Physica C 483 (2012) 113–118

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Physica C

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Doping effect on the anomalous behavior of the Hall effect in electron-dopedsuperconductor Nd2�xCexCuO4+d

T.B. Charikova a,⇑, N.G. Shelushinina a, G.I. Harus a, D.S. Petukhov a, A.V. Korolev a, V.N. Neverov a,A.A. Ivanov b

a Institute of Metal Physics RAS, Ekaterinburg, Russiab Moscow Engineering Physics Institute, Moscow, Russia

a r t i c l e i n f o

Article history:Received 13 March 2012Received in revised form 9 July 2012Accepted 1 September 2012Available online 7 September 2012

Keywords:Cuprate superconductorsMagnetoresistivityHall effect

0921-4534/$ - see front matter � 2012 Elsevier B.V. Ahttp://dx.doi.org/10.1016/j.physc.2012.09.001

⇑ Corresponding author. Tel.: +7 343 3783733; fax:E-mail address: charikova@imp.uran.ru (T.B. Chari

a b s t r a c t

Transport properties of Nd2�xCexCuO4+d/SrTiO3 single crystal films (Bkc; Jkab) are investigated in magneticfields B up to 9T at T = (0.4–4.2) K. An analysis of normal state (at B > Bc2) Hall coefficient Rn

H dependenceon Ce doping takes us to a conclusion about the existence both of electron-like and hole-like contribu-tions to transport in nominally electron-doped system. In accordance with Rn

HðxÞ analysis an anomaloussign reversal of Hall effect in mixed state at B < Bc2 may be ascribed to a flux-flow regime for two types ofcarriers with opposite charges.

� 2012 Elsevier B.V. All rights reserved.

1. Introduction

An observation that Hall effect in the mixed state can have asign opposite to that in the normal state has been reported forsome conventional superconductor since 1960s. The discoveryof a Hall effect sign change in the most of high-Tc cuprate super-conductors [1,2] stimulated a new interest to this anomalousbehavior.

The high-Tc superconductors are strongly type II and thus, inaccordance with the theory of Abrikosov, in the mixed state theirphysical properties are determined by the presence of quantizedflux lines or vortices. When an external current J is applied to thevortex system, the flux lines start to move under the action ofthe Lorentz force F ¼ J� B with a velocity vL. As a consequencean electric field E ¼ �vL � B is induced which has resistivity com-ponent Ex and Hall component Ey, the longitudinal resistivity,qxx ¼ Ex=J and Hall resistivity qxy ¼ Ey=J, thus appear.

Classical models which considered the hydrodynamical motionof vortices in response to a transport current in homogeneousstructures (flux-flow models) were created at 1960s [3,4]. In par-ticular, Bardeen and Stephen (BS) [3] have found that in the mixedstate a part of the current flows through the vortex core that re-sults in both dissipation and Hall effect. The core is treated as anormal conductor and thus in this model the Hall effect shouldhave the same sign as in the normal state.

ll rights reserved.

+7 343 3745244.kova).

Various theoretical models have been proposed to explain thesign change of the Hall resistivity in the mixed state (for an exten-sive review see [5]) but the origin of this phenomenon remainscontroversial.

If the proper pinning forces are included in the equation of vor-tex motion [6,7] it leads to a back-flow current and to a possibilityof Hall effect sign reversal. However, pinning forces are highlysample dependent and thus difficult to analyze.

Among the other models for the explanation of the Hall effectsign change in the mixed state a two-band model was created byHirsch and Marsiglio (HM) [8] with electrons and holes supposedto have rather different superconducting gaps. Such a model willsuggest a simple and natural explanation of the phenomenon butonly if there are grounds to consider that two types of carriers,namely, electrons and holes, really coexist in the systems studied.

There are much recent activities in investigation of possibleelectrons and holes coexistence in the normal state of cupratesuperconductors [9]. In particular, in electron-doped superconduc-ting cuprates, angle resolved photoemission spectroscopy (ARPES)[10–13] and transport studies [9,14–21] have shown that bothelectrons and holes play a role in the normal state properties.Two kinds of carriers in electron-doped cuprates with differentlanthanide cations seem to arise from the electronic structure nearthe Fermi surface (FS) of the CuO2 planes.

