critical field, the vortex-core chargewas calculated in s -wave superconductors with a cylin-drical Fermi surface [21] and a spherical Fermi surface[16] based on the AQC equations with the three forceterms. We found that the vortex-core charging in an s -wave superconductor with a spherical Fermi surfaceis dominated by the PPG force near zero temperatureand by the SDOS pressure near the transition tempera-ture when the magnetic penetration depth is larger thanthe coherence length, and the PPG force is dominant atlow temperatures even if the magnetic penetration depthis almost the same as the coherence length [16]. Thus,the PPG force contributes dominantly to the vortex-corecharge in a wide parameter range within the isolated vor-tex system. Masaki also studied the vortex-core chargingin s - and chiral p -wave superconductors with an isolatedvortex based on the AQC equations with the Lorentz andPPG forces, and pointed out that the angular parts com-ing from the phase of the pair potential in the PPG forceterms contribute dominantly to the vortex-core charg-ing [22]. The PPG force terms are divided into threeparts: the radial parts, which are the radial derivativesof the cylindrical coordinates around the vortex center,the angular parts, which are the angular derivatives, andthe vector potential terms coming from the gauge invari-ance of the AQC equations. Therefore, the vortex-corecharge is very small if the chirality and vorticity are an-tiparallel, i.e. L z = 0, since the phase coming from eachcancels out, where L z is the total angular momentum.If the chirality and vorticity are parallel, i.e. L z = 2,the vortex-core charge is enhanced compared to that in s - and antiparallel chiral p -wave superconductors since avortex has only the vorticity of L z = 1 in s -wave super-conductors. These results are consistent with those basedon the BdG equations obtained by Matsumoto et al . [5].While vortex-core charging due to the PPG force is yet tobe fully understood, it is well known that the vortex-corecharging due to the Lorentz force comes from the mag-netic Hall effect due to the Lorentz force acting on cir-culating supercurrents, and we can understand roughlythat the vortex-core charging due to the SDOS pressureis caused by the (effective) chemical potential differencebetween the core and its surrounding region by assuminga roughly normal metal at the core [16]. On the otherhand, one may think about the difference between thecharging due to the PPG force and the SDOS pressure,since the chemical potential is exactly equal in the normaland superconducting states of the homogeneous systemwhen considering only the PPG force [16]. We discussthis issue based on the recent results [16, 21, 22] and thecalculation in this present paper.The magnetic field dependence of the vortex-corecharge was calculated in s -wave superconductors with acylindrical Fermi surface [23] and in d -wave superconduc-tors with anisotropic Fermi surfaces used for cuprates [24]based on the AQC equations with only the Lorentz force.It was shown that the charge density at the core has alarge peak as a function of the magnetic field. There-fore, the vortex-core charge due to the Lorentz force maybe larger than that due to the PPG force in a strongmagnetic field region. The main purpose of this presentpaper is to develop a numerical method for calculatingthe temperature and magnetic field dependence of thevortex-core charge in the Abrikosov lattice [25] of type-IIsuperconductors microscopically within the AQC equa-tions, to study the forces responsible for the charging inthe Abrikosov lattice, and to clarify the temperature andmagnetic field dependence of the vortex-core charge inthe Abrikosov lattice.To this end, we calculate the temperature and mag-netic