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SciPost Phys. 8, 061 (2020)
Hall anomaly and moving vortex chargein layered superconductors
Assa Auerbach1? and Daniel P. Arovas2
1 Physics Department, Technion, Haifa, Israel2 Department of Physics, University of California at San Diego,
La Jolla, California 92093, USA
Abstract
Magnetotransport theory of layered superconductors in the flux flow steady state isrevisited. Longstanding controversies concerning observed Hall sign reversals are re-solved. The conductivity separates into a Bardeen-Stephen vortex core contribution, anda Hall conductivity due to moving vortex charge. This charge, which is responsible forHall anomaly, diverges logarithmically at weak magnetic field. Its values can be extractedfrom magetoresistivity data by extrapolation of vortex core Hall angle from the normalphase. Hall anomalies in YBa2Cu3O7, Bi2Sr2CaCu2O8−x , and Nd1.85Ce0.15CuO4−y dataare consistent with theoretical estimates based on doping dependence of London pene-tration depths. In the appendices, we derive the Streda formula for the hydrodynamicalHall conductivity, and refute previously assumed relevance of Galilean symmetry to Hallanomalies.
Copyright A. Auerbach and D. P. ArovasThis work is licensed under the Creative CommonsAttribution 4.0 International License.Published by the SciPost Foundation.
Received 16-01-2020Accepted 25-03-2020Published 17-04-2020
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doi:10.21468/SciPostPhys.8.4.061
Contents
1 Introduction 1
2 Flux Flow Steady State 3
3 Core and transport currents 5
4 Hall conductivity of charged vortices 7
5 Screened Ginzburg-Landau theory of the MVC 8
6 Extraction of Qv from experiment 11
7 MVC of cuprate superconductors 12
8 Inhomogeneous flow and fluctuations 13
9 Discussion 14
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A Dynamical Hall conductivity and Streda formula 16
B Invalidity of Galilean Hall conductivity in flux flow regime 17
References 18
1 Introduction
The Hall effect in the flux flow (FF) regime of superconducting films has long been an in-triguing and controversial subject. The pioneering theory of Bardeen and Stephen (BS) [1]predicted that the Hall sign is inherited from the normal phase which persists inside the vortexcores. Soon thereafter, in a challenge to BS theory, Hall sign reversals have been measuredin diverse superconductors [2–6]. This effect, named “Hall anomaly”, is illustrated in Fig. 1.Proposed explanations included the effects of disorder [2], thermal excitations [7], interlayervorticity [8], and vortex charge [6, 9, 10]. These effects were incorporated into the FF trans-port theory by vortex dynamics equations [9, 11, 12], and time dependent Ginzburg-Landautheory [13–15].
However, the microscopic origin of non-dissipative vortex forces and imaginary relaxationrates, as well as the definition of the relevant vortex charge in a screened environment, havebeen subjects of ongoing debates, lasting for more than 50 years.
In this paper, we take on the Hall anomaly problem with a new approach to flux flowtransport theory. This approach allows us to show that an additional Hall current is carried bymoving vortex charge (MVC), which does not vanish in a layered superconductor geometry.The MVC can be estimated from independently measured thermodynamic coefficients, and
Figure 1: Bardeen Stephen theory and Hall anomaly in the flux flow regime. For thisillustration, vortex cores are assumed to obey Drude theory of metals. The deviationof Hall resistivity from Bardeen Stephen theory, and its sign reversal, are ascribed tomoving vortex charge.
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compared to the value extracted from magnetoresistivities data.The strategy of this paper is as follows.
(i) We first revisit FF transport theory. While a vortex dynamical equation was used toexplain dissipative FF transport [16, 17], its form and some of its coefficients have notbeen microscopically derived. The source of the difficulty might be in computing theresistivity by imposing a bias current on superconductors without Galilean symmetry(see expanded discussion in Section 9).
Here, in contrast to previous approaches [1–17], we completely avoid bias currents,vortex forces and their phenomenological equations of motion. The mean vortex velocityin the steady state is constrained by the external electric field. The current is calculatedas a linear response to the applied electric field, and Galilean symmetry is not assumed.
(ii) We show that the conductivity (per layer) separates into two additive terms,
σαβ =�
Bc2(T )B
�
σcoreαβ + εαβ
2|e|Qv
h. (1)
e < 0 is the electron charge, and εαβ with α,β ∈ {x , y} is the antisymmetric tensor.The first term recovers BS formula [1] which describes the transport currents generatedinside the moving vortex cores. B and Bc2 are the magnetic field and upper critical fieldrespectively. The second term is due to the MVC, whose value is Qv.
(iii) We use the Streda formula [18, 19] to relate Qv to the extra charge induced into thesuperconducting layer by the addition of one vortex. In isotropic superconductors, Qv=0due to in-plane screening by co-moving charges. In layered superconductors, screeningin dopant (weakly superconducting) layers results in,
Qv =Q0 log(αBc2/B) . (2)
Q0 is proportional to the derivative of superfluid stiffness with respect to electron den-sity, and to the interlayer dielectric constant, which can be experimentally determinedwithout microscopic knowledge of the normal state correlations. α represents vortexcore properties and is of order unity. Generically, Q0 has opposite sign to first term inEq. (1), which can produce the Hall sign reversal depicted in Fig. 1.