ARPES studies in NdCeCuO [11,22,23] have revealed a smallelectron-like FS pocket in the underdoped region, and a simulta-neous presence of both electron- and hole-like pockets near opti-mal doping. The conclusions of the ARPES measurements and

Table 1Optimal annealing conditions, temperatures of the superconducting (SC) transitionand experimental values of the upper critical fields at the minimal temperature of themeasurements for Nd2�xCexCuO4+d/SrTiO3 epitaxial films with different ceriumdoping.

x Annealing conditions Tonsetc ðKÞ Tc ðKÞ Bc2 ðTÞ

0.14 T ¼ 780 �C 12.1 8.6 2.9 (T = 1.4 K)

p ¼ 1:33� 10�2 mbart ¼ 25 min

0.15 T ¼ 780 �C 21.1 20.0 7.1 (T = 0.4 K)

p ¼ 1:33� 10�2 mbart ¼ 60 min

0.17 T ¼ 780 �C 12.8 11.0 1.7 (T = 3 K)

p ¼ 1:33� 10�2 mbart ¼ 50 min

0.18 T ¼ 600 �C 7.7 6.4 0.9 (T = 1.6 K)

p ¼ 1:33� 10�5 mbart ¼ 25 min

114 T.B. Charikova et al. / Physica C 483 (2012) 113–118

first-principle calculations of the electronic structure on the elec-tron-doped high-Tc superconductors Ln1.85Ce0.15CuO4 (Ln = Nd,Sm and Eu), performed by Ikeda et al. [12,13] are in accordancewith the results of papers [11,22,23].

A spin density wave (SDW) model [24] was proposed whichgives qualitative explanation to ARPES observations. In this model,SDW ordering would induce FS reconstruction that results in anevolution from an electron pocket to the coexistence of electron-like and hole-like pockets and then into a single hole-like FS withincreasing of doping. At present the development of this theoreti-cal model is continued [25,26].

A model of FS reconstruction, induced by SDW, was equally wellused for interpretation both of high-field Hall effect and magneto-resistance in electron-doped Pr2�xCexCuO4 films [16] and of dopingdependence of Shubnikov-de-Haas (SdH) oscillations in Nd2�xCex-

CuO4 single crystals [19,20]. A two-carrier model with coexistingof electrons and holes turns out to be suitable also for describingof temperature and doping dependencies of normal state Hall ef-fect and thermopower in Nd2�xCexCuO4 [9,14], Pr2�xCexCuO4 [15]and La1.85Ce0.15CuO4 thin films [17,18].

Thus, it seems to us that it is just the time to make a new at-tempt of describing the behavior of the mixed state Hall effect inelectron-doped superconductors on the ground of a two-carriermodel. In this paper we study magnetic field dependencies of lon-gitudinal and Hall resistivities of electron-doped superconductorNd2�xCexCuO4+d with various Ce content both in the normal andthe mixed states.

