interlayer penetration depth in the pseudogap phase of cuprate superconductors

Interlayer penetration depth in the pseudogap phase

of cuprate superconductors

J. P. Carbotte1,2 and E. Schachinger3

1 Department of Physics and Astronomy, McMaster University, Hamilton, Ontario,

Canada N1G 2W12 The Canadian Institute for Advanced Research, Toronto, Ontario, Canada M5G

1Z83 Institute of Theoretical and Computational Physics, Graz University of Technology,

A-8010 Graz, Austria

E-mail: [email protected]

Abstract. The opening of a pseudogap in the electronic structure of the underdoped

high Tc cuprates has a profound effect on superconducting properties. Here we consider

the c-axis penetration depth. A phenomenological model of the pseudogap due to Yang,

Rice, and Zhang (YRZ) is used. It is based on the idea of a resonating valence bond

spin liquid. A simplifying limit, the arc model, is also considered as it provides useful

analytic formulas. The zero temperature value of the superfluid density ns(T = 0) is

greatly reduced with increasing values of the pseudogap (∆pg). This value reflects the

reconstruction of the Fermi surface from the large contour of Fermi liquid theory to ever

smaller Luttinger pockets as ∆pg becomes larger. Also, as temperature is increased

the ratio ns(T )/ns(0) as a function of the reduced temperature t = T/Tc decreases

more rapidly than in the corresponding Fermi liquid (∆pg = 0) as states which have

both superconducting and pseudogap become more significantly sampled.

PACS numbers: 74.20.Mn 74.72.-h 74.72.Kf

Submitted to: J. Phys.: Condens. Matter

Interplane penetration depth 2

1. Introduction

The underdoped cuprates show unusual behaviour in their transport properties

associated with the approach to a Mott antiferromagnetic insulating state as the hole

doping is reduced towards half filling. While the dc-resistivity within a CuO2 plane

exhibits metallic behaviour and decreases with decreasing temperature, the inter-layer

charge transport shows an upturn at low temperatures characteristic of an insulating

state [1, 2]. A Drude peak persists in the in-plane ac-dynamic conductivity [3, 4] even

beyond the superconducting dome in the antiferromagnetic region of the phase diagram.

On the other hand, the c-axis ac-conductivity acquires a more gap like behaviour at small

photon energies [5, 6, 7, 8] even in its normal state which becomes more pronounced

with decreasing temperature. This reduction in c-axis ac-conductivity is associated

with the formation of a pseudogap [9, 10]. There also exist corresponding differences

in superconducting properties. As an example, the temperature dependence of the

penetration depth in the extremely underdoped regime is observed to be linear with T

at small T bending over to a T 2 law due to impurities within the CuO2 plane [11] while

in the c-direction it follows a Tα law with α ≥ 3 [10, 12].

Recently Yang et al. [13] (YRZ) have proposed an ansatz for the self energy

of the charge carriers in the underdoped cuprates which describes the emergence of

a pseudogap of d-wave symmetry for doping levels x below a critical concentration

xc. As x is reduced below the quantum critical point (QCP) at x = xc, the large

Fermi surface of Fermi liquid theory undergoes a reconstruction into Luttinger pockets.

At sufficiently low x only hole pockets remain which are centered about the nodal

direction in the CuO2 Brillouin zone (BZ), and these Fermi pockets shrink progressively

in size although they never vanish. It is these pockets which account for the metallic

behaviour observed in the in-plane transport. While based on ideas of a resonating

valence bond spin liquid [14] (RVB) the model of YRZ remains phenomenological but

has been very successful in understanding many of the properties of the underdoped

cuprates [13, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25], many previously considered

to be anomalous. It seems clear that YRZ has captured a new element beyond BCS

theory and its extensions to include inelastic scattering [26, 27], energy dependence in

the underlying band structure,[28, 29, 30, 31] and anisotropies [32, 33, 34, 35]. Besides a

pseudogap of increasing magnitude as doping x decreases below the QCP x = xc = 0.2

the theory includes Gutzwiller factors. These factors account for the restriction that

double occupancy of an atomic site is forbidden because of the large on-site repulsive

Hubbard U . These factors also describe the narrowing of the energy bands expected in

highly correlated systems. The YRZ model is also different from other prominent models

of the pseudogap state such as the preformed pair model [36, 37] and the d-density wave

model [38, 39]. In particular the YRZ model was used successfully to understand the

temperature and doping dependence of the in-plane penetration depth [18].

A feature of the in-plane data that could be understood is the rapid decrease in

the magnitude of the zero temperature superfluid density while at the same time, the


slope of its linear in T low temperature reduction is relatively much less affected. As the

Mott insulating state is approached, well defined delocalized quasiparticles still exist in

the normal state but only on the surface of the Luttinger pockets. Thus, their number

is greatly reduced over the optimum doping case for which a large Fermi liquid type

Fermi surface exists. Consequently, the zero temperature superfluid density is greatly

reduced. But the low temperature linear law depends only on the number of thermal

quasiparticles that can be created by breaking Cooper pairs and, thus, depleting the

condensate. For very low T such excitations are, however, confined to the vicinity of

the nodal direction. This region is unaffected by pseudogap formation and, hence, is

only marginally changed by the existence of Luttinger hole pockets. In this paper we

will use the YRZ model to calculate the c-axis penetration depth and compare results

with the a-b plane case [18].

We use a coherent tunneling model for the charge transfer along the c-axis

perpendicular to the CuO2 planes. This model coupled with the YRZ pseudogap model

has also provided [40] a simple and appealing explanation for the observed puzzling

c-axis optical sum rule violation seen by Basov et al. [41] in the underdoped cuprates.

The usual Ferrell-Glover-Tinkham [42, 43] sum rule states that the missing area under

the real part of the superconducting state optical conductivity when compared with

the normal state is found in the superconducting condensate. For the c-axis in the

underdoped cuprates the lost optical spectral weight can instead be as small as half this

value [41].

