Page 1

arXiv:cond-mat/0102422v1 [cond-mat.supr-con] 23 Feb 2001

Metal-insulator crossover in superconducting cuprates in strong magnetic fields

P.A. Marchetti,aZhao-Bin Su,bLu Yuc,b

aDipartimento di Fisica “G. Galilei”,INFN, I–35131 Padova, Italy

dInstitute of Theoretical Physics, CAS, Beijing 100080, China

cAbdus Salam International Centre for Theoretical Physics, I-34100 Trieste, Italy

The metal-insulator crossover of the in-plane resistivity upon temperature decrease, recently ob-

served in several classes of cuprate superconductors, when a strong magnetic field suppresses the

superconductivity, is explained using the U(1) × SU(2) Chern-Simons gauge field theory. The ori-

gin of this crossover is the same as that for a similar phenomenon observed in heavily underdoped

cuprates without magnetic field. It is due to the interplay between the diffusive motion of the

charge carriers and the “peculiar” localization effect due to short-range antiferromagnetic order.

We also calculate the in-plane transverse magnetoresistance which is in a fairly good agreement

with available experimental data.

PACS Numbers: 71.10.Pm, 11.15.-q, 71.27.+a

The in-plane resistivity in heavily underdoped sam-

ples of cuprates (in particular LSCO) exhibits a mini-

mum and a crossover from metallic to insulating behav-

ior upon the temperature decrease [1]. Recently a similar

crossover was observed in several classes of superconduct-

ing cuprates [2–5] when a strong magnetic field (up to 60

Tesla) suppresses the superconductivity. The “obvious”

interpretation in terms of two-dimensional (2D) localiza-

tion or 2D insulator-superconductor transition is ruled

out, as the sheet resistance, defined as ρsh≡ ρab/a, where

a is the interlayer distance, at the crossover point is be-

tween 1/25 to 1/12 in units of h/e2[2–4], or, using a free

electron model, the estimated product kFl, where kF is

the Fermi momentum, l the mean free path, is between 12

and 25! The insulating ground state persists up to opti-

mal doping in LSCO [2] and in electron-doped supercon-

ductors Pr2−xCexCuO4(PCCO) [3], while in newly stud-

ied Bi2Sr2−xLaxCuO6+δ(BSLCO, or La-doped Bi-2201)

it persists only up to 1/8 hole-doping without show-

ing any signature of stripe formation. Thus ascribing

this metal-insulator (MI) crossover to a quantum critical

point related to charge density instability [6] is open to

objections [7]. There were several attempts to interpret

the insulating ground state in doped cuprates using var-

ious arguments on non-Fermi liquid behavior [8]. How-

ever, the crossover phenomenon as temperature varies

has not been addressed up to now, to the best of our

knowledge. It was thought earlier that the MI crossover

in the absence of magnetic field [1] and that induced

by magnetic field is of different origin (the sheet resis-

tance in the first case was substantially higher) [2]. This

is doubtful, because the recent experiments on YBCO

show a MI crossover in nonsuperconducting compounds

in the absence of magnetic field [9], and the same type

of MI crossover (with comparable sheet resistance at the

crossover)in the superconducting compounds of the same

series doped with Zn in the presence of magnetic field

[5]. We believe the two crossover phenomena are of the

same origin. We have used the U(1) × SU(2) Chern–

Simons (C.S.) approach to the t − J model, proposed

by us earlier [10] to explain the MI crossover in heav-

ily underdoped cuprates in the absence of magnetic field

[11]. In this Letter we generalize our formalism to in-

clude the effect of magnetic field and show that such a

MI crossover is a universal feature of doped cuprates,

and it is due to a “peculiar charge localization” effect

(using the wording of Ref. [4]), resulting from the inter-

play of the spin-excitation gap (corresponding to short-

range antiferromagnetic order (SRAFO)) and the holon

induced anomalous dissipation. Moreover, we will show

that the observed large positive in-plane transverse mag-

netoresistance (MR) at low temperatures [12,13] can be

semi-quantitatively explained within this formalism.

The U(1) × SU(2) C.S. gauge field approach is a par-

ticular scheme of slave-particle formalism to treat the

t − J model based on an exact identity [14], introduc-

ing a U(1) field gauging the global charge symmetry and

a SU(2) field gauging the global spin symmetry, both

with C.S. actions. Using an optimization procedure of

free energy [10], a careful mean field (MF) approxima-

tion gives the following results: The U(1) gauge field for

low doping δ develops a π-flux per plaquette converting

holons into Dirac fermions with a Fermi energy ǫF∼ tδ.