(iv) Qv can be extracted from experimental magnetresistivity data, and compared to Eq. (2),using a Hall angle extrapolation for estimating the BS term. Values extracted for holedoped [6, 20], and electron doped [20] cuprates are consistent with Eq. (2), where Q0can be fit to independent estimates of the doping derivative of London penetration depthand interlayer dielectric constant.
We briefly discuss effects of inhomogeneous pinning at low magnetic field, and supercon-ducting fluctuations. The paper ends with a summary of our results and their comparison toprevious theories.
2 Flux Flow Steady State
We consider a homogeneous thin film of a type-II superconductor, below the zero field tran-sition temperature T<T (0)c , A magnetic field B = Bz induces a two dimensional (2D) vortex
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density 1 nv = B/Φ0 = (2πl2B)−1, where Φ0 = hc/(2|e|) is the Josephson flux quantum. The FF
regime is defined by (see Fig. 1),
max§
Bmelt(T ) ,Φ0
2πλ2(T )
ª
< B(T ) < Bc2(T ), (3)
where Bc2 ≡ Φ0/2πξ2, ξ is the vortex core radius, and Bmelt is the vortex lattice melting
field [17, 21, 22]. For thin enough films, London’s penetration depth can easily exceed theinter-vortex separation λ � lB. Hence, the magnetic field is approximately uniform and amore appropriate term for “flux flow” would be “vorticity flow”.
In cuprates, and other highly anisotropic layered superconductors, Bmelt(T ) � Bc2(T ) atlow temperatures [23]. In the FF regime defined in Eq. (3), the critical current is zero, orimmeasurably small, such that we will later be able to apply linear response theory to computethe longitudinal and transverse conductivities.
The global phase field of the superconductor, outside the vortex cores, can be separatedinto the vorticity phase and transport phase:
φ(x) = φv(x) +φtr(x)
φv = sgn(e)Nv∑
i=1
arg(x−Xi) ,(4)
where Xi are the vortices’ positions, and∮
d` · ∇φtr = 0 on any orbit.Henceforth we fix the gauge choice to be A0(x, t) = 0 throughout the paper. For a config-
uration of static vortices φ(x) = 0. We introduce an external DC electric field into the vectorpotential as A = AB − cE t, where ∇×AB = Bz. If φ(x) remains time-independent, the 2Dcurrent density will increase linearly in time, viz.
j(t) =2e~ρs
�
∇φ −2e~c
A(t)�
∝ E t , (5)
where ρs is the 2D superfluid stiffness, and the mean free energy density will increase asf (t) ∼ ρs t2. The runaway energy will be cut off by destruction of the superfluid stiffness, orby mobilization of the vortices, which will result in ∇φ 6= 0.
Let us first consider a single moving vortex with velocity V as depicted in Fig. 2. Outside thevortex core of radius ξ, a dipolar electromotive force field (EMF) E = − ~
2e∇φv, is associatedwith the vortex motion. The EMF inside the metallic vortex core Ecore is determined by thevoltage drop between the core boundary points at ∂ξ =
�
x | |x−X|2 = ξ2
,
x2∫
x1
d` · Ecore =~2eφv
�
�
�
x2
−~2eφv
�
�
�
x1
, (6)
where xi ∈ ∂ξ. From the definition ofφv(x−V t), the core EMF is linearly related to the vortexvelocity by,
Ecore =~
2|e|ξ2z ×V . (7)
Since ∇2φv = 0 everywhere, E is divergence free, in analogy to an in-plane 2D “magneticfield”. The EMF produced by each moving vortex can be parameterized by a 2D “magneticmoment”,
µi =ξ2
2Ecore
i . (8)
1In this paper, a vortex (antivortex) is defined by the anticlockwise (clockise) circulation of the electrical current,rather than the winding of the phase (see Figure 2).
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Figure 2: Electromotive force field (EMF) E (red lines with arrows), created by asingle moving vortex of velocity V . The circulating supercurrent is depicted by awide blue arrow. A core EMF, Ecore, is imposed by the voltage drop (V1− V2) acrossthe metallic region (yellow disk) of radius ξ.
In analogy with magnetostatics [24], a finite density of 2D “moments” produces a 2D “mag-netization field” m = nvµ. Henceforth o ≡ 1
∆A
∫
∆Ad2 x o, denotes coarse graining of o over an
area ∆A which includes many vortices. The relation between the EMF and these “magneticmoments” is given by the “magnetic field” to “magnetization” ratio, E = 4πm. By Eqs. (7-8)this relation implies,
E = hnv
2|e|z ×V ≡
h2|e|
z ×J v = −V ×B/c . (9)
V denotes the average vortex velocity, and J v is the vorticity current. The last equality inEq. (9) uses nv = B/Φ0, and is known as Josephson’s relation [25]. In vortex dynamics ap-proaches [1,16], Josephson’s relation is used to express the EMF as a function of the computedvortex velocity.
Here we turn this relation on its head. By demanding a steady state dd t ⟨j⟩ = 0, the exter-
nally imposed electric field must be cancelled, on average, by − ~2e ⟨∇φ⟩, and hence E = E .
This constrains the average vortex velocity to be,
V (E) =c
B2E ×B. (10)
It should be emphasized that in our approach, the vortex velocity is independent of any Hamil-tonian parameters. The transport problem is formulated in terms of a linear response of thetransport current to the externally applied electric field E.