In Sec. 2 we present the formalism on which our work is based and in Sec. 3 we

give results. The work proceeds numerically with momentum integrals done through a

sampling of the entire CuO2 Brillouin zone (BZ). We also provide a simple limiting case

of the general formalism which reduces to an arc model. In this limit the pseudogap

is taken to be on the Fermi surface rather than on its own surface closely related to

the antiferromagnetic Brillouin zone. This has the advantage that analytic results can

be obtained which we found to be useful in providing physical insight into the more

complicated calculations. In Sec. 4 we summarize and give our conclusions.

2. Formalism

Levchenko et al. [44] found that a simple coherent tunneling model for charge transfer

from one CuO2 plane to another respecting the symmetry of the atomic arrangements

[45, 46, 47, 48, 49] involved, leads to a simple and compelling explanation of the observed

behaviour of the dc-resistivity in the underdoped cuprates within an arc model with

pseudogap. An extension of this work to the YRZ model by Ashby and Carbotte [50]

confirmed this good agreement with experiments [1, 2]. Here we use this same model,

with tunneling matrix element for transfer of an electron with momentum k in one plane

to an adjacent plane with momentum k′ given by

tk,k′ = t⊥δk,k′η2k. (1)


The Kronecker δ conserves the parallel momentum component, t⊥ sets the scale, and

ηk is a function which has d-wave symmetry [45, 46, 47, 48, 49] normalized to have a

maximum value of one:

ηk =1

2[cos(kxa)− cos(kya)] , (2)

with a the in-plane lattice parameter. This form has also been used in many other works

[51, 52, 53]. To describe the superconducting state of the underdoped cuprates we use

the YRZ model and with the c-axis penetration depth given by the formula [54]

1

λ2c(T )

= CπT∑ωn

∫ d2k

(2π)2F (k, iωn)F (k, iωn)η

4k. (3)

Here, the constant t⊥ is absorbed into the constant C, iωn = iπT (2n + 1) with

n = 0,±1,±2, ... are the Matsubara frequencies at temperature T and F (k, iωn) is

the Gor’kov anomalous Green’s function. The integral extends over the CuO2 Brillouin

zone. To specify F (k, iωn) we begin with the normal state which in the YRZ model

is unconventional and involves a pseudogap ∆pg(k, x) for doping values x less than a

critical value xc taken to be 0.2. At this point there is a QCP from the Fermi liquid

to the pseudogap phase. In Ref. [13] the pseudogap ∆pg(k, x) is taken to have d-wave

symmetry with [9, 55]

∆pg(k, x) = ∆0pg(x)ηk. (4)

The amplitude ∆0pg(x) is given as

∆0pg(x)/t0 = 0.6

(1− x

0.2

), x ≤ 0.2, (5)

where t0 is the bare band nearest neighbour hopping parameter. With pseudogap there

are two branches to the effective band-structure which have weightings W αk with α = ±

and energies Eαk given by

W αk =

1

2

(1 + α

ξkEk

), (6a)

and

Eαk = 0.5(ξk − ξ0k)± Ek, (6b)

with

Ek =√ξ2k +∆2

pg(k, x). (6c)

In Eqs. (2)

ξk = 0.5(ξk + ξ0k). (7)

The band structure dispersion curves renormalized by Gutzwiller factors which account

for no double on-site occupancies because of the large on-site Hubbard repulsion U , are

given by

ξk = − 2t(x)[cos(kxa) + cos(kya)]− 4t′(x) cos(kxa) cos(kya)

− 2t′′(x)[cos(2kxa) + cos(2kya)]− µp(x) (8)


up to third nearest neighbour hopping with

t(x) = gt(x)t0 + 2gs(x)Jχ/8, (9a)

t′(x) = gt(x)t′0, (9b)

t′′(x) = gt(x)t′′0. (9c)

The Gutzwiller factors are given by

gt(x) =2x

1 + x, and gs(x) =

4

(1 + x)2. (10)

Here, t0, t′0, and t′′0 are the bare band hopping parameters with t′0 = −0.3t0 and

t′′0 = 0.2t0, J is the magnetic energy of the t-J model taken to be J = t0/3, χ = 0.338

is the spin susceptibility, and in much of this work we will take t0 = 175meV.

The Umklapp surface energy is ξ0k = −2t(x)[cos(kxa) + cos(kya)] and ξ0k = 0 gives

the antiferromagnetic Brillouin zone. Finally, the chemical potential µp(x) is to be

determined to get the correct hole doping x based on the Luttinger sum rule (LSR).

This process was described by Yang et al. [13]. The superconducting state energies Eαk,s

are Eαk,s =

√(Eα

k )2 +∆2

sc(k, T, x) with the superconducting gap amplitudes

∆0sc(x)/t0 = 0.14[1− 82.6(x− 0.2)2] (11)

and ∆sc(k, T, x) taken to have d-wave symmetry and to follow the BCS mean field

temperature dependence

∆BCS(T )

∆BCS(T = 0)= tanh

[Tc

T

∆BCS(T )

∆BCS(T = 0)

]≡ ∆(T ), (12)

so that

∆sc(k, T, x) = ∆0sc(x)ηk∆(T ). (13)

Note that here we followed the model of Ref. [13] without change. Of course, for

several high Tc cuprate families optimum doping corresponds to x = 0.16. Also,

the QCP associated with the onset of the pseudogap may be at the overdoped edge

of the superconducting dome [55] rather than at optimum doping, but this remains

controversial.

