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15 Tunneling Spectroscopy of Conventional and Unconventional

Superconductors

J. Zasadzinski Illinois Institute of Technology, Physics Dept. Chicago, USA

15.1 Introduction .............................................................................833

15.2 Basic Tunneling Phenomenology ...........................................................835

15.2.1ElasticTunnelingCurrent............................................................835

15.2.2SuperconductingGapStructure.......................................................836

15.2.3BTKModel.........................................................................837

15.2.4 Novel d-waveEffects.................................................................838

15.2.5Strong-CouplingEffects:PhononStructure............................................839

15.2.6InelasticTunneling..................................................................840

15.3 Tunneling and Strong-Coupling Effects: Microscopic Picture ..................................841

15.4 Tunneling Spectroscopyof Conventional Superconductors ....................................844

15.4.1McMillan–RowellAnalysis...........................................................845

15.4.2ProximityEffectTunnelinginConventionalSuperconductors............................845

15.5 Tunneling in High Temperature Superconductors ............................................847

15.5.1 Nd

2−x

CuO

......................................................................848

15.5.2 Bi

CaCu

......................................................................850

15.5.3 Bi

CaCu

STMMapping........................................................856

15.5.4 Intrinsic c-Axis Junctions on Bi2212 . . . ................................................857

15.5.5 YBa

.........................................................................860

15.6 Heavy Fermion Superconductors ...........................................................861

15.7 Organic Superconductors ..................................................................862

15.8 Other Materials ...........................................................................863

15.9 Conclusions ..............................................................................864

References...............................................................................865

15.1 Introduction

Tunneling spectroscopy of conventional supercon-

ductors [1] such as Pb [2] leads to a completedescrip-

tionof thesuperconducting state.Fromthe tunneling

conductance,

vs. V (appropriately normalized),

one can obtain the quasiparticle density of states,

N(E). This gives an implicit measure of the com-

plex, superconducting gap parameter, (E). Using

Migdal–Eliashberg theory [3] the gap parameter can

be “inverted” by the iterative McMillan–Rowell pro-

cedure (MR) [2] to obtain the microscopic interac-

tions responsible for superconductivity, namely, the

electron–phonon spectral function, ˛

F(!), and the

renormalized coulomb pseudopotential, 

∗

.These

quantities can then be used to determine the tran-

sition temperature, T

, as well as the electron mass

enhancement, 1 + , which enters normal state ther-

modynamic properties such as the specific heat [4].

In some cases the ˛

F(!) obtained from tunneling

has been used to provide a very good fit of the tem-

perature dependent electrical resistivity far above

834 J.Zasadzinski

[5]. The fact that both superconducting and nor-

mal state quantities can be determined demonstrates

both the accuracy of strong–coupling superconduc-

tivity theory and the power of the tunneling method.

The various manifestations of the electron–phonon

interaction canbe foundin thereview articles ofCar-

botte [3] and Allen [4].

Using a straightforward extension of the basic

tunneling measurement to normal metal supercon-

ductor (N/S) bi-layers,the quantities ˛

F(!)and

∗

have been obtained for some transition metal ele-

ments(Nb,V,Ta),alloys(NbZr)andA-15compounds

such as Nb

Sn as well as nonsuperconducting mate-

rials such as Mg [1]. Furthermore, such proximity

effect junctions of the form Nb/Al/Al-oxide/Nb are

useful for a variety of devices from sensitive photon

detectors to fast switches and are the mainstay of a

successful superconducting electronics industry.

Considering the potential of tunneling spec-

troscopy for basic research and applications, it is

not surprising that many groups around the world

have attempted to fabricate junctions on exotic su-

perconductors including the high-T

cuprates and

bismuthates, organics, heavy fermion materials and

the recently discovered MgB

. Successful inversion

of tunneling spectra using a variation of the tra-

ditional MR procedure has been accomplished for

the bismuthate,Ba

1−x

BiO

,andthe relatively lower

,electron-doped cuprate Nd

1.85

0.15

CuO

[6].The

resulting ˛

F(!) spectra strongly suggest phonon

mediated pairing, however there remain unresolved

questions such as whether this interaction is com-

patible with d-wave symmetry [7]. For all of the

other classes of exotic materials the current state-

of-the-art tunneling spectroscopy does not include

a quantitative determination of the pairing mech-

anism. This is mostly a consequence of their re-

markable complexity. Tunneling in these materials

has, nevertheless, provided a wealth of important in-

formation on the superconducting gap magnitude,

pseudogap phenomena, pairing symmetry as well as

atomic scale imaging of crystal structure and defects.

For example, a highly unusual doping dependence

of the measured gap in the cuprate Bi

CaC

8+ı

(Bi2212)has been revealed by tunnelingand appears

to be related to pseudogap phenomena observed

Fig. 15.1. Schematic representation of junction types

above T

. Higher energy spectral features,analogous

to phonon structures, are reproducibly observed in

some cuprates as well as heavy fermion materials

which have been linked to spin excitations. The large

ratios of gap magnitude to T

in organic compounds

are similar to those found on underdoped cuprates

suggesting the possibility ofcommon origins of pair-

ing in hole doped cuprate,heavy fermion and organic

superconductors.