The current has two separate components: the metallic vortex cores which serve as idealcurrent pumps, and the MVC transported by the moving vortices (see Fig. 3).
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Figure 3: Flux flow steady state transport. The electric field dictates the mean vortexvelocity according to Josephson relation V = c
B2 E ×B. Each moving vortex core isan ideal current pump of both longitudinal and Hall components jcore. The core cur-rents combine to a transport current density jtr = jcore marked by long blue arrows.Moving vortex charges of average magnitude Qv produce an additional Hall current,which is responsible for the Hall anomaly.
3 Core and transport currents
Within the BS model, the vortex cores are metallic disks of radius ξ, with superconductingstiffness ρs=0 inside the core and constant outside. The electrons in the disk are subjected toa core EMF which interpolates between its boundaries which move at velocity V of Eq. (10),as depicted in Fig. 2. The scattering impurities are, naturally, at rest in the lab frame. Lorentztransformation of the EMF from the moving frame to the lab frame (to linear order in V /c)combined with the external field E, yields Ecore +E + V ×B/c = Ecore, where we have usedEq. (10). The core current in the lab frame is determined by the core conductivity tensor,
jcoreα =
∑
β
σcoreαβ Ecore
β . (11)
For a homogeneous superconducting film, we assume that all vortices move at the same av-erage speed and produce the same core current density. The moving vortex cores act as idealcurrent pumps of the DC transport current. Due to the finite viscosity of the core electrons, thecurrent density is continuous at the core boundaries,
jcore = jtr(x)≡2e~ρs∇φtr(x) (x ∈ ∂ξ) , (12)
where φtr was defined in Eq. (4). For weak enough electric field, additional vortex-antivortexpairs in the superconducting medium are not produced by the vortex motion and the transportcurrent remains laminar, i.e. ∇×∇φtr = 0. By charge conservation, ∇2φtr = 0 everywhereoutside the vortex cores. Thus, as a harmonic function, φtr is determined (up to a constant)by its gradients on the vortex core boundaries. The unique solution for the transport current,is a globally uniform current density jtr = jcore, as depicted in Fig. 3.
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Local fluctuations in core currents and charge density are eliminated in the coarse grainedDC current density jtr. Substituting Eq. (7) in Eq. (11), the BS conductivity is obtained:
σBSαβ =
�
Bc2(T )B
�
σcoreαβ , (13)
which constitutes the first term in Eq. (1).Note that the BS Hall angle tan(θBS
H ) = σBSx y/σ
BSx x , equals to the Hall angle of the metallic
core. In Drude metals, tan(θH) = ωcτtr, where ωc is the cyclotron frequency and τtr(T ) isthe transport relaxation time which varies on the temperature scales of the normal phase.The alternative Hall angle result of Nozieres and Vinen [11] is not validated by the derivationabove.
In the “dirty limit” ξ � ltr2, where ltr is the core mean free path, σcore(T, B) can be
approximated by extrapolating the normal phase conductivity σnormal(T, B) from T > Tc(B) toT < Tc(B). For cuprate superconductors and other cases of relatively short ξ, an extrapolationprocedure which exploits the continuity of the BS Hall angle will be proposed in Section 7.
Fig. 1 illustrates the expected BS resistivities for Drude-theory core conductivity. The ad-ditional effect of MVC is discussed in the following sections.
4 Hall conductivity of charged vortices
In this section we discuss the contribution of MVC to the Hall conductivity. In the static E=0case, in the absence of micrscopic particle-hole symmetry, one expects local vorticity currentsto induce charge density modulations δρvc(x−Xi), which would be centred around the vortexpositions {Xi}. Vortex charge has been previously proposed in the context of Hall anomalies[9,10], see discussion in Section 9. Vortex induced charge density modulations have also beenobserved experimentally [26]. If indeed each vortex drags (on average) an MVC of value Qv,there will be an additional Hall current given by,
jmvc =QvJ v =2|e|Qv
hE × z, (14)
which will produce the second term of Eq. (1). (See Fig. 3).The sign and magnitude of Qv are a-priori open to many options: The total conduction
electron density per vortex or some fraction of it [15]? The superconducting condensate den-sity per vortex? Coulomb screening could neutralize the total vortex charge, and must becarefully considered [27]. These dilemmas have long been debated.
Here, MVC is well defined using Kubo linear response theory. In the case of λ � lB, thevortex system behaves as an incompressible fluid due to the long range logarithmic interactionsbetween vortices. The coarse-grained charge density response to any local variation of themagnetic field δBv is given by
δρvc(x, t) =
∫
d2 x ′ d t ′ R(x−x′, t − t ′) δBv(x′, t ′) , (15)
where, by the aforementioned incompressibility, R is a local function of space and time.The dynamical Hall conductivity of the charged vortex fluid is proportional to the Fourier
transform of R [19],σmvc
x y (q,ω) = cR(q,ω) , (16)
2For simplicity we ignore proximity and Andreev reflection corrections at the vortex core boundaries, whichwould be small in the dirty limit of ξ� ltr.
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SciPost Phys. 8, 061 (2020)
which is shown in Appendix A. The second term in Eq. (1) is the DC limitσmvc
x y = limω→0σmvcx y (0,ω).