The Gor’kov anomalous Green’s function F (k, iωn) of Eq. (3) can be written in

terms of its spectral density B(k, ν) as

F (k, iωn) =∫ ∞

−∞dν

B(k, ν)

iωn − ν, (14)

with B(k, ν) given in terms of the quantities defines above as

B(k, ν) =∑α=±

gt(x)∆sc(k, T, x)

2Eαk,s

W αk

[δ(ν − Eα

k,s)− δ(ν + Eαk,s)

]. (15)

Substitution of Eq. (15( into Eq. (14) gives

F (k, iωn) = gt(x)∆sc(k, T, x)∑α=±

W αk

ω2n + (Eα

k,s)2, (16)


0 20 40 60 80 1000.0

5.0x10-5

1.0x10-4

1.5x10-4

-2 c(T

) (ar

b. u

nits

)

T (K)

x = 0.2 x = 0.18 x = 0.16 x = 0.14 x = 0.12 x = 0.1

pg = 0

Figure 1. (Colour online) Temperature dependence of the inverse square of the c-axis

penetration depth, λ−2c (T ), in arbitrary units. There are six pairs of curves as labeled

by colour and line type. The heavier lines include the effect of a pseudogap while the

light lines are with ∆0pg(x) set equal to zero.

and, hence, the penetration depth of Eq. (3) takes on the form

1

λ2c(T )

= CπT∑ωn

∑α,α′=±

∫ d2k

(2π)2η4kg

2t (x)

∆2sc(k, T, x)W

αkW

α′k

[ω2n + (Eα

k,s)2][ω2

n + (Eα′k,s)

2]. (17)

The same form applies also to the in-plane penetration depth 1/λ2ab(T ) [18] except that

the η4k factor which comes from the out of plane tunneling matrix element (1) is to

be replaced by the kx component of the Fermi velocity ∂ξk/∂kx where ξk is given by

Eq. (8), and the constant in front is also different.

3. Numerical results and analytic limits

In Fig. 1 we present λ−2c (T ) vs T for six values of hole doping 0.2 ≤ x ≤ 0.1 as indicated

in the figure where our colour and line type scheme is indicated. Each concentration

involves a pair of curves, one with the pseudogap (heavy lines) and one without (light

lines) included for comparison. To do the calculations the mean field temperature

dependence of the superconducting gap amplitude ∆0sc(T ) was used in all cases with

the ratio of twice the zero temperature gap amplitude to critical temperature Tc set

equal to 6, representative of a copper-oxide superconductor. For BCS d wave it would

be 4.3. The integral in Eq. (17) was evaluated numerically. Care must be taken because

of the factor η4k which is sharply peaked at the X, Y , and commensurate points of the

CuO2 BZ. Therefore, we sampled the integrand on the rather high number of 2048×2048

equally spaced points within the first quadrant of the CuO2 BZ (kx/a, ky/a ∈ [0, π]) to


(0

,x=0

.2)/

(0,x

)

x

c-axis, t = 1 c-axis, t = 1, pg = 0 ab-plane ab-plane, pg = 0

Experiment

pg = 2 YRZ

Figure 2. (Colour online) The ratio of the inverse of the penetration depth at zero

temperature λ(0, x) normalized to its value at x = 0.2 which in our model corresponds

to the QCP for pseudogap formation and is also optimum doping as in the paper by

Yang et al. [13]. Both, c-axis and a-b plane results are presented for comparison.

The dashed (red) and dashed-dotted (black) curves are with pseudogap amplitude

∆0pg(x) = 0 while the solid (black) and dashed-double dotted (red) ones are with

pseudogap. The shading helps to visualize the significant reduction in the ratio that

occurs as x is decreased, particularly for the c-axis (red shading) which at x = 0.1 is

more than three times that for the a-b-plane. The open circle shows a result when the

pseudogap amplitude is doubled its YRZ value for x = 0.14 and the solid (blue) circle

is the experimental value from Ref. [44].

calculate the estimate of the integral. An increase of this number of sampling points

changed the integral’s estimate only marginally. From this, we conclude the numerical

error to be less than two percent in all our results.

We note small changes in the temperature dependence of the c-axis superfluid

stiffness when a pseudogap is included but the most striking difference between the

two cases is the predicted strong reduction of its zero temperature value as ∆0pg(x)

is increased. This is further emphasized in Fig. 2 where the dimensionless ratio

λ(0, x = 0.2)/λ(0, x) vs doping x is plotted. The dashed (red) curve is our result

for no pseudogap, i.e. ∆0pg(x) = 0, while the dashed-double dotted (red) curve is with

∆0pg given by the YRZ parameters of Eq. (5). The red shading is included to emphasize

the large reduction in 1/λ2c(T = 0) brought about by the increasing pseudogap.

To understand these results better it is helpful to take the limit of YRZ theory

which corresponds to the arc model. In this limit the formula for the superfluid density

(or stiffness) greatly simplifies. We retain only the first neighbour hopping in the band

dispersion curve ξk of Eq. (8) and further take ξ0k to be ξk, that is, the pseudogap

is placed on the Fermi surface. In this case the energy Eαk of Eq. (6b) reduces to


±√ξ2k +∆2

pg(k, x) and Eαk,s =

√ξ2k +∆2

pg(k, x) + ∆2sc(k, T, x) which is now independent

of the index α and we will denote it simply by Ek,s. Equation (17) for 1/λ2c(T ) then

reduces to the simpler form

1

λ2c(T )

= CπT∑ωn

∫ d2k

(2π)2η4kg

2t (x)

∆2sc(k, T, x)

(ω2n + E2

k,s)2. (18)

To make more progress in our simplification of this formula we take the continuum limit

for the electronic band structure with constant density of states N(0, x) around the

Fermi surface and a circular Fermi surface (FS) with superconducting gap ∆sc(k, T, x) =

∆0sc(x)∆(T ) cos(2θ) with θ the angle on the FS. Furthermore, the pseudogap is also

taken to have the same form, namely ∆pg(k, x) = ∆0pg(x) cos(2θ) but with the further

assumption that it exists only in the antinodal regions. It is zero on the Fermi arcs

around the four equivalent nodal directions. We note that in this simplified band

structure model the chemical potential drops out explicitly. It does, however, determine

the value of N(0, x). The arcs extend in the upper right quadrant of the Brillouin zone

from π/4 to π/4 + θc as shown schematically in Fig. 3(a). With these assumptions

1

λ2c(T )

= CπT∑ωn

N(0, x)g2t (x)∫ π/2

0

dθ

π/2cos6(2θ)

× [∆0sc(x)]

2∆2(T )