It is impossible to adequately reviewall of the tun-

neling literature, even on cuprate superconductors

alone. The inclusion of conventional superconduc-

tors as well as other interesting materials makes the

task even more difficult. To narrow the scope,the fo-

cus of this review will be on those quasiparticle tun-

neling experiments which directly address the issue

of unconventionality,i.e.,gap size vs.T

,gap symme-

try, pseudogap and pairing mechanism. An attempt

has been made to select those results which are rep-

resentative and exhibit features reproduced by other

tunneling groups or are consistent with other (non-

tunneling) experiments. The brief review of conven-

tional superconductors is structured to serve as a

baseline for understanding the more complicated

tunneling spectra found in unconventional super-

conductors. Consequently a great deal of important

work on other aspects of the field (e.g. high-T

junc-

tion devices) has been left out.Furthermore,the large

volume of publicationsnecessitates the useof mono-

graphs and review articles as references whenever

possible.

A schematic of the junction types considered here

is shown in Fig.15.1.Figures15.1(a)–(c) arethe stan-

15 Tunneling Spectroscopy 835

dard junction geometries for all superconductors,

however almost all of the ˛

F(!) spectra have been

obtained using planar junctions.Figures15.1(d) and

(e)arejunction typesthat are associatedwithcuprate

superconductors.

15.2 Basic Tunneling Phenomenology

In this section an overview is given of the general

phenomena found for tunneling into s-wave and d-

wave superconductors. In addition, the theoretical

frameworks for understanding the various spectra

are discussed.

15.2.1 Elastic Tunneling Current

Analysis of experimental tunneling data often begins

with the standard expression for the junction current

between two metals separated by a thin barrier (typ-

ically 1–2 nm) either due to a vacuum as in STM or a

surface insulating layer as in planar or point contact

junctions [1]. Assuming elastic tunneling,one has

I =

2e



+∞



−∞

|T|

(E + eV)N

(E)[f (E)

− f (E + eV)]dE .

Here, N(E) is the electronic density of states (DOS)

and|T|

is thetunneling matrixelement.All informa-

tion has been lost about the various approximations

and assumptions that take the three-dimensional

sum over electron momenta, k, p in each electrode

and collapse it into a single integral over the quasi-

particle energy, E.Thuswhatisnotcleariswhether

N(E) represents the full quasiparticle DOS,including

band structure effects, or is just that part that is due

to correlationeffectswhich arise from e.g.,supercon-

ductivity, localization, chargedensity waves etc. This

is discussed in more detail later. For now we make

a standard assumption that the energy and voltage

dependence of |T|

is weak and can be taken out of

the integral to obtain

I =

2e



|T|

+∞



−∞

(E + eV)N

(E)[f (E) (15.1)

− f (E + eV)]dE .

This equation is used for both superconductor-

insulator-normalmetal (SIN) junctionsas well as SIS

junctions. Considering for the moment that N

is a

normal metal with a constant DOS near the Fermi

energy, one obtains,

I ∝ N

(0)

+∞



−∞

(E)[f (E)−f (E + eV )]dE .

Assuming T = 0 and taking the derivative, dI/dV,

in both the superconducting and normal states one

obtains,

 (V)=

(0)N

(eV)

(0)N

(0)

(eV)

(0)

For an s-wave, BCS superconductor,the DOS can be

written as a simple product

(E)=N

(0)

BCS

(E)=N

(0)

|E|

√

− 

where N

(0) is the DOS of the superconductor in

the normal state, assumed to be a constant given by

its value at the Fermi energy, E = 0. Under these

assumptions the normalized conductance,  (V)is

therefore a direct measure of the superconducting

part of the DOS



BCS

|E|

√

− 

. (15.2)

Note that the symbol  has been used to denote both

the normalized conductance  (V) and the supercon-

ducting DOS, 

BCS

, since they are equivalent at zero

temperature. For non-zero T it is

 (V)=



+∞

−∞



BCS

[f (E −eV)−f (E)]dE . (15.3)

The superconducting DOS is convoluted with the

derivative of the fermi function, a gaussian shaped

curve with a half-width of approximately 3.5 kT.

Thus the measured normalized conductance gives

a thermally smeared superconducting DOS.

836 J.Zasadzinski

15.2.2 Superconducting Gap Structure

The justification for all assumptions about the tun-

nel current lies in the degree to which the measured

normalized conductance agrees with equations of

the previous section. For example, the SIN gap re-

gion conductance for the bismuthate, Ba

1−x

BiO

(BKBO) is shown in Fig.15.2 where it is clear that an

ideal fit is obtained using Eqs. (15.2) and (15.3) [6].