Since R is a local function in space and time, R(q,ω) is a smooth function of q,ω, whoseorder of limits (q,ω)→ 0 commute. Therefore, we can reverse the order of limits, and obtainthe thermodynamic relation,
σmvcx y = c lim
q→0R(q, 0) = c
�
∂ ρvc
∂ B
�
µ≡
2|e|h
Qv , (17)
where the first equality is known as the Streda formula [18] (see Appendinx A). Eq. (17)defines Qv as
Qv ≡d
dNv
∫
d2 x ρvc(x) , (18)
that is to say, Qv is the change in total charge after inserting one additional flux quantum intothe system. For the calculation in the following section, we note that the background chargedensity which is not created by the vorticity, does not contribute to the MVC Hall current.
Streda formula is known for its application to gapped quantum Hall (QH) phases. There,the order of limits (q,ω)→ 0 also commute due to the locking of local charge and magneticflux variations. Intriguingly, the local flux-charge attachment property is shared between theQH and the vortex liquid phases [28]. The difference between the vortex fluid and the QHliquid is that Qv in the QH phases is quantized at particular rational multiples of e.
5 Screened Ginzburg-Landau theory of the MVC
Three dimensional screening ensures that the total charge accumulated around a static vortexvanishes, except near the surface [27]. Here we need to know whether for a moving vortex,the screening charges move with the vortex and cancel any contribution of δρvc to the Hallcurrent.
This question is answered by applying the theory of vortex charge screening of Khomskiiand Freimuth [9] to the layered superconductor. Since Thomas-Fermi screening length is muchshorter than ξ and lB, the local electrochemical equilibrium equation,
eδϕ(r) +δµ(r) = 0, (19)
relates between between the screening electrostatic potential ϕ, and the local chemical po-tential deviation δµ induced by the vorticity. Here, r = (x, z) is a three dimensional (3D)coordinate. The 3D charge density deviation is determined by Poisson’s equation,
∇2ϕ(r) = −4πε0δρ3D(r) , (20)
where ε0 is the local dielectric constant. Our goal is now to determine the profile of δµ(r).In the absence of pinning, our vortex fluid is described as a slowly flowing hexagonal vortex
lattice (VL). The quantities calculated will be accurate to leading order in the vortex velocity.We begin with the 2D Ginzburg-Landau (GL) free energy density of the superconducting con-densate,
f = a |Ψ|2 + 12 b |Ψ|4 + K
�
�(∇− 2ie~c A)Ψ
�
�
2, (21)
with A = 12 B z × x. In the superconducting phase, a < 0 and the coherence length is
ξ = (−K/a)1/2. Throughout we assume B � Bc2, in which case we may writeΨ(x) ≈ (−a/b)1/2 exp(iφ), where φ is given by Eq. (4) after setting φtr = 0. This form
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Figure 4: Ginzburg Landau free energy density of a vortex lattice, f vl(x , y) ofEq. (22), in arbitrary units. ξ is the vortex core radius, and blue arrows representa circulating current. The local chemical potential deviation δµ(x) = d f vl/dne, isapproximately proportional to f vl(x , y). By symmetry, the normal gradient ∇⊥δµvanishes on the unit cell boundaries. If the total charge per unit cell is non zero, itmust be screened in the third dimension, as depicted in Fig. 5
for Ψ(x) presumes that the vortex core size is zero, which we shall correct below. The GL freeenergy density is thus written as
f vl(x) =ρs
2
¦
−1
2ξ2+ (∇Φ)2
©
+ρsεcore
ξ2
∑
i
Θ(ξ− |x−Xi|) , (22)
where ρs = −2Ka/b is the superfluid stiffness, εcore is the dimensionless vortex core energy,and
Φ(x) =π
2BΦ0|x|2 −
∑
i
ln |x−Xi| . (23)
Setting ⟨∇2Φ⟩ = 0 (charge neutrality of a 2D Coulomb gas) forces the vortex density to berigidly determined by nv = B/Φ0. The profile of f (x), outside the vortex cores, is depicted inFig. 4.
The local chemical potential deviation in Eq. (19) can be derived from Eq. (22),
δµ(x, 0) =d f vl
dne(x) . (24)
In isotropic three dimensional superconductors, ∂z δµ(x, z) = 0. The in-plane screening caseis depicted in Fig. 5(a). The total 2D charge deviation in the unit cell area,
∫
uc
d2 x ∇2⊥δµ(x, z) = 0 , (25)
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SciPost Phys. 8, 061 (2020)
Figure 5: Three dimensional screening of vortex charge density. (a) Stacked super-conducting planes with in-plane screening. (b) Staggered dopant layers and super-conducting planes where an unscreened Hall current flows in the superconductingplanes.
which is the consequence of ~∇⊥δµvl(r) vanishing by symmetry on the 2D unit cell boundaries,see Fig. 4.