{ω2n + [∆0

sc(x)∆(T )]2 cos2(2θ) + ∆2pg(θ, x)}

, (19)

where by definition ∆pg(θ, x) = ∆0pg(x) cos(2θ) between θ ∈ [0, θc) and [π/2, π/2−θc) and

is zero on the arc θ ∈ [θc, π/2−θc]. The phase diagram used is shown in Fig. 3(b)with the

superconducting gap amplitude given by the (red) solid line dome and the pseudogap

amplitude by the (black) dashed stright line. It is also convenient to go over from many

imaginary Matsubara frequencies to a real axis formulation using the well known rule

limk′→k

πT∑ωn

F (k, iωn)F (k′, iωn) = limk′→k

∫ ∞

−∞

dνdν ′

ν − ν ′ [f(ν)− f(ν ′)]B(k, ν)B(k′, ν ′), (20)

where F (ν) is the finite temperature Fermi-Dirac distribution function. We get

1

λ2c(T )

= C∫ π/2

0

dθ

π/2cos6(2θ)

∫ ∞

−∞dε

[∆0sc(x)]

2∆2(T )

E2s (θ, T )

{− ∂

∂Es(θ, T )+

1

Es(θ, T )

}

× tanh

[βEs(θ, T )

2

], (21)

with β the inverse temperature. Constants have been absorbed in C and Es(θ, T ) ≡Es =

√ε2 + [∆0

sc(x)]2∆2(T ) cos2(2θ) + ∆2

pg(θ, x) outside the nodal arc and Es =√ε2 + [∆0

sc(x)]2 cos2(2θ) on the nodal Fermi arc. For the a-b plane penetration depth a

similar formula applies but with cos6(2θ) replaced by cos2(2θ) and the constant in front

is different. For example, it involves the in-plane Fermi velocities replacing the c-axis

tunneling matrix element.

At zero temperature Eq. (21) reduces further. Because the tanh(βEs/2) factor is

now one, the derivative piece gives zero and we are left with

1

λ2c(T = 0)

= C∫ θc

0

dθ

π/4cos6(2θ)

∫ ∞

−∞dε

[∆0sc(x)]

2

[ε2 + {[∆0sc(x)]

2 + [∆0pg(x)]

2} cos2(2θ)]3/2


0.0 0.1 0.2 0.30.0

0.1

0.2

0.3

0.4

0.5

0.6

0 (x)/t

0

x

0sc(x)/t0

0pg(x)/t0

(b)

(a)

Luttinger FS Gaped arc

Y(0, ) M( , )

X( ,0)(0,0)

c

c

k y/a

kx/a

Figure 3. (Colour online) (a) Arc model for Luttinger pockets [(gray) shaded region].

The two sides of the shaded region are Fermi surfaces and the (gray) line beyond is a

contour of closest approach which is gaped. The Fermi contour in the arc model is the

(red) solid line with θ ∈ [π/4, θc] and the (blue) dashed-dotted line is the extension,

but with a gap. (b) The superconducting gap amplitude ∆0sc(x) [(red) solid line] and

the pseudogap amplitude ∆0pg(x) [(black) dashed line] in units of t0 as a function of

doping x.


+ C∫ π/4

θc

dθ

π/4cos6(2θ)

∫ ∞

−∞dε

[∆0sc(x)]

2

{ε2 + [∆0sc(x)]

2 cos2(2θ)}3/2. (22)

We can perform the integral over energy using the fact that∫∞−∞dε [∆2/(∆2 + ε2)3/2] is

equal to two independent of ∆. Absorbing the two into the constant C we get

1

λ2c(T = 0)

= C

[∫ π/4

0

dθ

π/4cos4(2θ)−

[∆0pg(x)]

2

[∆0pg(x)]

2 + [∆0sc(x)]

2]

∫ θc

0

dθ

π/4cos4(2θ)

]. (23)

to get the a-b plane result we drop the cos4(2θ) factor and change the constant C which

is not of direct interest here. To within a constant

1

λ2ab(T = 0)

∼[1−

[∆0pg(x)]

2

[∆0pg(x)]

2 + [∆0sc(x)]

2]

4

πθc,

]. (24)

a result first obtained by Carbotte et al. [18] and

1

λ2c(T = 0)

∼{1−

[∆0pg(x)]

2

[∆0pg(x)]

2 + [∆0sc(x)]

2]

[4

πθc +

4

3πsin(4θc) +

1

6πsin(8θc)

].

}(25)

to get a rough estimate of the ratio λ2ab(T = 0)/λ2

c(T = 0) we take θc = π/8 and the

doping x = 0.12. The ratio ∆0pg(x)/∆

0sc(x) follows from Eqs. (5) and (11) and depends

only on x. This gives a ratio λ2ab(T = 0)/λ2

c(T = 0) = 0.26 while the more complete

calculations of Fig. 2 give 0.29. It is clear from our simple analytic formulas (24)

and (25) that the superfluid stiffness in the c-direction is more sensitive to pseudogap

formation than is the a-b plane due to the appearance of the sin(4θc) and sin(8θc)

factors in Eq. (25). An alternative way of presenting the data of Fig. 2 is to show the

critical temperature normalized to its optimum value Tc(x)/Tc(x = 0.2) as a function

of a similarly normalized superfluid stiffness λ2(T = 0, x = 0.2)/λ2(T = 0, x). This is

presented in Fig. 4 where in the inset we show the Fermi surface obtained for x = 0.2

[(black) dashed-dotted curve] and for x = 0.1 [(blue) solid curve] with back of the

Luttinger hole pocket (shaded region) as a (blue) dashed curve. In this case the FS has

been reconstructed from a large surface of Fermi liquid theory to a hole pocket by the

emergence of a pseudogap. In the main frame we show results both as a function of

c-axis [(red) dashed-double dotted with and (red) dashed without pseudogap] and a-b

plane [(black) solid line with and (black) dashed-dotted line without pseudogap]. The

first thing we note is that the relative behaviour of a-b plane compared to the c-direction

is quite different when a pseudogap is accounted for than when it is not. With ∆pg = 0

the curves cross while they do not when the pseudogap is included in which case the

c-axis curve remains above its a-b plane value. This represents a testable result of our

calculations. In the underdoped region, the (black) solid curve for the a-b plane shows a

quasilinear relationship between critical temperature and superfluid stiffness as is seen

in experiments [56, 57]. Here we predict a related trend when the c-axis stiffness is

considered instead.

Returning to Fig. 2, the (blue) solid circle is data by Bonn et al. [58] on the

zero temperature c-axis penetration depth in YBCO for doping corresponding in our

work to x = 0.14. It falls considerably below our (red) dashed-double dotted curve.