field dependence of the vortex-core charge in a two-dimensional s -wave superconductor due to the Lorentzand PPG forces using the AQC equations of supercon-ductivity in the Matsubara formalism. The SDOS pres-sure terms can be neglected in the case of superconduc-tors with a cylindrical Fermi surface [21]. A recently usedmethod may be more useful, but this formulation stillonly incorporates the Lorentz force [26]. We here per-form the numerical calculation of the vortex-core charg-ing combining the methods in Refs. [21] and [23].This paper is organized as follows. In Sect. II, wepresent the formalism based on the AQC equations withthe Lorentz and PPG forces in the Matsubara formal-ism, and show that the PPG force can be neglected inthe Meissner state. In Sect. III, we give numerical results for the charging in the vortex lattice of an s -wave super-conductor with a cylindrical Fermi surface, and discussthe vortex-core charging due to the PPG force. In Sect.IV, we provide a conclusion. II. FORMALISMA. Augmented quasiclassical equations
For clean s -wave superconductors in equilibrium, theAQC equations with the PPG and Lorentz forces in theMatsubara formalism are given by [16, 21] (cid:2) iε n ˆ τ − ˆ∆ˆ τ , ˆ g (cid:3) + i ~ v F · ∂ ˆ g + i ~ e ( v F × B ) · ∂∂ p F (cid:8) ˆ τ , ˆ g (cid:9) − i ~ ∂ ˆ∆ˆ τ · ∂ ˆ g∂ p F − i ~ ∂ ˆ g∂ p F · ∂ ˆ∆ˆ τ = ˆ0 , (1)where ˆ g = ˆ g ( ε n , p F , r ) and ˆ∆ = ˆ∆( r ) are the quasiclas-sical Green’s functions and the pair potential, respec-tively, ε n = (2 n + 1) πk B T is the fermion Matsubara en-ergy ( n = 0 , ± , · · · ) with k B and T denoting the Boltz-mann constant and temperature, respectively, v F and p F are the Fermi velocity and momentum, respectively, e < B = B ( r ) is the magnetic-flux density, ∂ is the gauge-invariant differential opera-tor. 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 first line in Eq. (1) corresponds tothe standard Eilenberger equations [27–30], the secondline is the Lorentz force terms [17, 18], and the third lineis the PPG force terms [16, 20, 21]. It may not be accu-rate to call the second line the Lorentz force since the vec-tor potential appears in the other terms [31], but we willdo so here for convenience. We also assume spin-singletpairing without spin paramagnetism. The matrices ˆ g , ˆ∆and ˆ τ are then expressible as [28]ˆ g = (cid:20) g − ifi ¯ f − ¯ g (cid:21) , ˆ∆ = (cid:20) ∗ (cid:21) , ˆ τ = (cid:20) − (cid:21) , (2)where the barred functions are defined generally by¯ X ( ε n , p F , r ) ≡ X ∗ ( ε n , − p F , r ), and the operator ∂ isgiven by ∂ ≡ ∇ on g or ¯ g ∇ − i e A ~ on f or ∆ ∇ + i e A ~ on ¯ f or ∆ ∗ , (3a)with A = A ( r ) denoting the vector potential.Following the procedure used in Ref. [7], we expand g and f formally in terms of δ as g = g + g + · · · and f = f + f + · · · , where g and f are the solutions ofthe standard Eilenberger equations. The standard Eilen-berger equations are given by [27–30] ε n f + 12 ~ v F · (cid:18) ∇ − i e A ~ (cid:19) f = ∆ g , (4a)∆ = Γ πk B T ∞ X n = −∞ h f i F , (4b) ∇ × ∇ × A = − i πeµ N (0) k B T ∞ X n = −∞ h v F g i F , (4c)with the normalization condition g = sgn( ε n ) (cid:0) − f ¯ f (cid:1) / [28, 29, 32]. Here, Γ ≪ h· · · i F is the Fermi surface average normalized as h i F = 1, µ is the vacuum permeability, and N (0) is thenormal density of states per spin and unit volume at theFermi energy. Equation (4) forms a set of self-consistentequations for f , ∆, and A .The equation for g can be obtained from Eq. (1) as[16, 21] v F · ∇ g = − e ( v F × B ) · ∂g ∂ p F − i ∂ ∆ ∗ · ∂f ∂ p F − i ∂ ∆ · ∂ ¯ f ∂ p F , (5)with g = − ¯ g . The electric field E = E ( r ) obeys [16, 21] − λ ∇ E + E = i πk B Te ∞ X n = −∞ h ∇ g i F , (6)where λ TF ≡ p ǫ d/ e N (0) is the Thomas–Fermi screen-ing length with ǫ and d denoting the vacuum permittiv-ity and the thickness, respectively [22, 33, 34]. This equa-tion enables us to calculate the electric field and chargedensity microscopically. B. Meissner state