The holons induce a vortex structure in the MF con-

figurations of the SU(2) gauge field, with vortices cen-

tered at the holon positions. These dressed holons in

turn are seen as slowly moving impurities by spin waves

giving rise to a spinon mass ms∼

this feedback is self-consistent because for low δ we have

ǫF ∼ tδ << ǫs∼ J?δ|lnδ|. We use J = 0.1eV, t/J = 3

of “gauge fixing” (using the N´ eel gauge) the SU(2) gauge

field becomes physical, describing the spin fluctuations.

The spinon action is given by a nonlinear σ-model with

a mass term (spinon gap) which in the CP1represen-

tation yields a new self-generated U(1) gauge field A

coupling holons and spinons ( This field is analogous to

the U(1) gauge field in the standard slave-particle ap-

proaches [15,16]). Due to coupling to holons (fermions

in our approach), this gauge field acquires an anoma-

?δ|lnδ|. Notice that

in our numerical computations. Due to a special choice

1

Page 2

lous dissipation term, the “Reizer singularity” [17], which

dominates the low-energy action for the transverse com-

ponent of the gauge field AT. For ω,|? q|,ω/|? q| ∼ 0 we have

?ATAT?(ω,? q) ∼ (χ|? q|2+ iκω

diamagnetic susceptibility and κ ∼ δ the Landau damp-

ing. The interplay of the two different energy scales, the

spinon gap and the holon induced anomalous dissipation,

is the key factor in our interpretation of the MI crossover

[11]. For the temporal component in the same limit, we

have ?A0A0?(ω,? q) ∼ (γ + ωp)−1, where γ is the fermion

density of states and ωpthe plasmon gap.

Now we consider the introduction of a magnetic field

H perpendicular to the plane.

[18], R = Rs+ Rh, i.e., the observed resistivity is the

sum of the holon and spinon contributions, can be gen-

eralized to this case. The external electromagnetic po-

tential Ae.m., corresponding to the constant magnetic

field H can couple with coefficient −ε to spinons and

1 − ε to holons. In principle, 0 ≤ ε ≤ 1 is arbitrary.

However, to be consistent with the requirement that the

physical inverse magnetitic susceptibility should be the

sum of the inverse of that of holons and spinons, i.e.,

χ−1= (χ∗

malized spinon and holon susceptibility, respectively (this

relation can be derived in the same way, as the Ioffe-

Larkin rule), we find ε = χ∗

was argued earlier using variational considerations [19].

Replacing these quantities by unrenormalized values, we

find 1 − ε ∼ χs/(χh+ χs) ∼

Coulomb gauge A0

e.m.= 0, the effective action for the

gauge field A can be written as:

|? q|)−1, where χ ∼ t/δ is the

The Ioffe–Larkin rule

s)−1+ (χ∗

h)−1, where χ∗

sand χ∗

hare the renor-

h/(χ∗

h+ χ∗

s). This value

J

t

?δ/|lnδ| << 1. In the

Seff(A) =

+(AT+ (1 − ε)Ae.m.)Π⊥

+(AT− εAe.m.)Π⊥

?

dx0d2x

?i

2[A0(Π0

h+ Π0

s)A0

h(AT+ (1 − ε)Ae.m.)

s(AT− εAe.m.)] +iσh(H)

2π

A0ǫij∂iAj

?

(1)

,

where Π0,⊥

bles due to holons and spinons, respectively, σh(H) is

the Hall conductivity due to holons. Note Ae.m.appears

in two places in this low energy effective action: one is

simply a shift of the transverse component of the gauge

field ATby (1 − ε)Ae.m.and −εAe.m.corresponding to

the minimal coupling to holons and spinons, respectively,

while the other is a C.S. term due to parity breaking in-

duced by H.

As remarked in [19], the leading effect of the integra-

tion over A0 is the renormalisation of the diamagnetic

susceptibility: χ → χ(H) = χ +

tive action. The holon contribution Rhcan be evaluated

using the Boltzmann equation, taking into account the

classical cyclotron effect, as in [19], obtaining

h,Π0,⊥

s

are unrenormalized polarization bub-

σ2

4π2γin the ATeffec-

h(H)

Rh= R0

h[1 + ((1 − ε)Hτ

mh

)2],

R0

h∼mh

8

1

τ∼ δ

?

1

ǫFτimp

+ (T

ǫF)4/3

?