T c(x

)/Tc(x

=0.2

)

2(T=0,x=0.2)/ 2(T=0,x)

ab-plane ab-plane, pg = 0 c-axis c-axis, pg = 0

c-axis, x = 0.14, pg = 2 YRZ

x = 0.2 x = 0.1

k y/akx/a(0,0) X( ,0)

M( , )Y( )

Figure 4. (Colour online) The normalized value of the critical temperature at doping

x to its value at optimum doping x = 0.2, i.e. Tc(x)/Tc(x = 0.2) as a function of the

zero temperature superfluid density normalized to its value at optimum doping x = 0.2.

The inset shows the Fermi contour for two values of doping. The dashed-dotted (black)

curve is for optimum doping with no pseudogap in which case one has the large Fermi

surface of Fermi liquid theory. The solid (blue) curve shows the reconstructed Fermi

surface for x = 0.1 with shading emphasizing the Luttinger pocket.

It appears that in experiment this quantity varies with x even more strongly than we

find here. The (blue) open circle results from our calculations when the pseudogap

value used in the YRZ model is simply doubled leaving everything else the same. This

brings theory and experiment into good agreement. In the phase diagram used by YRZ

which we reproduce here in Fig. 3(b) the QCP at which the pseudogap begins to open

is taken to coincide with optimum doping. However, analysis of many experimental

results on different samples with different measurement techniques have lead Hufner et

al. [55] to conclude that the pseudogap may indeed have its QCP at the top of the

superconducting dome in our phase diagram which we show schematically in Fig. 3(b)

and so the pseudogap in the YBCO sample of Bonn et al. [58] may in fact be a lot

larger than used in YRZ as our present calculations indicate.

In Fig. 5 we show the temperature dependence obtained for the inverse square of

the penetration depth normalized to its zero temperature value for the c-axis. We show

only the case of doping x = 0.1 with [(red) dashed-dotted curve] and without [(red)

dashed curve] pseudogap included. The (red) shaded region emphasizes the difference

in temperature dependence obtained. The (black) solid curve is for comparison and is

for optimum doping, no pseudogap, and falls very near the (red) dashed curve. This

shows that changes in band-structure with x due to the appearance of the Gutzwiller

factors gt(x) and gs(x) in the dispersion relations ξk of Eq. (8) do not affect much the


kx/a

k y/a

0.0

0.2

0.4

0.6

0.8

1.0X

MY

4k

0.0 0.2 0.4 0.6 0.8 1.00.0

0.2

0.4

0.6

0.8

1.0

x = 0.2 x = 0.1 x = 0.1,

pg = 0

2 c(0)/

2 c(T)

T/Tc

Figure 5. (Colour online) The temperature dependence T/Tc of the inverse square

of the out of plane c-axis penetration depth normalized to its zero temperature value.

The solid (black) curve is for x = 0.2 (optimum doping, no pseudogap) while the

dashed-dotted (red) curve is for x = 0.1 and includes a pseudogap. The dashed (red)

curve which is close to the optimally doped case is for x = 0.1 but with ∆0pg(x = 0.1)

set to zero. The shading emphasizes the large change brought about by the opening of

the pseudogap for x = 0.1. The inset shows the contours of η4k and the Fermi contour

[(white) heavy solid and dashed curves].

temperature dependence of the superfluid stiffness and a single curve is representative

for all values of doping. But the opening of the pseudogap which leads to profound

changes in the Fermi surface does imply significant modifications of this temperature

dependence. The reconstructed Fermi surface into Luttinger hole pockets is shown in

the insert as (white) solid and dashed lines in Fig. 5 where we also show contour plots

for the c-axis tunneling function η4k.

As we found for the zero temperature case, we can get additional insight into this

temperature dependence by considering the arc model. To get analytic results we also

need to retain only the leading corrections in temperature. This leading temperature

term comes entirely from the derivative in Eq. (21). The first temperature correction

to 1/λ2c(T ) is

∆

(1

λ2c(T )

)= C

∫ π/2

0

dθ

π/2cos6(2θ)

∫ ∞

−∞dε

[∆0sc(x)]

2

E2s (θ, T = 0)

{− 2βeβEs(θ,T=0)

[eβEs(θ,T=0) + 1]2

}, (26)

and the integral is dominated by the few thermal excitations which come almost

exclusively from the nodal direction where the superconducting gap is smaller than

kBT , with kB the Boltzmann constant. In this case we can replace Es(θ, T = 0) by√ε2 + [∆0

sc(x) cos(2θ)]2 and the important region of integration is around ε ∼ 0 and

cos(2θ) ∼ 0, i.e. in the region of the nodal direction on the Luttinger surface, as


illustrated in Fig. 3(a). The back side of the hole pocket on the antiferromagnetic

Brillouin zone is not important as it has nearly zero weight in the YRZ model. Writing

θ = π/4 + γ and working to lowest order in γ, we get

∆

(1

λ2c(T )

)= C

∫ ∞

−∞

dγ

π/2(2γ)6

∫ ∞

−∞dε

[∆0sc(x)]

2

ε2 + [2γ∆0sc(x)]

2

−2βeβ

√ε2+[2γ∆0

sc(x)]2(

1 + eβ√

ε2+[2γ∆0sc(x)]

2

)2

,(27)

where we have extended the integral over angles to the range γ ∈ [−∞,∞] because of

exponential damping in the last entry in Eq. (27). This allows us to use polar coordinates

(ζ, φ) instead of the variables ε and 2γ∆0sc(x) to get

∆

(1

λ2c(T )

)= −2Cc

(kBT

∆0sc(x)

)5 ∫ ∞

0dζ ζ5

eζ

(1 + eζ)2

∫ 2π

0dφ cos6(φ) (28)

for the c-axis superfluid stiffness. A similar expression can be derived for the a-b plane

where the constant in Eq. (28) has been denoted by Cc to distinguish it from Cab which

appears in the a-b case:

∆

(1

λ2ab(T )

)= − 2Cab

(kBT

∆0sc(x)

)∫ ∞

0dζ ζ

eζ

(1 + eζ)2

= − Cab2 ln 2

(kBT

∆0sc(x)

)(29)

which is to be contrasted with

∆

(1

λ2c(T )

)' −42.5Cc

(kBT

∆0sc(x)

)5

ζ(5), (30)

with ζ(5) the Riemann ζ function. Note that Eqs. (29) and (30) do not depend explicitly

on the pseudogap because these laws depend only of what happens on the Luttinger

Fermi surface in the nodal direction where there is no pseudogap as illustrated in Fig. 6.

