Poole Ch.P., Jr. Handbook of Superconductivity

536 Chapter 11: Electrical Properties

Hall Effect

The Hall effect employs crossed electric and magnetic fields to provide informa-

tion on the sign, concentration, and mobility of charge carriers. The experimental

arrangement illustrated in Fig. 11.1 shows a magnetic field B 0 applied in the z

direction perpendicular to a slab, and a battery that establishes an electric field Ey

that causes the current I --

JA

of density J -

nev d

to flow along y. The Lorentz

force qv x B 0 on each moving charge q causes a buildup of charge on the sides of

the plate, which produces an electric field

E x

and establishes a Hall voltage

A V x = Exd

perpendicular to the directions of I and B 0. In the superconducting

state a Hall voltage can arise from the transverse electric field

E x

induced by flux

motion. At equilibrium after the buildup of the charge the electric force

qE x

balances the magnetic force qv x B0, and the Hall coefficient RIj, defined as the

ratio

R H = Ex/JB z = 4-1/ne,

(1)

is positive when the majority carriers are holes, as with most cuprates, and

negative for electrons.

Fig. 11.1.

Z Bz

,

-i'---

1

~_~"

Experimental arrangement for Hall effect measurement showing an electrical current I flowing

through a plate of width d and thickness a in the presence of a uniform, transverse magnetic

field B z. The voltage drop V 2 - V 1 along the plate, the drop A Vx across the plate, and the electric

field E x across the plate are indicated. The figure is drawn for the case of negative charge

carriers, that is, electrons. [From Poole

et al.

(1995), p. 18.]

C. Tunneling 537

We can define the dimensionless Hall number, the Hall effect resistivity p~,

the Hall mobility PH (charge carrier drift velocity per unit electric field), and the

Hall angle O H as follows:

Hall number --

Vo/Rl_ie

(2)

pxy =

Ex/J

(3)

la n = [vl/Ey = RI_I/p

(4)

tan | =

Ex/Ey,

(5)

where V 0 is the volume per formula unit. Thus, the Hall effect distinguishes

electrons from holes, and when all charge carriers have the same sign Eq. (1)

provides the charge density n. When both positive and negative charge carriers are

present, their Hall effects partially (or totally) cancel.

C

Tunneling

Tunneling or barrier penetration is a process whereby an electron confined to a

region by an energy barrier can penetrate the barrier through a quantum

mechanical process and emerge on the other side. Tunneling is carried out

through an insulating layer, I, between two normal materials (N-I-N) such as

semiconductors, between a normal metal and a superconductor (N-I-S), and

between two superconductors (S-I-S). Tunneling through the barrier proceeds to

energy states that are empty so the Pauli exclusion principle is not violated, and

the total energy of the system is conserved in the process. Therefore, single-

electron tunneling occurs between levels with the same energy, and in two-

electron tunneling, involving for example the breakup of a Cooper pair, one

electron gains as much energy as the other loses. A positive bias +V on the

material lowers the Fermi energy level by

e V,

and electrons tunnel toward the

positive bias, with the tunneling current I flowing in the opposite direction.

Figures 11.2 and 11.3 depict tunneling at absolute zero when a normal

metal has its conduction band full below the Fermi level E F and empty above it,

and a superconductor has its energy states full below its energy gap Eg = 2A and

empty above. Figure 11.2 shows the N-I-S case in which electrons tunnel from

the superconductor to the normal metal for a negative bias V < -A/e, and from

the normal metal to the empty superconductor levels above the gap for a positive

bias V > A/e. At absolute zero no tunneling occurs for the intermediate range of

bias voltages -Ae < V < A/e, as indicated at the bottom of the figure. Figure

538

Chapter

11:

Electrical Properties

Fig.

11.2.

-V

1~"~ SC

N

(a)

V<- A/e

EF

N

(b)

o

2&

SC

o

SC

(c)

V=O

V>&/e

'1 ' "f

9 ~e i"

.....

J " LVe V

ps /

(g)

Normal metal to superconductor (N-I-S) tunneling at 0 K showing (a) super electron tunneling

(SC =~ N) for V <

-V/e,

(b) the absence of tunneling when the Fermi level is in the gap

(-A/e < V < A/e), and (c) normal electron tunneling (N =~ SC) for V > A/e. Arrows

indicate electron tunneling directions, which are opposite to current flow directions. The

current-voltage characteristic is given in (d). [From Poole

et al.

(1995), p. 407.]

11.3 shows the S-I-S case in which tunneling occurs in one direction for

V <-2A/e, in the reverse direction for V > 2A/e, and at absolute zero it

does not occur at all for the range -2A/e < V < 2A/e. At finite temperatures a

few electrons of a normal metal are found above

E v,

and some electrons of

superconductor are excited to levels above the gap, with the result that a weak

tunneling current flows for the range of biases where it is forbidden at 0 K. If the

gap has a different value for the each material in the S-I-S case, then the weak

tunneling current above absolute zero will exhibit a peak at the bias IA 2 - A 1 I/e,

and a sharp rise at (A 2 + A1)/e, as indicated in Fig. 11.4.