, (2)

where τ is the transport relaxation time, τimp the

impurity scattering time and mh ∼ δ/t the holon

mass.The spinon contribution Rs is evaluated here

using the Kubo formula for the spinon current: Rs =

limω→0ω(ImΠ⊥

polarization bubble at ? q = 0, renormalized by gauge

fluctuations.At large scales, for x0≫ |? x|, Π⊥

is approximately given by ?∂µG(x|A)∂µG(x|A)?, where

? ? denotes the A-expectation value and G(x|A) is the

spinon propagator.Using the Fradkin representation

[20,11] it can be transformed into a gauge invariant form

?∂µG(x|F)∂µG(x|F)?, where G(x|F) in terms of a path-

integral over 3-velocities, φµ(t) ≡ ˙ qµ(t),µ = 0,1,2, is

given by

s(ω))−1, where Π⊥

sdenotes the transverse

s(x)

G(x|F) ∼ i

?

with

?∞

0

dse−im2

ss

?

Dφµe

i

4

?s

0φ2

µ(t)dt

·d3peipx−ip2seiQij(p,s,φ|s′,λ)[Fij(p,s|s′,λ)+ǫijεH]

(3)

Qij(p,s,φ|s′,λ)[#] =

·(φi(s′) − 2pi)(φj(s′′) − 2pj)[#]

?1

0

dλλ

?s

0

ds′

?s′

0

ds′′

(4)

and

Fij(p,s|s′,λ) = Fij(x + λ

Fij≡ ∂iAj− ∂jAi.

After integration over AT, using the effective action, and

integration over the 3-velocities in the gaussian approxi-

mation, and over p and s by saddle point for low T, one

obtains at large scales

?s′

0

ds′′φ(s′′) − 2ps),

(5)

Π⊥

s(x) ∼

∂

· e−

∂xµ

e

T

−i

?

m2−

T

χ(H)f(α)+α2ε2H2

3q2

0

√

x2

0−|? x|2

4χ(H)q2

0g(α)

(x2

0−|? x|2)

m2

1

?(x2

0− |? x|2)

??2

, (6)

where α =

with the anomalous skin effect due to Reizer singular-

ity: q0∼?δ2T

is monotonically increasing, vanishing quadratically near

the origin and g is monotonically decreasing vanishing

at large argument.(See [21] for explicit expressions.)

The integration over |? x| and x0appearing in the Fourier

q0|? x|

2, q0 is the momentum scale associated

t

?1/3. f and g are functions describing the

effect of gauge fluctuations and for a real argument, f

2

Page 3

transformation are evaluated by saddle point for |? x|, at

large x0, and with scale renormalization by principal

part evaluation for x0[11,21]. The integrals are domi-

nated by a complex saddle point at |? x| = 2q−1

χ(H)q0|ms(T,H)|<∼T, Im(m2

?

0eiπ

4 for

s(T,H))<∼m2

cT

χ(H)−ε2H2

s, where

m2

s(T,H) = m2

s− i

3q2

0

?

, (7)

with c = −if(eiπ/4), a constant with real part ∼ 3 and a

small imaginary part. For the range of physical parame-

ters considered here (H<∼100 Tesla), these bounds gave

a temperature range validity lying between a few tens

and a few hundreds degrees.

The saddle point produces the following effects: it in-

duces “renormalization” of the spinon mass yielding a T

and H dependent damping: m2

rise to an attraction between spinon and antispinon lead-

ing to the formation of a damped spin wave; it introduces

a multiplicative renormalization of the correlation func-

tions which for Rsis given by Z(T,H)[m2

Z(T,H) = (c′

stant ∼ f′′(eiπ/4).

The final result in the range of T described above is

given by

s−→ m2

s(T,H); it gives

s(T,H)]

1

8, where

T

χ(H)q−3

0−2

3ε2H2q−5

0)1/2with c′a new con-

Rs∼ Z(T,H)|ms(T,H)|1/4

sinΘ(T,H)

4

, (8)

where ms(T,H) ≡ |ms(T,H)|e−iΘ(T,H). Basic features

of our formulas can be summarized as follows: for low

T, the effect of the spinon gap is dominating (Θ ց 0),

leading to an insulating behaviour; at higher tempera-

tures one finds a metallic behavior due to the dissipation

induced by gauge fluctuations, contained in |ms(T,H)|,

that becomes the dominant effect.

crossover is recovered, decreasing the temperature. The

minimum of R as a function of T, TMI(δ,H), is decreas-

ing with δ and increasing with H (see the MR curve

(R(H)−R(0))/R(0) in Fig. 1), in agreement with exper-

iment [5]. In the absence of magnetic field the crossover

is determined by the interplay between m2

T/χ ∼ Tmh ∼ λ−2, with ξ the magnetic correlation

length, λ the thermal de Broglie wave length.