On the other hand, the value of the zero temperature superfluid stiffness is strongly

affected by the pseudogap particularly so in the case of the c-axis as seen in Eq. (24)

and (25). This feature has been noted in experiment for the in-plane case [18].

The temperature dependence of the c-axis penetration depth can be affected by

various parameters other than the value of the pseudogap which we have noted not to

enter Eq. (30). The ratio of the gap to critical temperature 2∆0sc(x)/[kBTc(x)] ≡ R

however does enter Eq. (30) when we write it in terms of the reduced temperature

t = T/Tc which is the natural variable to use when discussing temperature dependence.

Increasing R decreases the coefficient of the T 5 dependence. Then ∆[1/λ2c(T )] has an

explicit [kBTc(x)/∆0sc(x)]

5 factor or (2/R)5 which needs to be kept in mind. A possible

extension of our tunneling model which should be considered when comparing with data

is that the c-axis matrix element given in Eq. (1) which results in a cos2(2θ) term in the

continuum limit, could also have a constant part. That is t⊥[t′+cos2(2θ)] would replace

t⊥ cos2(2θ). In this case the temperature law for the c-axis superfluid density would

have an additional linear in T part proportional to t′2 plus a T 3 piece proportional to

t′ as well as the dominant T 5 term. Here we are assuming t′ to be small. In Fig. 7


Figure 6. (Colour online) A schematic of the top right quadrant of the CuO2 BZ.

Energy E(k) is shown as a function of momentum, including both the superconducting

and the pseudogap. In the E(k) = 0 plane we show the Luttinger contours [(red)

and (green) solid contours] for the case x = 0.14. Beyond the Luttinger contours of

zero energy there is an extension that represents the contour of nearest approach for

E−k , which is gaped. When superconductivity is included, the Luttinger FS becomes

gaped except at the Dirac point that falls, as shown, on the heavily weighted part of

the Luttinger contour oriented toward the origin in nodal direction. A Dirac cone is

shown as the (blue) shaded region with its tip at the Dirac point. This region of k

space controls the value of the universal limits for dc electrical and electronic thermal

conductivity.

we show results for λ2c(0)/λ

2c(T ) vs t for doping x = 0.14 and compare various cases

of interest. The (black) solid curve is for comparison and was obtained assuming zero

pseudogap. The (red) dashed curve is for ∆0pg(x) = ∆Y RZ(x). It falls below the (black)

solid curve. Of course, this occurs because we are here comparing reduced quantities.

The temperature law of Eq. (30) does not care about ∆0pg(x) but to obtain the curves

in Fig. 7 we needed to divide by the zero temperature value of the superfluid stiffness

and this quantity does indeed depend on the pseudogap, see Eq. (25). This is the origin

of the deviations between the (black) solid curve and the (red) dashed curve in Fig. 7.

The (blue) dotted curve is for ∆0pg(x) = 2∆Y RZ(x) and we see a greater discrepancy

from the zero pseudogap case. The other two curves, (magenta) dashed-double dotted

and (green) dashed-dotted, respectively, include a constant part to the c-axis tunneling

matrix element with t′ = 0.2 and 0.4, respectively. This changes the character of the

temperature dependence at small reduced temperatures t which becomes more linear

like and as we will see next, does not agree with data.

Data for the temperature dependence of an YBCO6.6 sample by Bonn et al. [58]

which correspond to x = 0.14 in the present work is shown as (black) solid squares

in Fig. 8. The (black) dotted curve is the result of numerical calculations based on


0.0 0.2 0.4 0.6 0.8 1.00.0

0.2

0.4

0.6

0.8

1.0

2 c(

0)/

2 c(T)

T/Tc

pg = 0 pg = YRZ

pg = 2 YRZ

pg = 2 YRZ, t' = 0.2 pg = 2 YRZ, t' = 0.4

x = 0.14

Figure 7. (Colour online) The normalized c-axis penetration depth λ2c(0)/λ

2c(T )

versus the reduced temperature T/Tc for doping x = 0.14. The various curves are

colour and type coded as shown in the figure present results for various values of the

pseudogap amplitude ∆0pg = 0, its YRZ value, and twice this value. It also shows the

effect of adding on to the tunneling matrix element a constant piece t′.

0.0 0.2 0.4 0.6 0.8 1.00.0

0.2

0.4

0.6

0.8

1.0

2 c(0)

/2 c(

T)

T/Tc

2 (0)/(kBTc) = 9, pg = YRZ

2 (0)/(kBTc) = 9, pg = 2 YRZ

2 (0)/(kBTc) = 6, pg = YRZ

Experiment

Figure 8. (Colour online) The normalized c-axis penetration depth λ2c(0)/λ

2c(T )

versus reduced temperature T/Tc for doping x = 0.14. Three curves are considered

and compared with the results of Ref. [44]. The dotted (black) is for a gap ratio

2∆0sc(0)/(kBTc) = 9 with ∆0

pg(x) set at its YRZ value while the solid (blue) line is

for a case when the pseudogap amplitude is doubled. The dashed (red) curve is for

2∆0sc(0)/(kBTc) = 6 and a pseudogap amplitude set by Eq. (5).