In layered superconductors, neighboring dopant layers are positioned at z = ±ac/2, withδµ(x,±ac/2)∝ ρ
dops � ρs. The interlayer modulation can be approximately modelled by,
δµvl(x, z) =12
d f vl
dne(x)�
1+ cos(2πz/ac)�
+O(ρdops /ρs) . (26)
This modulation results in interlayer screening. The charge deviation given by Eq. (20) is
δρ3D(x, z) =ε0
4πe
�
∇2⊥ −
�
2πac
�2�
δµvl(x, z) . (27)
The MVC, by (17), is given by differentiating the total areal charge of a superconductingplane, with respect to vortex number:
Qv =d
dNv
∫
d2 x
ac/4∫
−ac/4
dz ρ3D(x, z) = −ε0
2eac
ddNv
ddne
∫
d2 x f vl(x) . (28)
We can discard the uniform condensation energy −ρs/4ξ2 in Eq. (22) which does not depend
on Nv. As shown in Eq. (25), the integral over the in-plane Laplacian ∇2⊥ vanishes over each
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unit cell. The relevant vortex charge density is obtained by the second contribution of Eq. (27).This charge density is screened on the neigboring dopant layers, see Fig. 5(b).
Averaging Eq. (22) over a unit cell (UC) of the vortex lattice can be performed analytically[29],
Qv = −ε0π
2eac
ddne
�
ρs (εM + εcore)
, (29)
where εM is the dimensionless Madelung energy,
εM =12
ln� Bc2p
3πB
�
− ln |η(τ)|2 , (30)
where (1,τ= exp(iπ/3)) are the complexified triangular lattice vectors, q = exp(2iπτ) and
η(τ) = |q|1/24∞∏
n=1
(1− qn) , (31)
is the Dedekind eta function. The dimensionless vortex core energy εcore is of order unity inBCS theory [30]. The log(1/B) dependence in Eq. (30) results from integration of the vorticitycurrent squared, |x−Xi|−2, over a unit cell area 2πl2
B.Combining Eqs. (29-31), we arrive at a compact formula for the MVC in terms of the GL
parameters,Qv(T, B) =Q0 log(αBc2/B) , (32)
where the temperature dependent parameters are,
Q0(T )/e = −πε0
4e2ac·
dρs
dne(33)
logα(T ) = −1.47+2πεcore −ρs
�
dρs
dne
�−1�∂ Bc2
∂ neB−1
c2 −2π
∂ εcore
∂ ne
�
.
α(T ) is of order unity, and depends on ξ, and εcore(ne). These quantities require a mi-croscopic theory of the core properties, but do not effect the value of Q0 and the logarithmicdependence of Qv on magnetic field in Eq. (32).
6 Extraction of Qv from experiment
One would like to extract MVC values from experimental Hall and longitudinal magnetoresis-tivities. The problem is that in unconventional superconductors, we often do not fully under-stand the behavior of the metallic core conductivities, which are required for the subtraction.Fortunately, the Hall conductivity in Eq. (1) exhibits a separation between the core and theMVC contributions which can be exploited.
The BS Hall angle inside the metallic cores reflects the extrapolated behavior of the normalphase. For temperatures not too far from Tc, it is reasonable to linearly extrapolate of thenormal state temperature dependences of ρx x ,ρy x to below Tc, as depicted in Fig. 6. Thisyields an extrapolated values of tanθH→ tan θH .
Thus, the BS Hall conductivity can be deduced by multiplying the measured σexpx x (which
is not affected by the MVC), by the extrapolated Hall angle,
σBSx y = σ
expx x (B, T ) tan θH(B, T ) . (34)
Using Eq. 1, we can extract the experimental values of the MVC by subtracting σBSx y from the
experimental Hall conductivity σexpxy ,
Qexpv (B, T )≡
h2|e|
�
σexpxy −σ
BSxy
�
. (35)
11
SciPost Phys. 8, 061 (2020)
Figure 6: Extrapolation (dashed lines) of the Hall angle from the normal state ofBi2Sr2CaCu2O8−x into the FF regime using the ratio of linearly extrapolated resistiv-ities (Inset) of the normal phase data of Ref. [6].
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0 -
|Qexp
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↵B
c2
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Qexp0 log
✓↵Bc2
B
◆
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Figure 7: Moving vortex charge Qv extracted from Zhao et al. [6]. The resistivitydata of two unit cell thickness film of Bi2Sr2CaCu2O8−xare inputs into the proceduredescribed in Section 7. Dashed lines are fits to theory, Eq. 32. Inset: Fitted pa-rameters αBc2 and Qexp
0 (T ), (which is negative). The weak temperature dependenceof Qexp
0 is consistent with a constant parameter γ which is roughy consistent withphenomenological Uemura’s relations [31,32] of Eq. (37).
7 MVC of cuprate superconductors
A crude theoretical estimation of Q0 of Eq. (33), can be obtained from measured doping andtemperature dependent London penetration depth λ, in bulk three dimensional cuprate su-
12
SciPost Phys. 8, 061 (2020)
Table 1: Vortex charge coefficients of cuprates. Using Eq. 35, Qexp0 is extracted from
Hall and magnetoresistivity data at fixed temperature T[K]. Qth0 , of Eq. 39, is fit to
Qexp0 using the listed values of εfit
0 . Data from refs. [6,20,34,35]. See text for details.