λ ≤ ξ, the “peculiar” localization effect due to SRAFO

is not “felt”, and a metallic behavior is observed. In the

opposite limit λ ≫ ξ the cuprate is insulating. The exter-

nal magnetic field effectively reduces the thermal energy,

or increases the thermal wavelength, so the crossover

temperature goes up.The resistivity is diverging at

T =

3cq2

0

, which approaches T = 0 as H vanishes.

This divergence is lying outside the region of validity of

our formulas and should be considered as an artifact.

Therefore a MI

s= ξ−2and

When

ε2H2χ(H)

100 200

Temperature (K)

300 400

−0.001

0

0.001

0.002

∆R(H)/R(0)

H=50T

H=30T

H=15T

FIG. 1. The calculated magnetoresistance for cases when

the quantum effects related to σh(H) are strong, for doping

δ = 0.05. It becomes negative near the minimum which itself

shifts to higher temperatures upon the field increase.

However, the shift in MI crossovertemperature leads to

a large positive (in–plane transverse) MR at low T which

is our main new result, and it was absent in the earlier

treatments [19]. The derived MR scales quadratically

with H (See Fig. 2) in agreement, in particular, with

data on LSCO [12,13], away from the doping δ = 1/8

where the stripe effects dominate. As remarked in [19],

the shift of χ induced by the C.S. term reduces the damp-

ing and the H2-term due to minimal coupling acts in the

same direction. In the region of T where dissipation dom-

inates this induces a reduction of resistivity, a tendency

contrasted by the classical cyclotron effect. One then has

two possible types of MR curves: one is always positive

but it exhibits a knee below the crossover temperature

between the mass gap and the dissipation dominated re-

gions (See Fig. 3). This behavior can be compared with

the one observed in LSCO reported in [13] and we find

a reasonably good agreement. If, on the contrast, the

quantum effects related to σh(H) are sufficiently strong,

02468 10

H (T)

0

0.002

0.004

0.006

0.008

∆R(H)/R(0)

02468 10

0

0.001

0.002

0.003

0.004

0.005

T=40K

T=70K

T=100K

T=40K

T=70K

FIG. 2. The calculated field dependence of the magnetore-

sistance for doping δ = 0.075, in comparison with experi-

mental data on La1.925Sr0.075CuO4+ǫ (inset), taken from Ref.

[13].

3

Page 4

0 100

Temperature (K)

200 300

0

0.001

0.002

0.003

0.004

0.005

∆R?(H)/R(0)

0 100200 300

0

0.001

0.002

0.003

0.004

0.005

H=10T

H=10T

FIG. 3. The calculated temperature dependence of the

magnetoresistance for doping δ = 0.075, in comparison with

experimental data on La1.925Sr0.075CuO4+ǫ (inset), taken

from Ref. [13].

a minimum develops, eventually leading to a negative

MR in some region around it. This is illustrated in Fig.1.

We should point out that the large positive MR at low

temperatures is foreseen in this theory for both cases.

A comment on the limit H ∼ 0 is in order, where we

recover the results of [10], in particular m2

icT/χ, Z(T,0) ∼

an inflection point at temperature T∗(δ) ∼ 200 − 300

K, (found also in the experimental curves), above which

the theoretical curves start to deviate strongly from the

experimental data. We propose to interpret this in-

flection point as a mid–point of a crossover to a new

“phase” where our MF treatment is not valid anymore.

If we identify our T∗(δ) with the crossover tempera-

ture T∗found in experiments, both MI crossover tem-

perature TMI(δ) ≡ TMI(δ,0) and T∗(δ) are found in

a reasonable agreement with experimental data (in the

range 0.02∼

of doping dependences:

χm2

in this limit can be written in terms of a dimension-

less variable x ≡ cT/χm2

?|lnδ|. Hence, if we neglect the contribution ∼ T4/3

˜R ≡ (R − R(TMI))/(R(T∗) − R(TMI)), this is a func-

tion only of x, thus exhibiting a “universal” behaviour,

as observed in YBCO [22].

As a final remark it might be worthwhile to notice

that the same U(1) × SU(2) approach is able to repro-

duce qualitatively [10,23] the behavior of the spin lattice

relaxation rate (1/T1T)63found in underdoped YBCO

[24] and a structure of Fermi arcs around (π

spectral density detected by ARPES [25].