2. Samples and equipment

The series of Nd2�xCexCuO4+d/SrTiO3 epitaxial films (x = 0.14,0.15, 0.17, 0.18) with standard (001) orientation were synthesizedby pulsed laser evaporation [21]. The original target (the sinteredceramic tablet of Nd2�xCexCuO4+d of the given composition) wasevaporated by a focused laser beam and the evaporated targetmaterial was deposited on a heated single-crystal substrate. Thesubstrate material was SrTiO3 with (100) orientation and dimen-sions of 5� 10� 1:5 mm. The substrate temperature was 800 �C,the pressure during the deposition was 1.067 mbar, the residualgas was nitrous oxide (N2O). Then the films were subjected to heattreatment (annealing) under various conditions to obtain sampleswith various oxygen content. As a result, three types of sampleswith x = 0.15 were obtained: as-grown samples, optimally reducedsamples (optimally annealed in a vacuum at T = 780 �C fort = 60 min; p ¼ 1:33� 10�2 mbar) and nonoptimally reduced sam-ples (annealed in a vacuum T ¼ 780 �C for t = 40 min;p ¼ 1:33� 10�2 mbar), five types of samples with x = 0.14: opti-mally annealed in a vacuum at T ¼ 780 �C for t = 25 min;p ¼ 1:33� 10�2 mbar and four nonoptimally reduced samples (an-nealed in a vacuum T ¼ 780 �C for t = 5, 20, 30, 64 min;p ¼ 1:33� 10�2 mbar), ten types of samples with x = 0.17: opti-mally annealed in a vacuum at T ¼ 780 �C for t = 50 min;p ¼ 1:33� 10�2 mbar) and nonoptimally reduced samples (an-nealed in a vacuum T ¼ 780 �C for different times;p ¼ 1:33� 10�2 mbar) and five types of samples with x = 0.18:optimally annealed in a vacuum at T ¼ 600 �C for t = 35 min;p ¼ 1:33� 10�5 mbar and four nonoptimally reduced samples (an-nealed in a vacuum T ¼ 600 �C for t = 10, 15, 25, 60 min;p ¼ 1:33� 10�5 mbar) The film thickness was 1200–3800 Å. Theannealing conditions and transition temperatures of optimally re-duced films and experimental values of the upper critical fieldsare presented in Table 1.

A part of experimental results on the galvanomagnetic proper-ties of these films are presented earlier [27]. In particular, in ourworks [27a ] the upper critical fields Bc2 were determined for

Nd1.85Ce0.15CuO4+d single crystal films by so named ‘‘resistive’’method: from the shift of the resistive transition into the supercon-ducting state in an external magnetic field. For all of investigatedNd1.85Ce0.15CuO4+d samples an increase of the external magneticfield resulted in a real shift of the resistive transition to lower tem-peratures without appreciable broadening. Although superconduc-ting fluctuations led to a rounding of the transition, it was stillpossible to deduce a critical field from the position of the transitionby extrapolating its steep part to the normal state value. In [27a]we have presented Bc2 ðTÞ dependences corresponding to 0.5 qn

criterion.Now we report on the investigation results of the magnetic field

dependent Hall coefficient as a function of doped electron concen-tration in optimally reduced films. The sharp resistive transitioninto the superconducting state and the high critical temperatureof the optimally reduced samples confirm the high quality of thefilms under investigation. Hall effect and magnetoresistance weremeasured by a standard dc technique using a 12T Oxford Instru-ments superconducting magnet in the temperature rangeT ¼ ð0:4=4:2Þ K.

3. Experimental results and discussion

The in-plane longitudinal qxx and Hall qxy components of theresistivity are measured as the functions of magnetic field B per-pendicular to the ab-plane up to 9T in single crystal films of elec-tron-doped superconductor Nd2�xCexCuO4+d at T = (0.4–4.2) K.Fig. 1 shows the field dependences of the resistivities qxx and qxy

for optimally reduced Nd2�xCexCuO4+d films with x = 0.14 (a) and0.15 (b) and Fig. 2 shows qxxðBÞ and qxyðBÞ dependences for opti-mally reduced Nd2�xCexCuO4+d films with x = 0.17 (a) and 0.18(b) for T = 4.2 K. Here Bp is a vortex-depinning field and Bc2 is theupper critical field, determined as it is shown in Fig. 1b, that, dis-tracting the rounding of a transition due to a superconducting fluc-tuation, marks a field of the transition to the normal state both ofqxx and qxy. Arguments in favor of ‘‘onset’’ criterion for an estima-tion of Bc2 based on peculiar features of the vortex system behaviorin a high-temperature superconductor are given by Gantmaheret al. [28] for Nd2�xCexCuO4+y single crystals and by Jaroszynskiet al. [29] for NdFeAsO0.7F0.3 single crystals. The obtained Bc2 valuesare presented in Table 1. The region Bp < B < Bc2 corresponds to amixed (vortex) state where finite resistivity is a consequence ofvortex moving under the action of the Lorentz force.