Eq. (17) assuming a gap to critical temperature ratio R = 9. Theory falls above the

data. It can be brought into better agreement by increasing the pseudogap of YRZ by

a factor of 2 [(blue) solid line]. On the other hand, R = 6 is possibly more realistic and

with the pseudogap given by YRZ it provides a reasonable fit to the data [(red) dashed

line]. But it should be remembered that to fit the measured zero temperature value of

1/λ2c(T = 0) we had to increase ∆0

pg(x) to twice the YRZ value. Doing so would bring

the (red) dashed curve below the data in the region of t ' 0.6. To make it agree better

with the data we would have to correspondingly increase R. As the value of R is not

well known for the underdoped cuprates we did not do this. Instead we conclude from

this comparison that our model predicts a behaviour which is in reasonable qualitative

agreement with experiment considering the uncertainty in the value of R and, indeed,

of the pseudogap itself. The slight overestimate of theory as compared to experiment

in the region t ' 0.2 could be an indication of a small constant c-axis tunneling matrix

element t′ in addition to a dominant t⊥ cos2(2θ) term.

4. Summary and conclusion

We used the model of the pseudogap state by Yang et al. [13] to calculate the c-

axis penetration depth in the underdoped region of the hole doped cuprate phase

diagram. While in part phenomenological, the YRZ model is grounded in the ideas of

the resonating valence bond spin liquid. It has been remarkably successful in providing

a qualitative explanation of several in-plane properties of the superconducting state of

the underdoped cuprates previously considered anomalous.

To treat c-axis properties the YRZ model needs to be supplemented by some model

for the charge transfer between planes. Here we employed a coherent tunneling matrix

element used previously to discuss dc-transport and c-axis sum rule which is know

from experiment to violate the well known Ferrell-Glover-Tinkham expectation. Both,

the doping dependence of the superfluid stiffness and its temperature dependence are

considered. The value of the zero temperature inverse square of the penetration depth,

1/λ2c(T = 0), is greatly suppressed by the emergence of the pseudogap which reconstructs

the large Fermi surface of Fermi liquid theory into small Luttinger hole pockets. The

suppression of 1/λ2c(T = 0) is found to be much larger than the corresponding a-b

plane 1/λ2ab(T = 0). In sharp contrast to this finding, the magnitude and temperature

power law of the low temperature dependence of the superfluid stiffness is completely

unaffected by the pseudogap. The physical reason this happens is that in the nodal

direction, the heavily weighted part of the Luttinger pocket remains Fermi liquid like and

at low temperatures only states in this region of momentum space become significantly

thermally activated.

Comparison of our theoretical results with experimental data on the zero

temperature value of 1/λ2c(T = 0) in underdoped YBCO6.6, an underdoped sample

studied by Bonn et al. [58], lead to the conclusion that a pseudogap value larger than

suggested in the work of YRZ is needed. This finding supports the compilation presented


by Hufner et al. [55] who surveyed many experimental data obtained by different

techniques and suggests that the pseudogap opens at the top of the superconducting

dome rather than at optimal doping assumed in the model phase diagram of YRZ. The

temperature dependence of 1/λ2c(T ) of the YBCO6.6 is reproduced qualitatively in our

calculations. We describe how the change in the ratio of superconducting gap to critical

temperature affects this temperature dependence as well as how changes in the c-axis

tunneling matrix element impact on it.

Acknowledgments

Research supported in part by the Natural Sciences and Engineering Research Council

of Canada (NSERC) and by the Canadian Institute for Advanced Research (CIFAR).

References

[1] T. Ito, K. Takenaka, and S. Uchida. Phys. Rev. Lett., 70:3995–3998, 1993.

[2] K. Takenaka, K. Mizuhashi, H. Takagi, and S. Uchida. Phys. Rev. B, 50:6534(R)–6537, 1994.

[3] W. J. Padilla, Y. S. Lee, M. Dumm, G. Blumberg, S. Ono, Kouji Segawa, Seiki Komiya, Yoichi

Ando, and D. N. Basov. Phys. Rev. B, 72:060511(R), 2005.

[4] Y. S. Lee, Kouji Segawa, Z. Q. Li, W. J. Padilla, M. Dumm, S. V. Dordevic, C. C. Homes, Yoichi

Ando, and D. N. Basov. Phys. Rev. B, 72:054529, 2005.

[5] C. C. Homes, T. Timusk, R. Liang, D. A. Bonn, and W. N. Hardy. Phys. Rev. Lett., 71:1645–1648,

1993.

[6] C. C. Homes, T. Timusk, D. A. Bonn, R. Liang, and W. N. Hardy. Physica C: Superconductivity,

254(3–4):265–280, 1995.

[7] D. N. Basov, T. Timusk, B. Dabrowski, and J. D. Jorgensen. Phys. Rev. B, 50:3511–3514, 1994.

[8] D. N. Basov, H. A. Mook, B. Dabrowski, and T. Timusk. Phys. Rev. B, 52:R13141–R13144, 1995.

[9] T. Timusk and B. Statt. Rep. Prog. Phys., 62:61, 1999.

[10] Daniel E. Sheehy, T. P. Davis, and M. Franz. Phys. Rev. B, 70:054510, 2004.

[11] D. M. Broun, W. A. Huttema, P. J. Turner, S. Ozcan, B. Morgan, Ruixing Liang, W. N. Hardy,

and D. A. Bonn. Phys. Rev. Lett., 99:237003, 2007.

[12] A. Hosseini, D. M. Broun, D. E. Sheehy, T. P. Davis, M. Franz, W. N. Hardy, Ruixing Liang, and

D. A. Bonn. Phys. Rev. Lett., 93:107003, 2004.

[13] Kai-Yu Yang, T. M. Rice, and Fu-Chun Zhang. Phys. Rev. B, 73:174501, 2006.

[14] P. W. Anderson. Science, 235:1196, 1987.

[15] B. Valenzuela and E. Bascones. Phys. Rev. Lett., 98:227002, 2007.

[16] B. Valenzuela and E. Bascones. Phys. Rev. B, 78:174522, 2008.

[17] E. Illes, E. J. Nicol, and J. P. Carbotte. Phys. Rev. B, 79:100505(R), 2009.

[18] J. P. Carbotte, K. A. G. Fisher, J. P. F. LeBlanc, and E. J. Nicol. Phys. Rev. B, 81:014522, 2010.

[19] J. P. F. LeBlanc, E. J. Nicol, and J. P. Carbotte. Phys. Rev. B, 80:060505, 2009.

[20] Kai-Yu Yang, H.-B Yang, P.D. Johnson, T.M. Rice, and Fu-Chun Zhang. EPL, 86:37002, 2009.

