Influence of temperature on resistivity/conductivity; superconductivity

kitefleaUrban and Civil

Nov 15, 2013 (3 years and 8 months 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
,


,

楳⁡慳畲潦o瑨t⁥慳攠睩w栠
睨楣栠慮a敬散瑲潮o捡渠浯n攠瑨t潵杨o瑨t獯s楤i


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






卵灥牣潮摵d瑩t楴y




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

献⁄整牭楮攠⁴桥敬散瑲潮t
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,


瑨攠
捯湤畣瑩楴礠潦o瑨攠浡瑥物t氬l慮搠䔠瑨E
敬e捴物c晩敬搠睩瑨楮w瑨攠浡瑥物t氮

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


楳i瑨攠敬e捴牯渠浯扩汩瑹.

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(

)

-
癥⁔R

卬潰S‽

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(

)

汮l

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(


縠ㄯ1





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(


縠ㄯ1





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:













敡捨c浥浢敲m潦⁴桥 灡楲p浵獴m桡癥 敱畡氠慮搠 †

潰灯獩s攠浯浥湴畭




獯s浯浥湴畭m潦⁴桥 灡楲⁲p浡楮猠

畮捨慮来搠

批 捯汬c獩潮猬s椮i⸠湯 獣慴瑥物湧r潣捵牳

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

cable made by Pirelli Cables

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†

渠楳⁣潮獴慮琠i湤n




T
-
1


Hence




T

35





偬潴P 汮l

⤠孯)渨

⥝)
癳⸠ㄯ吠T牥⁩浰潲慮a†††
because they can be
used to obtain E
g


ln(

)

-
癥⁔R

卬潰S‽

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(


縠ㄯ1





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(


縠ㄯ1





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(


縠ㄯ1





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(


縠ㄯ1





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


楳⁣潮獴i湴n

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