The evolution of the Hall coefficient value in the normal stateabove the upper critical field Bc2 is traced with a variation of Ce

a

b

Fig. 1. In-plane resistivity qxx and Hall resistivity qxy versus magnetic field B plotsfor optimally annealed single crystal films of Nd2�xCexCuO4+d with x = 0.14 (a) and0.15 (b). Bp is a vortex-depinning field and Bc2 is the upper critical field.

a

b

Fig. 2. In-plane resistivity qxx and Hall resistivity qxy versus magnetic field B plotsfor optimally annealed single crystal films of Nd2�xCexCuO4+d with x = 0.17 (a) and0.18 (b).

Fig. 3. Hall coefficient RH in the normal state of Nd2�xCexCuO4+d (NCCO) systems asa function of Ce content. The dash-doted theoretical curve corresponds to the one-carrier model. The solid circles are our experimental data, solid lines are guides foreyes.

T.B. Charikova et al. / Physica C 483 (2012) 113–118 115

doping. It is found that low temperature normal state Hall coeffi-cient Rn

H is negative for underdoped (x = 0.14) and optimally doped(x = 0.15) films, positive for highly overdoped (x = 0.18) films andhas Rn

H ffi 0 for slightly overdoped (x = 0.17) films in accordancewith previous results for normal state RH of Nd2�xCexCuO4+d atT > Tc [30–32] (Fig. 3).

In other electron-doped superconductors the situation is thesame: in Pr2�xCexCuO4 Rn

H < 0 for x = (0.11–0.15) and RnH > 0 for

x = (0.16–0.19) at 0.35 K and B = 14 T [11,15]; in La2�xCexCuO4 thesign of the Hall resistivity changes from negative for x = (0.06–0.105) to positive for x = (0.12–0.17) at 2 K and B = 10 T [17,18].

Such a behavior of RnHðxÞ may be naturally explained by a

coexistence of electrons (of concentration n) with holes (of concen-tration p) even in nominally electron-doped cuprate system.Proceeding from a relation n � x for ideal electron doping Luo [9]carried out a division of electron and hole contributions and cameto a conclusion about rather rapid increase of p(x). The two types ofcarriers in cuprates may originate from electron-like and hole-likeparts of the Fermi surface in agreement with the ARPES results[11].

As in the most of cuprates we have found an anomalous changeof the Hall effect sign in the mixed state Rf

H just below Bc2: fromnegative to positive for x = 0.14; 0.15 and from positive to negativefor x = 0.18 (see Figs. 1, 2. This Hall anomaly has been previouslyinterpreted using a lot of different assumptions, most of whichwere closely connected with the peculiarities of magnetic fluxdynamics in the mixed state of superconductors [5–7,33,34].

We have adopted a semi phenomenological description of amixed state Hall effect by flux-flow model of Bardeen and Stephen[3] modified by coexistence of electrons and holes. In analogy withHM [8] we have proposed that two types of carriers have ratherdifferent superconducting gaps and thus different upper criticalfields BðiÞc2; i ¼ 1;2 for electrons and holes, respectively. Really, tak-ing into account of an antiferromagnetic spin fluctuations leadsnaturally to a two-band/two-gap model in superconducting stateof the electron-doped cuprate [25].

Magnetic field dependencies of resistivity qxxðBÞ and Hall coef-ficient RHðBÞ, usually observed in semiconductors with two (or sev-eral) types of carriers [35], are associated both with strong-fieldcondition xcs� 1 (xc being a cyclotron frequency and s beingan momentum relaxation time) at least for one of the types andwith widely different carrier mobilities.

In contrast, in our situation magnetic field dependencies of qxx

and RH , even with xcs� 1 for each types of carriers, are causedby conditions of the flux-flow regime and the interplay between

a

b

Fig. 4. The experimental behavior of RH as a function of magnetic field (points) inelectron-doped superconductor Nd2�xCexCuO4+d with x = 0.14 and 0.15 and calcu-lated RHðBÞ dependences (solid lines) for reasonable values of parameters (see Table2).