[21] J. P. F. LeBlanc, J. P. Carbotte, and E. J. Nicol. Phys. Rev. B, 81:064504, 2010.

[22] A. J. H. Borne, J. P. Carbotte, and E. J. Nicol. Phys. Rev. B, 82:024521, 2010.

[23] Kai-Yu Yang, Kun Huang, Wei-Qiang Chen, T. M. Rice, and Fu-Chun Zhang. Phys. Rev. Lett.,

105:167004, 2010.

[24] A. Pound, J. P. Carbotte, and E. J. Nicol. Europ. J. Phys. B, 81:69, 2011.

[25] H.-B. Yang, J. D. Rameau, Z.-H. Pan, G. D. Gu, P. D. Johnson, H. Claus, D. G. Hinks, and T. E.

Kidd. Phys. Rev. Lett., 107:047003, 2011.


[26] J. P. Carbotte, F. Marsiglio, and B. Mitrovic. Phys. Rev. B, 33:6135–6140, 1986.

[27] F. Marsiglio, R. Akis, and J. P. Carbotte. Phys. Rev. B, 45:9865–9871, 1992.

[28] E. Schachinger, M. G. Greeson, and J. P. Carbotte. Phys. Rev. B, 42:406–411, 1990.

[29] P. Arberg, M. Mansor, and J. P. Carbotte. Solid State Comm., 86(10):671–673, 1993.

[30] B. Mitrovic and J. P. Carbotte. Can. J. Phys., 61:758–783, 1983.

[31] B. Mitrovic and J. P. Carbotte. Can. J. Phys., 61:784–801, 1983.

[32] D. Branch and J. P. Carbotte. Phys. Rev. B, 52:603–614, 1995.

[33] C. O’Donovan and J. P. Carbotte. Phys. Rev. B, 52:16208–16212, 1995.

[34] H. K. Leung, J. P. Carbotte, D. W. Taylor, and C. R. Leavens. Can. J. Phys., 54:1585–1599,

1976.

[35] B. Mitrovic, C. R. Leavens, and J. P. Carbotte. Phys. Rev. B, 21:5048–5054, 1980.

[36] Mohit Randeria, Nandini Trivedi, Adriana Moreo, and Richard T. Scalettar. Phys. Rev. Lett.,

69:2001–2004, 1992.

[37] V.J. Emery and S.A. Kivelson. Nature (Lonodon), 374:434, 1995.

[38] Sudip Chakravarty, R. B. Laughlin, Dirk K. Morr, and Chetan Nayak. Phys. Rev. B, 63:094503,

2001.

[39] Jian-Xin Zhu, Wonkee Kim, C. S. Ting, and J. P. Carbotte. Phys. Rev. Lett., 87:197001, 2001.

[40] J. P. Carbotte and E. Schachinger. Phys. Rev. B, 86:224512, 2012.

[41] D. N. Basov, S. I. Woods, A. S. Katz, E. J. Singley, R. C. Dynes, M. Xu, D. G. Hinks, C. C.

Homes, and M. Strongin. Science, 283:49–52, 1999.

[42] Richard A. Ferrell and Rolfe E. Glover. Phys. Rev., 109:1398–1399, 1958.

[43] M. Tinkham and R. A. Ferrell. Phys. Rev. Lett., 2:331–333, 1959.

[44] A. Levchenko, T. Micklitz, M. R. Norman, and I. Paul. Phys. Rev. B, 82:060502(R), 2010.

[45] S. Chakravarty, A. Subo, P. W. Anderson, and S. Strong. Science, 261:337, 1993.

[46] O. K. Anderson, A. I. Liechtenstein, O. Jepen, and F. Paulsen. J. Phys. Chem. Solids, 56:1573,

1995.

[47] T. Xiang and J. M. Wheatley. Phys. Rev. Lett., 77:4632–4635, 1996.

[48] T. Xiang, C. Panagopoulos, and J. R. Cooper. Int. J. Mod. Phys. B, 12:1007, 1998.

[49] K. G. Sandeman and A. J. Schofield. Phys. Rev. B, 63:094510, 2001.

[50] P. E. C. Ashby and J. P. Carbotte. arXiv 1211.0922, 2011.

[51] L.B. Ioffe and A.J. Millis. Science, 285:1241–1244, 1999.

[52] Wonkee Kim and J. P. Carbotte. Phys. Rev. B, 61:R11886–R11889, 2000.

[53] Wonkee Kim and J. P. Carbotte. Phys. Rev. B, 62:8661–8664, 2000.

[54] R. J. Radtke and K. Levin. Physica C, 250:282, 1995.

[55] S. Hufner, M.A. Hossain, A. Damascelli, and G. A. Sawatzky. Rep. Prog. Phys., 71:062501, 2008.

[56] C. C. Homes, S. V. Dordevic, M. Strongin, D. A. Bonn, Ruixing Liang, W. N. Hardy, Seiki Komiya,

Yoichi Ando, G. Yu, N. Kaneko, X. Zhao, M. Greven, D. N. Basov, and T. Timusk. Nature

(London), 430:539–541, 2004.

[57] Y. J. Uemura, G. M. Luke, B. J. Sternlieb, J. H. Brewer, J. F. Carolan, W. N. Hardy, R. Kadono,

J. R. Kempton, R. F. Kiefl, S. R. Kreitzman, P. Mulhern, T. M. Riseman, D. Ll. Williams,

B. X. Yang, S. Uchida, H. Takagi, J. Gopalakrishnan, A. W. Sleight, M. A. Subramanian, C. L.

Chien, M. Z. Cieplak, Gang Xiao, V. Y. Lee, B. W. Statt, C. E. Stronach, W. J. Kossler, and

X. H. Yu. Phys. Rev. Lett., 62:2317–2320, 1989.

[58] D. A. Bonn, S. Kamal, A. Bonakdarpour, Ruixing Liang, W. N. Hardy, C. C. Homes, D. N. Basov,

and T. Timusk. Czech. J. Phys., 46:3195–3202, 1996.

interlayer penetration depth in the pseudogap phase of cuprate superconductors

Documents