# Influence of temperature on resistivity/conductivity; superconductivity

Urban and Civil

Nov 15, 2013 (5 years and 1 month ago)

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1

Electrical Engineering 2

Microelectronics 2

(Room G08, SMC; email
-

pjse)

Lecture 6

Dr. Peter Ewen

2

MOBILITY

The
MOBILITY
,

,

Electron movement is hindered by collisions, i.e. by
SCATTERING
. Electrons are scattered by:

(A) The thermal vibrations of the atoms (i.e.
PHONONS
)

this is termed
LATTICE SCATTERING
.

3

Im

(B)
DEFECTS

(e.g. impurities)

DEFECT SCATTERING

or
IMPURITY SCATTERING
.

V

I

Fig. 30: Defect examples

D

V: a
vacancy

a place where
an atom is missing.

I: an
interstitial

an extra
atom that is residing in the
space between regular

atoms.

D: a
dislocation

here a
region of the crystal has
slipped relative to a
neighbouring region.

Im: an
impurity

e.g. a donor
or acceptor atom in the case
of an extrinsic semiconductor

4

d
/

E

def

Note:

A mobility can also be defined for holes.

Units for mobility are:
m
2
V
-
1
s
-
1

Mobility depends on temperature. For lattice
scattering, mobility decreases as temperature
increases because there are more thermal
vibrations around at higher temperatures.

For lattice scattering:

l

= K
l
T
-
3/2

l

indicates lattice scattering

K
l

constant

T

temperature in

K

d

drift velocity
E

electric field

5

+

Fig. 31

Low temperature

electron

moving slowly

+

ionised donor atom

+

High temperature

electron

moving fast

For impurity scattering:

i

= K
i
T
3/2

For a metal:

T
-
1

6

CONDUCTION BY CHARGE DRIFT

Electron motion during
time interval, t, in
zero

field.

E

(
F

㴠=
E
)

Electron motion during
time interval, t, in field
E
.

DRIFT VELOCITY
:

v
d

=
l

/ t

l

7

CONDUCTIVITY

J =

E Ohm’s Law

Metals:

㴠湥

Semiconductors:

㴠湥

e

+ pe

h

(intrinsic or extrinsic)

special cases

Intrinsic Semiconductors:

㴠敮
i
(

e

+

h
)

(n = p = n
i

for intrinsic)

n
-
type:

n
n
e

e

≈ N
D
e

e

(n
n

>> p
n
for n
-
type)

p
-
type:

p
p
e

h

≈ N
A
e

h

(p
p

>> n
p
for n
-
type)

conductivity

E

electric field

J =
I

/ A

current density

8

LECTURE 6

Influence of temperature on

resistivity/conductivity

䵥瑡汳

Intrinsic semiconductors

Extrinsic semiconductors

9

12. Mobility

When 3 V is applied across the faces of a
2 mm thick wafer of pure Si at 300 K the
electrons are found to drift a distance of
10
-
3
m in 5

mobility.

Assuming that lattice scattering
predominates over impurity scattering, what
would the electron mobility be at 100 K?

10

12. Mobility

1
1
2
3
6
3
13
.
0
10
2
3
10
5
10

s
V
m
x
V
t
l
E
v
d

2mm

3V

11

2
/
3
2
/
3
2
/
3
300
13
.
0

T
K
T
K
l
l
l
l

Assuming that lattice scattering
predominates over impurity scattering, what
would the electron mobility be at 100 K?

2
/
3

T
l

2
/
3
T
i

1

T

1
1
2
2
/
3
2
/
3
2
/
3
68
.
0
3
13
.
0
100
300
13
.
0
)
100
(

s
V
m
l

12

13. Ohm's Law in terms of current density

Show that Ohm's Law can also be written as

J =

䔠E

where J is the current density,

13

13. Ohm’s Law in terms of current density

E
J
E
x
V
l
V
J
A
I
l
V
A
I
A
l
V
I
A
l
R
R
V
I

)

;

1

;

(But

1
:
g
Rearrangin
/

Hence
But
A

l

V

I

R

14

14. Conductivity of a metal

Show that the conductivity of a metal is given
by

= ne

where n is the (free) electron concentration
and

15

dt
dQ
I

,
definition
By
14. Conductivity of a metal

the
volume

of electrons passing point P in time t will be v
d
t

A

Let:

A

be the cross sectional area of the wire

n

be the free electron concentration in the wire

v
d

be the electron drift velocity

metal wire

I

Consider a fixed point, P, along the wire and work out the amount
of charge passing this point in time, t:

P

v
d
t

A

the
amount

of charge passing point P in time t will be v
d
t

A

n

e

the
number

of electrons passing point P in time t will be v
d
t

A

n

16

)

(since

)
J

(since

Hence

)

definition
by

(since

E
J
ne
A
I
E
ne
J
E
ne
A
I
E
v
EA
ne
A
nev
I
dt
dQ
I
tAne
v
Q
d
d
d

the
amount
, Q, of charge passing point P in time t will
be
v
d
t

A

n

e

17

Variation of resistance with temperature for a solid

Resistance

Temperature

Resistance

Temperature

Resistance

Temperature

Resistance

Temperature

Resistance

Temperature

Resistance

Temperature

18

For a metal:

㴠湥

=

-
1

= 1 / ne

Fig. 33: Resistivity

vs temperature

for a metal

But for a metal

n is constant

T
-
1

Hence

T

19

Intrinsic Semiconductors

㴠敮
i
(

e

+

h
)

ln(

o
)

ln(

)

-

E
g
/2k

1/T

High T Low T

Fig. 34(a): Conductivity vs

inverse temperature for an

intrinsic semiconductor

e

and

h

T
-
3/2

o
exp (

E
g
/2kT)

ln(

⤠㴠汮=

o
)

E
g
/2kT

n
i

T
3/2
exp (

E
g
/2kT)

20

Intrinsic Semiconductors

㴠ㄯ

†††
㴠ㄯ[

o
exp (

E
g
/2kT)]

o
exp (E
g
/2kT)

or

ln(

⤠㴠=渨

o
) + E
g
/2kT

ln(

)

o
)

-
ve TCR

Slope = E
g
/2k

1/T

High T Low T

Fig. 34(b): Resistivity vs

inverse temperature for an

intrinsic semiconductor

21

Extrinsic Semiconductors

ln(

⤠⼠

m

Fig. 35: Resistivity
vs temperature for
n
-
type Si

Consider n
-
type Si:

= 1/

1/(n
n
e

e
)

Temperature /

C

-
200
-
100 0 100 200 300

10
5

10
4

10
3

10
2

10

1

Room Temperature

Temperature, T /

C

䕬散瑲潮t捯湣c湴牡瑩潮Ɱ m
-
3

N
D

n
i

-
200
-
100 0 100 200 300

Intrinsic Si

ln(

T
3/2

Degenerate case

10
20

10
21

10
22

10
23

10
24

10
25

10
19

Doping

density (m
-
3
)

For low/moderate doping

T
-
3/2

TCR
+ve

TCR

-
ve

TCR zero

22

1.
The table gives the resistance of four different samples at
various temperatures. There is:

one metal,

one intrinsic semiconductor,

one n
-
type semiconductor with N
D
= 10
25

m
-
3
,

one p
-
type semiconductor with N
A

= 10
20

m
-
3
.

Which sample is the p
-
type semiconductor?

Temp. (K)

Resistance (Ω)

Sample A

Sample B

Sample C

Sample D

200

2.0

1
x
10
6

2.8
x
10
5

80

300

3.2

3
x
10
6

321

80

500

5.6

5
x
10
4

1.42

80

23

For a metal:

㴠=e

=

-
1

= 1 / ne

Fig. 33: Resistivity

vs temperature

for a metal

But for a metal

n is constant

T
-
1

Hence

T

Sample A is a metal

because
resistance

T

24

Extrinsic Semiconductors

ln(

⤠⼠

m

䙩朮F㌵㨠剥獩獴3癩y
vs temperature for
n
-
type Si

Consider n
-
type Si:

= 1/

1/(n
n
e

e
)

Temperature /

C

-
200
-
100 0 100 200 300

10
5

10
4

10
3

10
2

10

1

Room Temperature

Temperature, T /

C

䕬散瑲潮t捯湣c湴牡瑩潮Ɱ m
-
3

N
D

n
i

-
200
-
100 0 100 200 300

Intrinsic Si

ln(

T
3/2

Degenerate case

10
20

10
21

10
22

10
23

10
24

10
25

10
19

Doping

density (m
-
3
)

For low/moderate doping

T
-
3/2

Sample

C

Sample

D

25

SUPERCONDUCTIVITY

Resistivity

T敭灥牡瑵牥

0

0

T
c

T
c

critical

temperature

Fig. 36: Loss of resistance
in a superconductor at low
temperatures

Discovered in 1911 by
H. Kamerlingh
-
Onnes

Before 1986 T
c

< 23 K

1986 T
c
~ 35 K

1987
T
c
~ 90 K

20??
T
c
~ 300 K

1994
T
c
~ 153 K

26

1913

H. Kamerlingh
-
Onnes,

discoverer of the effect in Hg.

Nobel

Prize
-
winners

in the field of

Superconductivity

1972

John Bardeen, Leon Cooper and Robert
Schrieffer, theoreticians who developed the
first correct theory of the effect. (This was
Bardeen’s second Nobel Prize

he also got one
in 1956 for co
-
inventing the transistor.)

John Bardeen, William Shockley
and Walter Brattain

inventors
of the transistor, for which they
were awarded the Nobel Prize
for Physics in 1956.

27

1913

H. Kamerlingh
-
Onnes,

discoverer of the effect in Hg.

Nobel

Prize
-
winners

in the field of

Superconductivity

1972

John Bardeen, Leon Cooper and Robert
Schrieffer, theoreticians who developed the
first correct theory of the effect. (This was
Bardeen’s second Nobel Prize

he also got one
in 1956 for co
-
inventing the transistor.)