Compounds T[K] Qexp0 /e λ/a Qth
0 /e ε0
Bi2Sr2CaCu2O8−x 75 −0.45 684 −0.05ε0 9YBa2Cu3O7 91 −1.68 410 −0.13ε0 12.4
Nd1.85Ce0.15CuO4−y 11 +0.97 533 +0.086ε0 11.3
perconductors. The two dimensional (per layer) superfluid stiffness ρs, is related to λ by
ρs =Φ2
0
4π3
ac
λ2= 6.3× 10−4λ−2 × ac [eV/cm] . (36)
ac is the c-axis layer separation, which will later drop out of Q0.In the underdoped regime x ≤ 0.15, the doping and temperature dependent ρs of cuprates
[31–33] is roughly captured by an empirical formula
ρs(x , T ) = ρs(xopt, 0)
�
|x |xopt
�
− γkB T , (37)
for |x |< xopt, where x = 1−nea2 is the doping concentration per copper, xopt ≈ 0.16 is optimaldoping, and a ' 3.8Å. is the copper-copper distance in the superconducting planes. Note thatx is positive (negative) for hole (electron) doped materials, and that γ is weakly doping andtemperature dependent, which is consistent with the empirical Uemura scaling [31,32], givenby ρ(0)s ≈ γTc.
Thus,dρs(T )
dne' −sgn(x) a2
ρs(xopt, 0)
|xopt|, (38)
which by Eqs. (33) and (36) yields,
Qth0 = −3650 e sgn(x)
�
ε0
xopt
��
aλ
�2
, (39)
whose values for hole-doped Bi2Sr2CaCu2O8−x and YBa2Cu3O7, and electron-dopedNd1.85Ce0.15CuO4−y , are listed in Table 1.
The experimental values of Qexp0 in Table 1, are extracted from the data of Refs. [6, 20].
The agreement is quite reassuring: we can fit Qexp0 = Qth using ε0 = 9 − 11.3. This short
wavelength parameter is difficult to obtain experimentally. Dielectric constants are expectedto be similar in the different cuprates, due to similar local environments. These values are notextremely different from ε0 = 4.5 which was used to fit ellipsometry data of Bi2Sr2CaCu2O8−xin Ref. [36].
8 Inhomogeneous flow and fluctuations
This paper has implicitly assumed weak effects of disorder and pinning in the FF regime.Eq. (1) applies to homegenous vortex motion, ‘deep’ in the FF regime as defined by Eq. (3).
13
SciPost Phys. 8, 061 (2020)
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pinn
ed
vortice
s
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Figure 8: Inhomogeneous flux flow. The pinned vortices do not contribute to thevoltage drop, and the transport currents avoid their metallic cores. The externalvoltage V extdrops only on the flux flow strip of width Lff, which effectively enhancesthe Bardeen Stephen conductivity by a factor of L/Lff, but not the moving vortexcharge contribution, see Eq. (40).
Short range (relative to ξ) disorder determines the normal state conductivities of σcore.Long range disorder may broaden the melting transition at weak fields, due to formation of"vorticity rivers" of total cross section Lff, between pinned regions of cross section L − Lff. InFig. 8 we depict two domains, which can be generalized to describe realistic systems withmultiple rivers and pinned domains.
The FF conductivity, Eq. (1) is readily modified to take into account such inhomogeneousflow. The mean vortex velocity in the rivers, is enhanced by a geometric factor V → V L
Lff .Thus, the core currents of the moving vortices are also enhanced by that factor. Due to thelaminar inter-vortex current flow, the identity jtr = jcore for the average current density stillholds. The transport current through the superconductor bypasses the resistive cores of thepinned vortices, see Fig. 8. The BS conductivity is therefore multiplied by a factor of L/Lff.
The MVC Hall current, however, remains unchanged, because it is proportional to the totalvorticity current, and the effective conductivity due to partial pinning is
σFF−partialαβ
=�
Bc2
B
��
LLff
�
σcoreαβ + εαβ
2|e|Qv
h. (40)
The partial pinning parameter Lff/L < 1 depends on the sample inhomogeneities, and isan increasing function of field, temperature, and current. A signature of the partial pinningregime, would be a non linear current-voltage relation.
We note partial pinning enhances both the BS Hall and longitudinal conductivities. There-fore, we can still extract Qexp
v from σexpx x ,σexp
x y , and the extrapolated Hall angle, using Eq. (35).The primary difficulty at low fields is to measure the extremely low values of ρx x and ρx y inthe linear response regime.
Fig. 1 depicts the resistivities of Bardeen-Stephen mean field theory, in which Bc2 is asharp phase transition. In reality, for quasi-2D films and especially for few monolayers as
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in Ref. [6], Tc(B) (or Bc2(T )) denote a broad crossover between the flux flow and normalmetal regimes. Evidently, short range phase correlations which are required to define vortices,persist in a limited range of T > Tc, and B > Bc2(T ). In the thin Bi2Sr2CaCu2O8−x films,Tc = TKT is of Kosterlitz-Thouless type, where, as Halperin and Nelson [37] have shown,mobile vortices exist in some range above TKT . Consequently, if these vortices are chargedby Qv 6= 0, they can produce a Hall sign reversal even slightly above TKT. In thicker films,superconducting order pairing fluctuations [17,38] play an important role above Tc. We notethat for cuprates, the normal state (a.k.a. the “bad metal”) longitudinal and Hall conductivitiesare poorly understood.
9 Discussion
Hall anomalies in superconductors have been an open theoretical problem for more than fourdecades. Some of the confusion in the field originates in the formulation of vortex forces andviscosities. Vortex dynamics approaches calculate the vortex velocity (and the associated EMFit generates) as a response to the force exerted by a bias current, in addition to other forces. Theconductivity tensor depends on the coefficients of these forces. However, conductivities canonly be methodically computed by Kubo formulas, where the current is derived as a responseto an applied electric field (as in our approach), and not vice versa.