Since the angular parts which originate from the phaseof the pair potential in the PPG force terms contributedominantly to the charging in an isolated vortex of type-II superconductors, one may predict that the PPG forceacts on supercurrents even in the Meissner state andcauses the Hall effect near a surface. However, we showbelow that the PPG force does not act on supercurrentsin the Meissner state. To this end, we derive an expres-sion for the Hall electric field in the Meissner state bysolving the AQC equations to study the action of thePPG force on supercurrents. We first express the pairpotential and the anomalous Green’s function as ∆( r ) = | ∆( r ) | e iϕ ( r ) and f ( ε n , p F , r ) = ˜ f ( ε n , p F , r )e iϕ ( r ) , re-spectively, substitute them into Eq. (4a), and neglectthe spatial derivative of ˜ f . Then we obtain the Doppler-shifted Green’s functions in the standard Eilenberger equations as g = ˜ ε n p ˜ ε n + | ∆ | , (7a)˜ f = | ∆ | p ˜ ε n + | ∆ | , (7b)where ˜ ε n is defined by ˜ ε n ≡ ε n + im v F · v s with thesuperfluid velocity v s ≡ ( ~ / m )( ∇ ϕ − e A / ~ ) and theelectron mass m . We also substitute Eq. (7) into Eq. (5)and then obtain v F · ∇ g = − e ( v F × B ) · ∂∂ p F ˜ ε n p ˜ ε n + | ∆ | . (8)Thus, the PPG force terms all cancel each other out.Since the Doppler shift method is an approximation thatneglects the low energy excitations such as the vortex andsurface states, we find that the low energy excitations areimportant for the PPG force to work.We next assume that the gap amplitude is spatiallyconstant as | ∆( r ) | = | ∆ | , and expand g , f , and v F · ∇ g up to the first-order in v s as g = ε n p ε n + | ∆ | + im v F · v s | ∆ | ( ε n + | ∆ | ) / , (9a)˜ f = | ∆ | p ε n + | ∆ | − im v F · v s ε n | ∆ | ( ε n + | ∆ | ) / , (9b) v F · ∇ g = − ie ( v F × B ) · ∂∂ p F m v F · v s | ∆ | ( ε n + | ∆ | ) / . (9c)Substituting Eqs. (9a) and (9b) into Eqs. (4b) and (4c),respectively, and using h v F i F = , we obtain the gapequation and the London equation [28]. Furthermore,considering the region outside the vortex core or a sur-face without any spatial variation in the gap amplitude,substituting Eq. (9c) into Eq. (6), and using g = − ¯ g ,the equation for the electric field in the Meissner state isgiven by − λ ∇ E + E = B × R H j , (10)where j and R H are the current density and the Hall co-efficient tensor, respectively, in the Meissner state givenby j = meN (0)(1 − Y ) h v F v F i F v s , (11a) R H = 12 eN (0) (cid:28) ∂∂ p F v F (cid:29) F h v F v F i − , (11b)with Y = Y ( T ) denoting the Yosida function [7, 28, 35]defined by Y ≡ − πk B T ∞ X n =0 | ∆ | ( ε n + | ∆ | ) / . (12) FIG. 1. Gap amplitude | ∆( r ) | at temperature T = 0 . T c in units of the zero temperature gap ∆ on a square grid with x and y ranging from [ − ξ , +2 ξ ] for the average flux densities (a) ¯ B = 0 . B c2 , (b) ¯ B = 0 . B c2 , and (c) ¯ B = 0 . B c2 .FIG. 2. Magnetic-flux density B ( r ) at temperature T = 0 . T c in units of B ≡ ~ / | e | ξ on a square grid with x and y rangingfrom [ − ξ , +2 ξ ] for the average flux densities (a) ¯ B = 0 . B c2 , (b) ¯ B = 0 . B c2 , and (c) ¯ B = 0 . B c2 . This is the same as the result obtained in Ref. [7], de-spite taking into account the PPG force. Therefore, theLorentz force acts on supercurrents in the Meissner stateas calculated in the previous work [7], but the PPG forcedoes not. We can also show that the PPG force doesnot act on the shielding currents in anisotropic [34] andchiral [22] superconductors based on the correspondingAQC equations.