Thus, BKBO with a T

near 28 K appears to be an

isotropic, s-wave superconductor. The strong cou-

pling ratio,

2

, is in the range 3.7–3.8, which is

slightly more than the BCS weak coupling limit, and

indicates moderately strong coupling.

While the issue of pairing symmetry in high-T

cuprates is not entirely settled, there is certainly con-

siderable evidence to suggest that the order param-

Fig. 15.2. Normalized tunneling conductance and BCS fit

for Ba

1−x

BiO

from Huang et al. [6]

eter has d

−y

(d-wave) symmetry. Thus it is useful

to consider the d-wave analogue of the BCS density

of states. For conventional superconductors, the en-

ergy gap, , is a constant in k space (isotropic s-

wave pairing symmetry).However,for superconduc-

tors with d-wave pairing symmetry, the energy gap

is anisotropic on the Fermi surface. In a two dimen-

sional superconductor with a BCS-type interaction,a

simple d-wave DOS suggested by Won and Maki [8]

is given by

 (E, k)=Re

E − i 



(E − i  )

− (k)

, (15.4)

where (k)=cos(2)isthek dependent energy

gap and  is the polar angle in k-space. Note that

when d-wave symmetry is assumed, the parameter

 refers to the maximum value of the gap. The DOS,

 (E), is found by an integral over the polar angle .

Here, E is replaced by E − i ,where is a smear-

ing parameter to account for quasiparticle lifetime.

This parameter was originally introduced into the

BCS DOS by Kaplan et al. [9] as a convenient way

to approximately account for scattering and recom-

bination effects on the quasiparticle lifetime within

conventional Eliashberg theory. Experimental verifi-

cation of this modified DOS was made by Dynes et

al.[10],where was shownto besignificant only near

. However, this  term has taken on an expanded

role, serving as a phenomenological smearing term

for the superconducting DOS, irrespective of origin.

In Fig. 15.3 is shown the density of states DOS for

a d-wave superconductorwith a maximum gap mag-

Fig. 15.3. Characteristic fea-

tures of (a)BCSdensityof

states and (b)SISspectrumfor

d-wave superconductors

15 Tunneling Spectroscopy 837

Fig. 15.4. Normalized conductance (dots)andd-wave fit

(solid line) for Tl2201 from Ozyuzer et al. [12]

nitude of 20 meV and the expected SIS spectrum for

two identical superconductors. The geometry of the

SIS junction is very important as will be discussed

later. Here it is assumed the barrier normal is along

the c-axis, similar to the intrinsic junction shown in

Fig. 15.1(e). The characteristic feature of the DOS is

a cusp at the Fermi energy (zero bias) and a rela-

tively small ratio of the peak height to background

(PHB) which is about 2. The SIS curve is character-

ized by a similar PHB ratio and a sharp drop of the

conductance at the voltage, V =

.

These characteristics are certainly distinctive and

shouldbe observedin tunnel junctionson d-wavesu-

perconductors. However, cuprates superconductors

generally exhibit much more complicated spectra

as will be shown. The one high-T

cuprate system

that most closely and reproducibly displays the d-

wave characteristics in the tunneling conductance is

CuO

[11], an example of which is shown in

Fig. 15.4 [12].

15.2.3 BTK Model

Many junctionson novel superconductors utilize the

point contact geometry (see Fig. 15.1(c)) where the

force from a sharp, normal metal tip on a supercon-

ductor can be varied. The contact between tip and

Fig. 15.5. Calculated conductances for a point contact be-

tween a superconducting and normal metal electrode in

the BTK model for various Z parameters. Dark solid lines

areforthespecialcasesofZ =0, 1,  1.Dashed lines show

smooth evolution of the curves from Z =0to3

sample is a nanoconstriction and can be of the tun-

neling type if the surface oxides of the electrodes

provide a suitable barrier, but can also be metallic.

The latter occurs if the surface barrier is perforated

or if the electrodes are noble metals without a native

insulating surface. The model of Blonder–Tinkham–

Klapwijk (BTK) [13] provides a smooth interpo-

lation between the SIN tunneling and SN metallic

regimes. A delta function tunnel barrier is assumed

and the strength is given by a dimensionless parame-

ter, Z. The transmission probability is assumed to be

the same for all electrons impinging on the interface

between the electrodes.

The calculated conductance spectra for various Z

parameters is shown in Fig. 15.5 from the work of

Schmidt [14]. For the pure metallic contact (Z =0)

there is an increase in conductance near zero bias

byafactorof2overthehighbiasvalue.Thisisa

consequence of Andreev reﬂection. An electron in

the normal electrode incident on the superconduc-

tor cannot propagate and gets reﬂected back as a

hole, producing a dissipationless pair current in the

superconductor. In steady state, electrons and holes

are movingin opposite directions,addingto the con-

ductance of the normal electrode. Physically, it is as