The thickness of the insulating layer is a critical factor in the operation of

the junction. If a superconducting rod is cut at some point, and then joined

through an intervening insulating section, there are three cases to consider: (1)

C. Tunneling

539

Fig. 11.3.

-v

SC

0

U

SC

(a) (b)

V <- 261o V = 0

o

2A

SC

0

[_J .v

(c) sc

V > 2Ale

'l " "f

.2A/, /

S "

2AI e

V

t t

(g)

Superconductor to superconductor (S-I-S) tunneling at 0 K showing (a) super electron

tunneling (SC =~ N) for V < --2A/e, (b) the absence of tunneling for bias voltages in the

range (-2A/e < V < 2A/e), and (c) reverse direction superelectron tunneling for V > 2A/e.

Arrows show tunneling directions. The current-voltage characteristic is given in (d). [From

Poole

et al.

(1995), p. 408.]

The insulating region can be so thick that the two separated superconducting

sections lose contact, and no tunneling occurs. (2) The insulating section, called a

weak link, can be intermediate in thickness so that the superconductors are

weakly coupled and electrons can tunnel. (3) There can be close to direct

touching, or perhaps with an intervening monolayer of foreign atoms, so that

strong comact is maintained across the interface. When two superconductors that

differ in T C are in strong contact, electron pairs leak back and forth and can cause

the two materials to exhibit one intermediate To, a phenomenon called the

proximity effect.

t~

Fig. 11.4.

(a)

/

J

T=O

/

t

/

1

/

t

J

/

t

/

f

(b)

I

T>O

/

t

f

I

,r

I

,r

I

/,/,/

I

J

/

|

.....

= .......

.

,

_

~,-~,

~, ./~ ~,-~___~, 5, .~,

9 9 V e 9 V

Tunneling current vs bias voltage for S-I-S case involving two superconductors with the energy gaps A 2 > A 1 . No tunneling occurs for T -- 0 until the bias

V = (A2 + A1)/e is reached, and weak tunneling appears at the bias V = (A 2 - A1)/e for T > 0, with the onset of strong tunneling at V = (A 1 + A2)/e. [From

Poole

et al.

(1995), p. 416.]

D. ]osephson Effect

541

D

Josephson Effect

The dc Josephson effect involves Cooper pair tunneling from one superconductor

to another across an insulating barrier at zero bias (V = 0) (see Likharev, 1986;

Van Duzer and Turner, 1981). The wavefunctions for the two sides of the junction

can be written in the form

W1 -- (nsl)

1/2ei01

(6)

~2 -- (ns2) 1/2ei02

(7)

4~ = 02 -

01, (8)

where nsl and ns2 are the densities of super electrons in the two superconductors,

and 4) is the phase difference across the barrier. The Josephson expressions

d

2e

dt 05 - ~ V (9)

J = Jc sin ~b (10)

relate ~b to the current density J through the junction and to the voltage V across

it. The super electron densities determine the magnitude of the critical current

density Jc,

Jc cx (nsl ns2) 1/2 (11)

Thus, the presence of a phase difference (8) across the junction causes dc current

to flow spontaneously at zero voltage.

When a constant voltage is applied across the junction, Eq. (9) can be

integrated directly to give

~b(t) -- 4)o + 2rcvjt, (12)

where vj is known as the Josephson frequency

vj -- 2eV/h = V/ap o.

(13)

aP o = h/2e

is the quantum of flux, and the ratio

vj/V

has the value

vj/V = 2e/h

= 483.6 MHz//~V

(14)

This ac Josephson effect has the following expression for the oscillations of the

critical current density:

J -- Jc sin(2rcvjt + ~b0). (15)

542 Chapter 11" Electrical Properties

These oscillations occur for a constant applied voltage. The current density

amplitude Jc reaches a maximum value when the applied voltage is the gap

voltage, V = 2A/e. There is also an inverse ac Josephson effect whereby the dc

voltage is induced across the junction when an ac current is caused to flow

through it, or when an electromagnetic field, as from microwaves, is incident on

it.

A practical Josephson junction can be represented as driven by a current

source I c sin ~b from Eq. (10) in parallel with a conductance G to account for

quasiparticle tunneling and a capacitor C to take into account displacement

current, as illustrated in Fig. 11.5. The current flow in this equivalent circuit is

described by the differential equation

I(t) - I c sin ~b +

GV + C~

dV

dt "

(16)

With the aid of the Josephson relation (9) this becomes

I - hCdZ~P hGddp

2--e dt----T+-~e---~ +

Ic sin ~b" (17)

We can define a critical voltage Vc -

Ic/G,

an associated Josephson frequency co c

from Eq. (14),

co c -- (2e/h)

V c , (18)

and a dimensionless admittance ratio tic,

fie - ~

(19)

Fig. 11.5

( )coO ,

/.Source

I

J

'7

I csln

Parallel circuit representing a Josephson junction driven by the dc current source

I(t)

on the

left. The right-hand side shows the junction current source I c sin ~b in parallel with a capacitor

C and a conducatance G. [From Poole

et al.