III. NUMERICAL RESULTSA. Numerical procedures
We solve Eqs. (4), (5), and (6) numerically for a trian-gular vortex lattice of an s -wave type-II superconductorwith a cylindrical Fermi surface based on the methodsin Refs. [21] and [23]. We take the magnetic field to bealong the axial direction of the cylinder. The correspond-ing vector potential is expressible in terms of the averageflux density ¯ B = (0 , , ¯ B ) as ¯ A ( r ) = ( ¯ B × r ) / A ( r ), where ˜ A denotes the spatial variation of the flux density.Functions ˜ A ( r ) and ∆( r ) for the triangular lattice obeythe following periodic boundary conditions [36–38]:˜ A ( r + R ) = ˜ A ( r ) , (13a)∆( r + R ) = ∆( r )e i | e | ~ ¯ B · ( r × R )+ iπn n , (13b)where R = n a + n a with the integers n and n ,and a = a (1 / , √ / ,
0) and a = a (0 , ,
0) are thebasic vectors of the triangular lattice with the length a determined by the flux-quantization condition ( a × a ) · ¯ B = h/ | e | .To start with, we solve the standard Eilenberger equa-tions (4) self-consistently for the vortex lattice using theRiccati method [28, 39–41]. The solution is substitutedinto the right-hand side of Eq. (5), which is solved byusing the standard Runge–Kutta method. We next ob-tain the electric field substituting the solution of Eq. (5)into Eq. (6) and solving Eq. (6), and then calculate thecharge density ρ using the Gauss’ law ρ = ǫ d ∇ · E numerically. The results present below are for λ TF =0 . ξ , λ = 5 ξ and δ = 0 .
03, where λ is the mag- (i) PPG (DM) , ¯ B = 0 . B c2 (h) PPG (DM) , ¯ B = 0 . B c2 (g) PPG (DM) , ¯ B = 0 . B c2 (d) PPG (PM) , ¯ B = 0 . B c2 (e) PPG (PM) , ¯ B = 0 . B c2 (f) PPG (PM) , ¯ B = 0 . B c2 (a) Lorentz , ¯ B = 0 . B c2 (b) Lorentz , ¯ B = 0 . B c2 (c) Lorentz , ¯ B = 0 . B c2 FIG. 3. Charge density ρ ( r ) due to the Lorentz force ((a), (b), and (c)), and the paramagnetic ((d), (e), and (f)) and diamagnetic((g), (h), and (i)) terms in the PPG force at temperature T = 0 . T c in units of ρ ≡ ∆ ǫ d/ | e | ξ on a square grid with x and y ranging from [ − ξ , +2 ξ ] for the average flux densities ¯ B = 0 . B c2 , ¯ B = 0 . B c2 , and ¯ B = 0 . B c2 from right to left,respectively. PM and DM denote paramagnetic and diamagnetic, respectively. netic penetration depth at zero temperature defined by λ ≡ (cid:2) µ N (0) e h v i F (cid:3) − / . The average magnetic fluxdensity, which is a parameter, is normalized by usingthe upper critical field B c2 = µ H c2 obtained from theHelfand–Werthamer theory [42, 43]. B. Results
Figures 1 and 2 plot the spatial variations of the gapamplitude | ∆( r ) | and the z -component of the magnetic- flux density B ( r ), respectively, at temperature T = 0 . T c for the average flux densities ¯ B = 0 . B c2 , ¯ B = 0 . B c2 ,and ¯ B = 0 . B c2 , respectively. It is shown that the gapamplitude away from the vortex center and its slope atthe vortex center become small together, and the B/ ¯ B at vortex center is also small, i.e. the flux density be-comes spatially uniform at strong magnetic fields com-pared with weak fields. We also find that the distancebetween the vortices becomes closer and the six-fold sym-metrical anisotropy becomes stronger at strong magneticfields. Thus, we have reproduced the previous work pro- -60-40-20 0 20 40 60 0 0.2 0.4 0.6 0.8 1 -1-0.5 0 0.5 1 1.5 0 0.2 0.4 0.6 0.8 1 (a) T = 0 . T c Lorentz Lorentz ¯ B/B c2 ρ ( ) / ρ ρ ( ) / ρ ¯ B/B c2 (b) T = 0 . T c PPG (PM)
PPG (DM)
PPG (PM)
PPG (DM)
FIG. 4. Charge density at the vortex center ρ ( ) due to the Lorentz force (green circular points), and the paramagnetic (bluesquare points) and diamagnetic (red triangular points) terms in the PPG force in units of ρ ≡ ∆ ǫ d/ | e | ξ as a functionof the magnetic field calculated for temperatures (a) T = 0 . T c and (b) T = 0 . T c . PM and DM denote paramagnetic anddiamagnetic, respectively. posed by Ichioka et al. [38].In Fig. 3, we show plots for the spatial dependence ofthe charge density due to the Lorentz force, and the spa-tial derivative terms of the pair potential and the termsfor the product of the vector and pair potential in thePPG force, respectively, at temperature T = 0 . T c forthe average flux densites ¯ B = 0 . B c2 , ¯ B = 0 . B c2 ,and ¯ B = 0 . B c2 , respectively. The