Table 2The values of parameters obtained from fitting of Eqs. (5)–(7) to the experimentaldata on qxxðBÞ and RHðBÞ.

x n (cm�3) p (cm�3) c lncm2

Bc

� �lp

cm2

Bc

� �b

0.14 2:3� 1020 4:9� 1019 0.21 27 55 2

0.15 4:3� 1021 5:4� 1021 1.3 18 12 0.6

0.17 1:7� 1021 6:7� 1021 4.0 60 30 0.5

0.18 3:5� 1020 2:8� 1021 8.0 100 50 0.5

116 T.B. Charikova et al. / Physica C 483 (2012) 113–118

two types in a magnetic field arises due to widely different rates oftheir transition into the normal state (different BðiÞc2).

In the framework of a conventional Drude model the low-fieldHall coefficient for two types of carriers with partial conductivitiesri and Hall coefficients RðiÞH Ri (i = 1, 2) is given by [35]:

RH ¼r2

1R1 þ r22R2

r2 ; ð1Þ

where the total conductivity r ¼ r1 þ r2. For electrons (i = 1) withconcentration n and mobility ln and holes (i = 2) with concentrationp and mobility lp in the normal state we have:

r01 ¼ enln; r02 ¼ eplp;

R01 ¼ �1en

; R02 ¼1ep: ð2Þ

In the HM paper [8] the equation for the longitudinal resistivityqxx in the mixed state was taken as in the BS model [3], but in theequation for a Hall angle it was supposed that the ratio of the nor-mal part of electrons (holes) in the mixed state to the total numberof electrons (holes) is proportional to B=BðiÞc2. In contrast to that weuse the standard BS dependencies on B both of qxx and qxy for eachof the types of carriers in the mixed state:

qfxx ¼ q0

xx B

Bc2;

qfxy ¼ xcsq0

xxB

Bc2¼ q0

xy B

Bc2; ð3Þ

where q0xx ¼ 1=r0 and q0

xy ¼ R0 B are the normal state resistivitiesand R0 is the normal state Hall coefficient.

As in the low-field limit rf ¼ 1=qfxx and Rf ¼ qf

xy=B, thus wepropose:

rfi ¼ r0i

Bi

B; Rf

i ¼ R0i BBi; ð4Þ

where for simplicity Bi BðiÞc2.Then we have three regions of magnetic field:

(a) B < B1, where both of electrons and holes are in the mixedstate;

(b) B1 < B < B2, where electrons are in the normal state butholes are in the mixed one;

(c) B > B2, where both electrons and holes are in the normalstate.

From Eqs. (1), (2) and (4) after some algebra we obtain simpleformulas, suitable for description of experimental dependencies(qxx ¼ 1=r):

ðaÞ RjR0nj

¼cb2 B2

B1� 1

cb B2B1þ 1

� �2 BB1;

rr0n¼ 1þ cb

B2

B1

� � B1

B; ð5Þ

ðbÞ RjR0nj

¼cb2 B2

B � 1

cb B2B þ 1

� �2 ;rr0n¼ 1þ cb

B2

B; ð6Þ

ðcÞ RjR0nj

¼ cb2 � 1

ðcbþ 1Þ2;

rr0n¼ 1þ cb; ð7Þ

where jR0nj 1=en; r0n ¼ enln; c ¼ p=n and b ¼ lp=ln.Fig. 4 shows a comparison of experimental dependencies of RH

with ones calculated by means of formulas (5)–(7) for x = 0.14 (a)and x = 0.15 (b) films. An effect of pinning is taken into accountphenomenologically by a counting of field B in formulas (5)–(7)from the depinning field Bp and by a supposition that atB < Bpqxx ¼ 0 and RH ¼ 0.

For reasonable description of qxxðBÞ and RHðBÞ dependences bythe theoretical expressions (5)–(7) a field of anomalous peak max-

imum Bmax should be taken as a critical field for electrons:B1 ¼ Bmax, and an empirical value of the upper critical field Bc2

(see Table 1) should be taken as a critical field for holes: B2 ¼ Bc2.Thus, only carrier concentrations n; p and mobilities ln;lp comeout as the fitting parameters. The values of parameters obtainedby fitting procedure both for qxxðBÞ and for RHðBÞ are presentedin Table 2.