The vortex dynamical equation has seen endless debates since first introduced by BS [1]and Nozieres and Vinen [11] to describe flux flow. Missing in this equation are effects of re-alistic band structure, impurities, time-retardation and the multi-vortex interactions. Whilethe friction term can be derived from the longitudinal core conductivity by equating the Joulepower dissipation [1], non-dissipative forces have been difficult to justify microscopically [39].Hall voltage producing forces were related to imaginary relaxation time of the time dependentGinzburg Landau equation [13], and to the topological phase-density action [15]. The signand magnitude of such a terms were determined by Ref. [15] by appealing to Galilean invari-ance. However, this argument is not valid to flux flow in superconductors for two fundamentalreasons:
1. Superconductors in a periodic potential with impurities are not Galilean invariant.Kelvin’s circulation theorem is valid for a perfectly Galilean liquid, where vortices mustmove at the velocity of the background current, i.e. “go with the flow”. This rule wouldindeed produce the Galilean Hall conductivity of σx y = cρ/B. However, de Gennes andNozieres [40] pointed out that superconducting vortices may only exhibit “go with theflow” effects for ωcτ = tan(θH) � 1. As seen in Fig. 6, the experimental values oftan(θH)∼ 10−3 are well outside this regime. A counterexample is provided around halffilling ρa2 ' 1 (a is a lattice constant), where the σx y vanishes and changes sign dueto particle hole symmetry of the bandstructure.
2. As shown by several arguments in Appendix B, the onset of short range superconductingstiffness prevents the vortices from “going with the flow”. The vortex liquid is incom-pressible due to its long range interactions. As a consequence, the Hall conductivity isgiven by the non Galilean invariant Streda formula, Eq. (46): σx y = c (∂ ρ/∂ B) 6= cρ/B.
In this paper the vortex velocity is not driven by various forces, but dictated by the electricfield via Josephson relation (10). The moving vortices’ cores act as ideal transport currentpumps. The derivation of Eq. (1) vindicates BS result for the magnetic field dependent Hallconductivity (in the absence of MVC), vis-a-vis Nozieres and Vinen’s [11] Galilean symmetrymotivated prediction.
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An important feature of the present approach is the separability of the conductivity tensorin Eq. (1). This feature allows us to subtract the core conductivity term, which is poorly under-stood in many systems. The normal state conductivities are highly sensitive to material specificdisorder, superconducting fluctuations (discussed in Section 8) , and inelastic scattering. Incuprates, it is fair to say, little is yet understood about its normal state transport coefficients.
Hall sign reversals are a result of Qvσcorex y < 0. Whether there are one or two sign rever-
sals, depends on the individual temperature dependence of each of the two terms in Eq. (1).σcoreαβ
depends on quasiparticle scattering time, while the MVC is determined by the superfluidstiffness which saturates at low temperatures. A second (lower temperature) sign reversal,such as observed in Ref. [6], indicates that quasiparticle scattering is dominated by inelastic(electron-electron and electron phonon) processes.
Previous authors [9, 10] have discussed the role of vortex charge in Hall anomalies, withemphasis on the vortex core charge density. Coulomb screening however can completely neu-tralize the total vortex charge in an isotropic superconductor, as depicted in Fig. 5(a). Herewe have considered the geometry of dopant layers with interlayer screening. This allows thevortices to drag the MVC within the superconducting layers, while the vertically displacedscreening charges reside in neighboring low mobility dopant layers. The major contributionfor the MVC arises from vorticity kinetic energy density far outside the cores. Integration ofthis contribution over inter-vortex separation yields the log(Bc2/B) dependence of |Qv| at lowfields, which is consistent with the experimental data in Fig. 7.
Eq. (1) has been applied to analyse cuprate flux flow transport data in Section 7. Wefind reassuring agreement between estimates of the doping-dependent London penetrationdepth, interlayer dielectric constant, and the values of Q0 extracted from experiments, as givenin Table 1. Independent measurements of vortex charging in these systems by resonance,capacitance and force microscopy, would be very informative.
At very strong magnetic fields, Hall sign reversals in cuprates have been attributed to Fermisurface reconstructions in the normal metallic phase [41,42], which exhibits quantum oscilla-tions. It would be interesting to investigate the crossover between Hall anomalies at relativelyweak field magnetic fields, and Fermi-liquid regimes at much higher fields.
Acknowledgements
We thank Amit Kanigel, Bert Halperin, Michael Reznikov, and Ari Turner, for useful inputand discussions. We also thank Referees 1 and 2 for urging us to clarify our stand on as-pects of previous theoretical work. We acknowledge support from US-Israel Binational Sci-ence Foundation grant 2016168, Israel Science Foundation grant 2021367, Aspen Center forPhysics, Grant NSF-PHY-1066293, where parts of this work were done, and the workshop atICTS/topmatter2019/12.
A Dynamical Hall conductivity and Streda formula
We restrict ourselves to a 2D system with area A. The dynamical Hall conductivity is definedas
σx y(ω) =~
AZ
∑
n,m
�
e−βEn − e−βEm
(Em − En)(Em − En − ~ω)
�
Im�
⟨n| j xq |m⟩⟨m| j
y−q|n⟩
�
, (41)
where Z = Tr e−βH , and {En, |n⟩} is the spectrum of H.