contribution of thespatial derivative and vector potential terms in the PPGforce comes from the PPG force acting on paramagneticand diamagnetic supercurrents, respectively. Thereforewe call the spatial derivative and vector potential termsthe paramagnetic and diamagnetic terms, respectively.We see that the charges are accumulated at the corewhere the pair potential is zero. The charge redistri-bution due to the Lorentz force and the paramagneticterms in the PPG force are very similar, but the chargedue to the paramagnetic terms in the PPG force is largerat weak magnetic fields compared to that due to theLorentz force. On the other hand, the charge due tothe diamagnetic terms in the PPG force has an oppositesign and almost the same magnitude as that due to theLorentz force. We also find that larger charge accumu-lates between the vortices at strong magnetic fields sincethe distance between vortices becomes smaller.Figure 4 shows the magnetic field dependence of thecharge density at the vortex center for temperatures T = 0 . T c and T = 0 . T c , respectively. We confirmthat the vortex-core charge due to the Lorentz force hasa large peak of upper convexity as shown in the previouswork [23]. However, the vortex-core charge due to theLorentz force calculated by us is about 10 times largerthan that calculated in the previous work [23], even if weuse the same quasiclassical parameter. This is becausethe method in the previous work neglected the compo-nent perpendicular to the Fermi velocity of in ∇ g . Wehave here adopted a direct method of solving the equa- tion for g [Eq. (5)] in Refs. [16, 21]. We also notethat the vortex-core charge has the λ TF -dependence andis not linear with respect to δ , with δ = λ TF /ξ , evenwithin the AQC theory. On the other hand, the vortex-core charge due to the diamagnetic terms in the PPGforce has an opposite sign and almost the same magni-tude as that due to the Lorentz force at all temperaturesand magnetic fields within our calculation. Therefore,the charge due to the Lorentz force and the diamagneticterms in the PPG force almost cancel each other out.Consequently, the total charge is almost the same as thatdue to the paramagnetic terms in the PPG force. We alsofind that the vortex-core charge due to the paramagneticand diamagnetic terms, respectively, in the PPG forcealso has a peak at about half the upper critical field asa function of the magnetic field. This may be due tothe fact that the paramagnetic and diamagnetic super-currents also have peaks because it is expected that themagnetic field dependence of the charge due to the PPGforce is the same as that of the supercurrents. Basedon this, we can confirm that both the charge due to allthe PPG force terms and the total supercurrents aroundthe core decrease monotonically with an increase in mag-netic field. We can also explain that the peak of the totalcharge comes from the competition between the chargesdue to the PPG force, which is dominant at weak fields,and the Lorentz force, which is dominant at strong fields.Thus, measurements should be performed at low temper-atures and about half the upper critical field to detect thevortex-core charge experimentally.To compare our result with the vortex-core charge es-timated by the NMR/NQR measurements [13], we nextcalculate the order of magnitude for the accumulatedcharge around a vortex. We here adopt the core regionof radius 0 . ξ and the thickness d = 10 ˚A to roughly es-timate the peak value of the accumulated charge Q . Thevortex-core charge in YBCO is given by Q ∼ − | e | for the following appropriate parameters: k − ≃ . ≃
28 meV [44], and ξ ≃