1987

Alexander Muller and Georg
Bednorz, IBM Zurich, discoverers of
the effect in ceramic materials at 35 K.

20??

“For the discovery of a room
temperature superconductor, this

year’s Nobel Prize in Physics goes to...”

28

Superconductivity occurs in metals, alloys,
semiconductors and ceramics.

Nature

444, 465
-
468 (23 November 2006)

T
c
= 0.35K

29

Electron pairs (“Cooper pairs”) form:

Leon Cooper

30

The Cooper Pairs take a finite time to respond
to any change in electric field so for an A.C.
signal the current will lag the voltage

the
superconductor has an inductance L associated
with it

Below T
c

Cooper pairs exist alongside ordinary
electrons. The superconductor can be
represented by:

㴠〠景爠䐮䌮

≈ 0 for A.C.

Represents the
normal electrons

Represents the

Cooper Pairs

Fig. 37

resistanceless

wire

L (perfect inductor)

R

31

The Meissner Effect

Fig. 38: Expulsion of magnetic flux for T < T
c

Magnetic field

T
c

57.3

Magnetometer

Superconductor

32

APPLICATIONS OF SUPERCONDUCTIVITY

High
-
Field Electromagnetics

Superconducting magnet from

a particle accelerator.

Mag
-
lev train based on

superconducting magnets.

MRI body scanner using

superconducting magnets

(
M
agnetic
R
esonance
I
maging)

33

Device/Chip Interconnects

Faster operation (no RC time delays)

Less heat generation (no Joule heating)

Lossless Power Transmission

connections between chips on pcb

24KV, 2400A superconducting power

on
-
chip interconnects

34

SUMMARY

INFLUENCE OF TEMPERATURE ON

RESISTIVITY
/ CONDUCTIVITY

The temperature dependence of the resistivity (or
conductivity) of a solid is governed by the
temperature dependence of the mobility and carrier
concentration.

METALS

-
1

= 1 / ne

㨠†

T
-
1

Hence

T

35

⤠孯)⁬渨

⥝)

because they can be
used to obtain E
g

ln(

)

-

E
g
/2k

1/T

High T Low T

INTRINSIC SEMICONDUCTORS

ln(

⤠孯爠汮)

⥝楳i灲潰潲瑩潮慬o瑯tㄯ吠†1††††

†††
⡩(攮e
INVERSE TEMPERATURE
)

36

ln(

⤠⼠

m

T敭灥牡瑵牥e⼠

C

-
200
-
100 0 100 200 300

10
5

10
4

10
3

10
2

10

1

Room Temperature

Intrinsic Si

ln(

T
3/2

10
19

Doping

density (m
-
3
)

EXTRINSIC SEMICONDUCTORS

The temperature
dependence of the
resistivity depends
on the
temperature
range

and
doping
concentration.

For light/moderate
doping & temperatures
below the onset of
intrinsic behaviour:

T
3/2

37

ln(

⤠⼠

m

T敭灥牡瑵牥e⼠

C

-
200
-
100 0 100 200 300

10
5

10
4

10
3

10
2

10

1

Room Temperature

Intrinsic Si

ln(

T
3/2

10
19

Doping

density (m
-
3
)

EXTRINSIC SEMICONDUCTORS

The temperature
dependence of the
resistivity depends
on the
temperature
range

and
doping
concentration.

For temperatures
above the onset of
intrinsic behaviour
the resistivity is
essentially that of the
intrinsic material.

38

ln(

⤠⼠

m

T敭灥牡瑵牥e⼠

C

-
200
-
100 0 100 200 300

10
5

10
4

10
3

10
2

10

1

Room Temperature

Intrinsic Si

ln(

T
3/2

10
20

10
21

10
22

10
19

Doping

density (m
-
3
)

EXTRINSIC SEMICONDUCTORS

The temperature
dependence of the
resistivity depends
on the
temperature
range

and
doping
concentration.

As doping is
increased,

decreases and the
plots move down
the ln(

⤠慸)献

39

ln(

⤠⼠

m

T敭灥牡瑵牥e⼠

C

-
200
-
100 0 100 200 300

10
5

10
4

10
3

10
2

10

1

Room Temperature

Intrinsic Si

ln(

T
3/2

Degenerate case

10
20

10
21

10
22

10
23

10
24

10
25

10
19

Doping

density (m
-
3
)

EXTRINSIC SEMICONDUCTORS

The temperature
dependence of the
resistivity depends
on the
temperature
range

and
doping
concentration.

For heavy doping,
the temperature
dependence of the
mobilities for lattice
and impurity
scattering cancel
and

40

SUPERCONDUCTIVITY

D.C. resistivity drops to

ZERO

at
T
c

the
CRITICAL

TEMPERATURE

(There is a

small A.C. impedance.)

T
c
's are now > liquid nitrogen temperature 77K

Superconductivity occurs in metals, alloys
semiconductors and ceramics.

The
MEISSNER EFFECT

can be used to detect
the onset of superconductivity.

Device applications
-

interconnects

Resistivity

T敭灥牡瑵牥

0

0

T
c

T
c

critical

temperature