(1995), p. 430.]

D. Josephson Effect

543

Fig. 11.6.

(a)

21 e

I::

lc

..... ! ' | -

,~

o

l , o~

~

o

9 , I i

.....

Ir 21r

(b).,

21 e

-

/

~ |c -

I I

,,

Ir 21el G

Vollage V

(c)

" i ' I'l /

/-

21, - I PC-" ~176

i

0 !r

I,,,,

" .... l , . . J, I

I = I G 21 r

Voltage V

Vollage V

Normalized current-voltage characteristics, I vs V, where V - (V) is the average voltage, for

the Josephson junction circuit of Fig. 11.5 with (a) negligible capacitance, tic << 1, (b)

appreciable capacitance, /~c = 4, and (c) dominating capacitance, ]~c =~ oo, where the

admittance ratio is defined by/~c =

~

[From Poole

et aL

(1995), p. 432.]

Solutions to this second-order differential equation (17) are plotted in Fig. 11.6

for small (//c << 1), medium (tic = 4), and large (/~c >> 1) values of the admit-

tance ratio. This figure is a plot of current vs the average voltage V = (V), and

Fig. 11.7 shows the oscillations of the voltage at points A and B of Fig. 11.6a.

Note the presence of hysteresis in the plots of Fig. 11.6.

544

Chapter

11:

Electrical Properties

Fig. 11.7.

o

it V(T)/Ri,,

I I i f

SO. O0 100. O0

o

4~

o

%.00 '

I 1 " I I ' I

ISO,OO ~o0.00 ~ 2sO.OO ~ 3oo.00 ~ 3SO.OO ~ ~oo.oo

T

Voltage oscillations across the Josephson junction of Fig.

11.5

for the negligible capacitance

case

fie << 1,

and small and large dc bias voltages as marked at points A and B, respectively, of

Fig.

11.6a.

[From Barone and Paterno

(1982), p. 128.]

Magnetic Field and Size Effects

When a magnetic field is applied to a Josephson junction, the behavior of the

junction will depend on its size relative to the Josephson penetration depth 2j

defined by

2j --

(Oo/2rC#oJcdeff ) 1/2,

(20)

where def f is the effective thickness of the junction. This penetration depth is

approximately equal to the length L of a Josephson junction when the stored

magnetic field energy UB equals the energy Uj associated with the current flow.

A long Josephson junction, with L >> 2j, has its phase q5 determined by the

sine Gordon equation, which has solitary wave (soliton) solutions. A Short

Josephson junction, with L << 2j, can exhibit the current oscillations shown in

Fig. 11.8 arising from the Josephson junction (Fraunhofer) diffraction expression

I - I c

sin q5 o sin(nO/O~

rc~/~o , (21)

F. Superconducting QUantum Interference Device [SQUID] 545

Fig. U.8.

Imax/l C

_ ;_4

0 1 :2 3 4 5 ~0

Josephson Fraunhofer diffraction pattern showing the dependence of the maximum zero-

voltage current

Imax/I c

on the normalized flux ~/A% through the junction. [From Van Duzer

and Turner (1981), p. 155.]

where I c =

AJ c

is the critical current and I has a maximum value for the phase

difference q50 =

n/2.

The figure shows how the tunneling current varies with

increasing magnetic flux 9 through the junction. Extremely small Josephson

junctions called nanojunctions exhibit new phenomena. The change in voltage

A V = e/C

arising from single-electron tunneling can become comparable with a

typical junction voltage and produce a blockage of current flow (Coulomb

blockade). Fluctuations can appear that produce a so-called Coulomb staircase

on an I vs V characteristic plot. A dc current biased ultrasmall Josephson junction

can exhibit correlated tunneling of Cooper pairs leading to what are called Bloch

oscillations at the frequency v B -

I/2e.

Superconducting QUantum Interference Device (SQUID)

A Superconducting QUantum Interference Device or SQUID measures flux

changes in a loop containing weak links to determine the strength of an applied

magnetic field. In one arrangement, called a dc SQUID, the current change

through a pair of two weak links is detected as a voltage and amplified. Another

type, called an rf SQUID, consists of a loop with one weak link coupled to an LC

tuned circuit driven by an rf current source. A change of flux in the loop changes

the loading of the tuned circuit, and this is detected by measuring the change in rf

voltage across the circuit. Since a SQUID easily detects a change in one quantum

of flux in an area with dimensions in the centimeter range, it is said to measure a

macroscopic quantum phenomenon (vide Poole

et al.

(1995), p. 451).

Poole Ch.P., Jr. Handbook of Superconductivity

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