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
㴠
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
㴠
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)]
㴠
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
㴠

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
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