We thank J.H. Dai for collaboration in the early stage

of this work, and Y. Ando and T. Xiang for very helpful

discussions. The work of P. M. is partially supported

by RTN Programme HPRN-CT2000-00131, while L.Y.

acknowledges the partial support by INTAS Georgia 97-

1340.

s(T,0) = m2

s−

1

√δ. In this limit the resistivity exhibits

< δ∼

< 0.08), due to a delicate cancellation

T

∼

s

Tδ

tδ|lnδ|=

T

t lnδ.Rs

sapart from an overall factor

due to holons and define a “normalised resistivity” by

2,π

2) in the

[1] H. Takagi et al., Phys. Rev. Lett. 69, 2975 (1992); B.

Keimer et al., Phys. Rev. B 46, 14034 (1992).

[2] Y. Ando et al., Phys. Rev. Lett. 75, 4662 (1995); G.S.

Boebinger et al., ibid, 77, 5417 (1996); Y.Ando et al., J.

Low Temp. Phys. 105, 867 (1996).

[3] P. Fournier et al., Phys. Rev. Lett. 81, 4720 (1998).

[4] S. Ono et al., Phys. Rev. Lett. 85, 638 (2000)

[5] K. Segawa and Y. Ando, Phys. Rev. B 59, R3948 (1999).

In Zn-doped samples, the MI crossover becomes observ-

able upon the increase of Zn-doping. This is consistent

with our picture, as the magnetic correlation length is

getting shorter in this case, although we can not directly

compare these results, because this factor has not been

explicitly taken into account in the present calculation.

[6] C. Castellani et al., Z. Phys. B 103, 137 (1997).

[7] S. Chakravarty et al., cond-mat/0005443.

[8] P.W. Anderson et al., Phys. Rev. Lett. 77, 4241 (1996);

C.M. Varma, Phys. Rev. Lett. 79, 1535 (1997); A.D.

Mirlin and P. W¨ olfle, Phys. Rev. B 55, 5141 (1997).

[9] Y. Ando et al. , Phys. Rev. Lett. 83, 2813 (1999).

[10] P. A. Marchetti, Z-B. Su and L. Yu, Phys. Rev. B 58,

5808 (1998); Mod. Phys. Lett. B 12, 173 (1998).

[11] P.A. Marchetti, J.H. Dai, Z.B. Su, and L. Yu, J. Phys.:

Condens. Matter 12, L329 (2000).

[12] T. Kimura et al., Phys. Rev. B 53, 8733 (1996); Y. Abe

et al., Phys. Rev. B 59, 14753 (1999).

[13] A. Lacerda et al., Phys. Rev. B 49, 9097 (1994).

[14] J. Fr¨ ohlich et al., Nucl. Phys. B 374, 511 (1992); J.

Fr¨ ohlich and P. A. Marchetti, Phys. Rev. B 46, 6535

(1992); P.A. Marchetti, cond-mat/9812251.

[15] P. A. Lee and N. Nagaosa, Phys. Rev. Lett. 64, 2450

(1990); Phys. Rev. B 46, 5621 (1992).

[16] L. B. Ioffe and P. B. Wiegmann, Phys. Rev. Lett. 65, 653

(1990).

[17] M. Reizer, Phys. Rev. B 40, 11571 (1989).

[18] L. Ioffe and A. Larkin, Phys. Rev. B 39, 8988 (1989).

[19] L. Ioffe, G. Kotliar, Phys. Rev. B 42, 1038 (1990); L.

Ioffe, P. Wiegmann, Phys. Rev. B 45, 519 (1992).

[20] E.S. Fradkin, Nucl. Phys. 76, 588 (1966); H.M. Fried and

Y.M. Gabellini, Phys. Rev. D 51, 890 (1995).

[21] P.A. Marchetti, J.H. Dai, Z.B. Su, and L. Yu, to be pub-

lished.

[22] B. Wuyts et al., Phys. Rev. B 53, 9418 (1996); L. Trap-

peniers et al. cond-mat/9910033.

[23] L. de Leo, Laurea Thesis, University of Padova, 2000; L.

de Leo, P.A. Marchetti, Z. B. Su, L. Yu, in preparation.

[24] See, e.g. C. Berthier et al., Physica C 235-240. 67 (1994).

[25] A.G. Loeser et al., Science 273, 325 (1996); H. Ding et

al., Nature (London) 382, 51 (1996).

4