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SciPost Phys. 8, 061 (2020)
Charge conservation yields,
ρq =i~[H,ρq] = −iq · jq . (42)
Maxwell’s equation j = c∇×M, relates the magnetization to the current density, and in Fourierrepresentation, jq = ic q ×Mq . Without loss of generality, we can choose q = (qx , 0, 0) andM = M z z, and relate the matrix elements,
⟨n| j xq |m⟩= −
1~qx(En − Em)⟨n|ρq|m⟩
⟨m| j y−q|n⟩= icqx⟨m|M z
−q|n⟩.(43)
Inserting the current matrix elements (43) in (41) yields
σx y(q,ω) =c
AZ
∑
n,m
�
e−βEn − e−βEm
(Em − En − ~ω)
�
Re�
⟨n|ρq|m⟩⟨m|M z−q|n⟩
�
≡ cδρ
δB(q,ω) .
(44)
The DC Hall conductivity is given by the DC transport limit,
σx y = limω→0
limq→0
σx y(q,ω) . (45)
The Streda formula is the static thermodynamic susceptibility,
σStredax y = lim
q→0limω→0
σx y(q,ω) = c�
∂ ρ
∂ B
�
µ,T. (46)
When the limits (q,ω)→ (0, 0) of Eq. (44) commute, σx y = σStredax y . This condition is satisfied
in gapped quantum Hall phases, and for the vortex liquid discussed in Section 4. In both ofthese systems the response of δρ(x, t) to δB(x′, t ′) defined by R(x− x′, t − t ′) in Eq. (15) islocal. For resistive gapless metals, the two limits are different, and Streda’s formula does notdescribe the Hall conductivity.
B Invalidity of Galilean Hall conductivity in flux flow regime
To discuss the hydrodynamic contribution to the Hall current we consider “coreless” vorticesby setting σcore
x y → 0 in Eq. (1). Here we do not assume underlying particle hole symmetry,since we wish to discuss the approximately Galilean invariant superconductor.
In a normal Galilean invariant liquid with total charge density ρ, vortices “go with flow” asdictated by Kelvin’s circulation theorem. That is to say the averaged particles’ velocity shouldbe equal to that of the vortices, i.e. jHall = ρV . In the presence of an electric field, Josephson’smean vortex velocity is V = cE × z/B – see Eq. (10) – and the Galilean invariant (GI) Hallconductivity would be
σGIx y =
cρB
. (47)
In this Appendix, we show that Eq. (47) is not valid in the flux flow regime of superconductorseven in an approximate Galilean Hamiltonian. The conceptual point is that Eq. (47) doesnot capture the effects of short range superconducting stiffness ρs 6= 0, which differentiatesbetween flux flow and normal metal regimes.
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SciPost Phys. 8, 061 (2020)
1. The current induced by a moving vortex in the (Galilean invariant) dynamical Gross-Pitaevskii theory was calculated in the superfluid phase by Arovas and Freire [43] usingthe mapping to dual (2+1)D electrodynamics:
jv∝z × (r−V t)|r−V t|2
+O(V 2) . (48)
This current integrates to zero, and does not produce a global Hall current jHall = ρV ,which is required for Eq. (47).
2. Consider a charged two dimensional bosonic superfluid of density n and charge q, andHamiltonian
H =∑
i
(pi −qc A)
2
2m∗(49)
on a narrow ring, subjected to a perpendicular magnetic field A = − B2r × z. A vortex
lattice of density nv = B/Φ0 is produced in the bulk of the ring. Now a radial electric fieldE ‖ r is adiabatically turned on. The vortex lattice will start moving around the annulusat Josephson’s velocity V = cE × z/B. We can understand the null drag effect of themoving vortex lattice by analogy to that of moving potential term
∫
d2 x ϕ(x−V t)ρ(x).
At low velocities, the rigid condensate “sticks" to the lab frame in which it was prepared,and is not dragged by the moving potential (in other words: it is hard to pump a super-fluid). This statement can be verified by the following Gedankenexperiment: we boostthe Hamiltonian to the rotating frame of velocity V , which is implemented by insertingan effective Aharonov-Bohm (AB) flux ΦAB = (m∗c/q)V L. Superconducting stiffness en-sures that the circulating persistent current in the moving frame would be j = −nqV ata finite flux. The persistent current does not decay even in the presence of the (now)static potentialϕ(x). (This is to be contrasted with the metallic phase, where the currentdecays to zero by scattering at all fluxes ΦAB = integer×Φ0.) Boosting back to the non-rotating lab frame, by setting ΦAB→ 0, restores the motion of the potential, and rewindsthe current back to zero. Thus the moving potential does not drag the condensate.
3. Due to the local stiffness (rigidity) of the superconductor, the vortex liquid is incompress-ible due to the logarithmic interactions between vortices. This justifies the interchangeof q,ω order of limits of the hydrodynamical Hall conductivity, resulting in the applica-bility of the Streda formulaσx y = c (∂ ρ/∂ B). The Streda formula for the vortex liquid isproven in Appendix A. Thus, the distinction between a Galilean normal metal and a lowertemperature flux flow regime can be captured by the inequality of ρ/B 6= (∂ ρ/∂ B).
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