30 ˚A [13]. Theamount of charge estimated based on our present cal-culation is an order of magnitude larger than the chargereported in Ref. [23], owing to the difference in the calcu-lation method and the Thomas–Fermi screening lengthdescribed above. The order of magnitude of the esti-mated charge in YBCO is roughly consistent with theexperimental results by Kumagai et al. [13].We finally discuss what the vortex-core charging due tothe PPG force is. The vortex-core charge due to the PPGforce at weak magnetic fields has the following charac-teristics: (i) the dominant contribution from the angularparts, i.e. the angular derivatives of the cylindrical coor-dinates around the vortex center [22], (ii) the PPG forceacts on only supercurrents in the vortex state in the core,and (iii) it is relatively large even in the high- κ limit andin the isolated vortex system [16, 21]. The PPG forceterms are dominated by the angular parts coming fromthe phase of the pair potential, and they all cancel eachother out away from the core as shown in Subsect. II B.Hence the presence of supercurrents is essential to PPGforce charging in the vortex state. Furthermore, since thevortex-core charge is relatively large even in the high- κ limit and in the isolated vortex system, the large vortex-charging is caused by the PPG force acting on circulatingsupercurrents in the vortex state even in an area that canbe considered zero magnetic field. Thus, we can deducethat the vortex-core charging due to the PPG force isthe anomalous Hall effect on supercurrents in the vor-tex state. Fujimoto showed that the Berry phase effectassociated with a spatially modulated superconductingorder parameter, i.e. the PPG force, gives rise to a fic-titious Lorentz force acting on quasiparticles, based onthe augmented quasiclassical theory similar to ours [45].Therefore, we may also infer that the vortex-charging dueto the PPG force is caused by the Berry phase effect onsupercurrents in the vortex state. IV. CONCLUSION
We have developed a numerical method for the study ofcharging in the vortex lattice state of type-II supercon-ductors based on the AQC equations with the Lorentzand PPG forces. Using it, we have calculated the chargedistribution in the vortex lattice of s -wave superconduc-tors with a cylindrical Fermi surface. We have shownthat the vortex-core charge due to the Lorentz force andthe terms for the product of the vector and pair potential coming from diamagnetic supercurrents in the PPG forcealmost cancel each other out, and the total vortex-corecharge becomes almost the same as that due to the spatialderivative terms of the pair potential coming from para-magnetic supercurrents in the PPG force. We have alsofound that low temperatures and magnetic fields abouthalf the upper critical field are suitable for the experi-mental measurement of electric charge in the vortex core.Moreover, we have shown that the PPG force does notcontribute to the charging in the the Meissner state. Onthe other hand, the PPG force contributes to the vortex-core charging even at zero field such as the isolated vortexsystem and the high- κ limit, and is dominated by theterms of the pair potential phase, which are related tosupercurrents. Therefore, it can be understood that thevortex-core charging due to the PPG force is caused bythe anomalous Hall effect on supercurrents in the vortexstate. We again emphasize that only the Lorentz forceacts on supercurrents away from the vortex core and theLorentz force may be important for transport phenomena[20, 46].There still remains many interesting problems in rela-tion to the study of vortex lattice systems using the AQCequations. For example, our present method can be usedto study the flux-flow Hall effect in the vortex lattice bycombing the AC response theory based on the standardEilenberger equations [10, 11], and to also calculate thevortex lattice in He [47–50].Kumagai et al. estimated the vortex-core charge incuprate superconductors by the NMR/NQR measure-ments. However, they used the local electric field gra-dient obtained from changes in the nuclear quadrupoleresonance frequency to estimate the vortex-core chargeexperimentally. To the best of our knowledge, direct ob-servation of the vortex-core charge such as the atomicforce microscopy measurement has not been achieved yet.We hope that our present study will stimulate more de-tailed experiments on vortex-core charging.
ACKNOWLEDGMENTS
We thank T. Uchihashi, J. Goryo, R. C. Regan, A.Kirikoshi, and E. S. Joshua for useful discussions andcomments. H.U. is supported in part by JSPS KAK-ENHI Grant No. 15H05885 (J-Physics). The computa-tion in this work was carried out using the facilities ofthe Supercomputer Center, the Institute for Solid StatePhysics, the University of Tokyo. [1] F. London,
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