We see that it turned out possible to describe qualitatively thebehavior of RHðBÞ both in the normal and mixed states of Nd2�xCex-

CuO4+d films with x = 0.14 and 0.15 for b � 1 and reasonable valuesof parameters n and p: for a small magnetic fields Bp < B < B1 (re-gion (a)) the electrons will become ‘‘normal’’ more rapidly than theholes, B1 < B2 (Appendix A) and thus the Hall coefficient have agood chance to be positive, switching to negative values atB ¼ Binv in a region (b) and then tends to its normal (negative) va-lue in a region (c). In particular, a field of RH sign inversion is givenby Binv ¼ cb2 B2. Thus for b ffi 1 and n ffi pðc ffi 1Þ the sign changeof the Hall coefficient may be used for empirical evaluation ofthe upper critical field.

For overdoped systems where Hall coefficient is positive in thenormal state and negative in the mixed state (see Fig. 2 informulas, analogous to (5)–(7), for reasonable description of

a

b

Fig. 5. The experimental behavior of RH as a function of magnetic field in electron-doped superconductor Nd2�xCexCuO4+d with x = 0.17 and 0.18 (points) and calcu-lated RHðBÞ dependences (solid lines) for reasonable values of parameters (see Table2).

T.B. Charikova et al. / Physica C 483 (2012) 113–118 117

experimental data it should be taken B2 < B1 (see Appendix A) andthus Bmax ¼ B2;Bc2 ¼ B1 and Binv ¼ B1=cb2.

In Table 2 the parameters n; p and ln;lp obtained from fitting ofcalculated dependencies of qxxðBÞ and RHðBÞ to the experimentalones for Nd2�xCexCuO4+d with x = 0.17 and 0.18 (Fig. 5 are alsopresented.

It is seen that we obtained b ffi 1 for all the samples and n� pfor x = 0.14, n ffi p for x = 0.15, p > n for x = 0.17 and p� n forx = 0.18. Such a trend is in qualitative accordance with estimatesfrom Rn

HðxÞ dependencies [9], from ARPES data [11], from frequen-cies of SdH oscillations [19,20] for different Nd2�xCexCuO4 samplesand from high-field qxyðBÞ dependencies for Pr2�xCexCuO4 systems[16].

Thus, an anomalous sign reversal of Hall effect in the mixedstate of investigated systems may be described rather well by aflux-flow model for two types of carriers with opposite charges.

4. Conclusions

We have analyzed the Ce doping dependence of the normalstate Hall coefficient in optimally reduced Nd2�xCexCuO4+d singlecrystal films and, on the grounds of this analysis, have recruiteda two-carrier model for describing of the mixed state Hall coeffi-cient. Our scheme is based on a simple Drude model for normalstate and on a semi phenomenological Bardeen–Stephen modelfor the mixed one modified by coexistence of electrons and holes.

This description may be considered as an illustration of apossibility for the natural explanation of Hall effect sign reversalsystematically observed in a mixed state of cuprate superconduc-tors. Such an approach may occur actual in the light of much recentefforts on experimental (ARPES) and theoretical (SDW-model)

investigations of the electronic structure near Fermi surface inthe CuO2 plane of high-Tc superconducting systems.

Acknowledgment

This work was done within RAS Program (Project No.01.2.006.13394) with partial support of RFBR (Grant No. 12-02-00202).

Appendix A

A.1. The ratio of electron and hole critical fields

In the ‘‘dirty limit’’ that takes place in our system [31] as well asin PrCeCuO [36], for the upper critical field at T � Tc one has [37]:

Bc2 ffiU0

�hD kTc �

Dr; ð8Þ

where U0 ¼ pc�h=e;D – diffusion coefficient, r – conductivity and D –the amplitude of the SC gap function. Here we use the correlationsTc � D for various cases of s-wave, d-wave, extended d-wave gapsymmetry [38] and r � D. The situation with piecewise FS (withtwo types of FS parts) in superconducting state of electron dopedcuprates is properly considered, for example, by Liu and Wu [25].The model suggests two dx2�y2 -wave gaps, possibly with differentamplitude, Da for electron-like band and Db for hole-like one. Thefitting of the model to experimental data for x < 0:17 leads toDb > Da, that is the hole states (b) would have a higher gap ampli-tude than the electron states (a) in the excellent accordance withour assignment B2 > B1 (in our designations: 1-electrons, 2-holes).

In electron doped cuprates the dramatic change of a Fermi-surface topology associated with the emergence of the singlehole-like Fermi surface around the (p;p) position takes place at atransition from the optimal doping level to the overdoped region[11]. Concrete result of this FS reconstruction is a disappearanceof an electron channel of conduction (rn ! 0) and transition tothe conduction by the one (hole) type of carriers. For the objectof our research it has the following consequences:

1. The sign of the Hall effect in the normal state of NdCeCuO [30–32], PrCeCuO [11,15]and LaCeCuO [17,18] changes to positivevalue in overdoped region (see Fig. 3. We have Rn

H > 0 forx = 0.18 as well.

2. The inversion of the ratio of electron (B1) and hole (B2) criticalfields takes place so far the electron channel does not disappearcompletely. According to (8) we have:

B1

B2¼ rp

rn Dn

Dp; ð9Þ

where Dn and Dp are electron and hole gap amplitudes, respectively.Then if rn ! 0 magnetic field B1 becomes higher than B2ðB1 > B2Þfor any finite ratio of the gap amplitudes in accordance with ourassumption for x P 0:17.

References

[1] S.J. Hagen, A.W. Smith, M. Rajeswari, J.L. Peng, Z.Y. Li, R.L. Greene, S.N. Mao, X.X.Xi, S. Bhattacharya, Qi Li, C.J. Lobb, Phys. Rev. B 47 (1993) 1064. and referencestherein.

[2] A.W. Smith, T.W. Clinton, Wu Liu, C.C. Tsuei, A. Pique, Qi Li, C.J. Lobb, Phys. Rev.B 56 (1997) R2944. and references therein.

[3] J. Bardeen, M.J. Stephen, Phys. Rev. 140 (1965) A1197.[4] P. Nozieres, W.F. Vinen, Philos. Mag. 14 (1966) 667.[5] E.H. Brandt, Rep. Prog. Phys. 58 (1995) 1465 (Section 7.5).[6] Z.D. Wang, C.S. Ting, Phys. Rev. Lett. 67 (1991) 3618.[7] Y.X. Jia, J.Z. Liu, M.D. Lan, R.N. Shelton, Phys. Rev B 47 (1993) 6043.[8] J.E. Hirsch, F. Marsiglio, Phys. Rev. B 43 (1991) 424.

118 T.B. Charikova et al. / Physica C 483 (2012) 113–118

[9] N. Luo, arXiv:cond-mat/0003074v2.[10] N.P. Armitage, F. Ronning, D.H. Lu, C. Kim, A. Damascelli, K.M. Shen, D.L. Feng,

H. Eisaki, Z.-X. Shen, P.K. Mang, N. Kaneko, M. Greven, Y. Onose, Y. Taguchi, Y.Tokura, Phys. Rev. Lett. 88 (2002) 257001.

[11] N.P. Armitage, P. Fournier, R.L. Greene, Rev. Mod. Phys. 82 (2010) 2421.[12] M. Ikeda, T. Yoshida, A. Fujimori, M. Kubota, K. Ono, K. Unozawa, T. Sasagawa,

H. Takagi, J. Supercond. Nov. Magn. 20 (2007) 563.[13] M. Ikeda, T. Yoshida, A. Fujimori, M. Kubota, K. Ono, Hena Das, T. Saha-

Dasgupta, K. Unozawa, Y. Kaga, T. Sasagawa, H. Takagi, Phys. Rev. B 80 (2009)014510.

[14] P. Fournier, X. Jiang, W. Jiang, S.N. Mao, T. Venkatesan, C.J. Lobb, R.L. Greene,Phys. Rev. B 56 (1997) 14149.

[15] Y. Dagan, R.L. Greene, Phys. Rev. B 76 (2007) 024506.[16] Pengcheng Li, F.F. Balakirev, R.L. Greene, Phys. Rev. Lett. 99 (2007) 047003.[17] K. Jin, B.Y. Zhu, B.X. Wu, L.J. Gao, B.R. Zhao, Phys. Rev. B 78 (2008) 174521.[18] L. Zhao, G.D. Liu, Supercond. Sci. Technol. 23 (2010) 042001.[19] T. Helm, M.V. Kartsovnik, M. Bartkowiak, N. Bittner, M. Lambacher, A. Erb, J.

Wosnitza, R. Gross, Phys. Rev. Lett. 103 (2009) 157002.[20] M.V. Kartsovnik, T. Helm, C. Putzke, F. Wolff-Fabris, I. Sheikin, S. Lepault, C.

Proust, D. Vignolles, N. Bittner, W. Biberacher, A. Erb, J. Wosnitza, R. Gross,New J. Phys. 13 (2011) 015001.

[21] T.B. Charikova, A.I. Ponomarev, G.I. Kharus, N.G. Shelushinina, A.O. Tashlykov,A.V. Tkach, A.A. Ivanov, JETP 105 (2007) 626.

[22] H. Matsui, K. Terashima, T. Sato, T. Takahashi, S.-C. Wang, H.-B. Yang, H. Ding,T. Uefuji, K. Yamada, Phys. Rev. Lett. 94 (2005) 047005.

[23] H. Matsui, K. Terashima, T. Sato, T. Takahashi, M. Fujita, K. Yamada, Phys. Rev.Lett. 95 (2005) 017003.

[24] J. Lin, A.J. Millis, Phys. Rev. B 72 (2005) 214506.

[25] C.S. Liu, W.C. Wu, Phys. Rev. B 76 (2007) 014513. Figs. 1c, d.[26] E.Z. Kuchinskii, M.V. Sadovskii, JETP Lett. 88 (2008) 224.[27] (a) T.B. Charikova, N.G. Shelushinina, G.I. Kharus, A.A. Ivanov, JETP Lett. 88

(2008) 123;(b) T.B. Charikova, N.G. Shelushinina, G.I. Kharus, O.E. Sochinskaya, A.A.Ivanov, Phys. Solid State 51 (2009) 2229;(c) T.B. Charikova, N.G. Shelushinina, G.I. Harus, V.N. Neverov, D.S. Petukhov,O.E. Sochinskaya, A.A. Ivanov, Fizika Nizkikh Temp. 37 (3) (2011) 344.

[28] V.F. Gantmacher, G.A. Emel’chenko, I.G. Naumenko, G.E. Tsydynzhapov, JETPLett. 72 (2000) 21.

[29] J. Jaroszynski, F. Hunte, L. Balicas, Youn-jung Jo, I. Raicevic, A. Gure-vich, D.C.Larbalestier, F.F. Balakirev, L. Fang, P. Cheng, Y. Jia, H.H. Wen, Phys. Rev. B 78(2008) 174523.

[30] S. Uchida, H. Takagi, Physica 162–164 (1989) 1677.[31] A.I. Ponomarev, A.N. Ignatenkov, T.B. Charikova, A.O. Tashlykov, K.S. Redkina,

L.D. Sabirzyanova, G.I. Harus, N.G. Shelushinina, A.A. Ivanov, Phys. Met. Metall.95 (6) (2003) 551.

[32] X.Q. Xu, S.N. Mao, Wu Jiang, J.L. Peng, R.L. Greene, Phys. Rev. B 53 (1996) 871.[33] M.N. Feigelman, V.B. Geshkenbein, A.I. Larkin, V.M. Vinokur, cond-mat/

9503082.[34] D.I. Khomskii, A. Freimuth, Phys. Rev. Lett. 75 (1995) 1384.[35] Frank J. Blatt, Theory of Mobility of Electrons in Solids, Academic Press

Publishers, New York, 1957. Eq. (III.51).[36] Y. Dagan, R. Beck, R.L. Greene, Phys. Rev. Lett. 99 (2007) 147004.[37] P.G. de Gennes, Superconductivity of Metals and Alloys, W.A. Benjamin, Inc.,

1966.[38] R. Prozorov, Supercond. Sci. Technol. 21 (2008) 082003.