Transcript alp diffusion

Introductory Nanotechnology
~ Basic Condensed Matter Physics ~
Atsufumi Hirohata
Department of Electronics
Quick Review over the Last Lecture
Classic model : Dulong-Petit empirical law
c V, mol
3R
0
Einstein model :
E
D
T
Debye model :
•  E : Einstein temperature
•  D : Debye temperature
• c V, mol ~ 3R for  E << T
• c V, mol ~ 3R for  D << T
• c V, mol  exp (-  E / T) for  E << T
• c V, mol  T 3 for  D << T
Contents of Introductory Nanotechnology
First half of the course :
Basic condensed matter physics
1. Why solids are solid ?
2. What is the most common atom on the earth ?
3. How does an electron travel in a material ?
4. How does lattices vibrate thermally ?
5. What is a semi-conductor ?
6. How does an electron tunnel through a barrier ?
7. Why does a magnet attract / retract ?
8. What happens at interfaces ?
Second half of the course :
Introduction to nanotechnology (nano-fabrication / application)
What Is a Semi-Conductor ?
Elemental / compound semiconductor
•
•
Intrinsic / extrinsic semiconductors
•
•
n / p-dope
Temperature dependence
•
Schottky junctions
•
pn junctions
What is semi-conductor ?
Band diagrams :
Allowed
Allowed
Allowed
Forbidden
Forbidden
Allowed
Forbidden
Allowed
metal conductors
Forbidden
Allowed
Allowed
semiconductors
insulators
With very small energy,
electrons can overcome
the forbidden band.
EF
Energy Band of a semiconductor
Schematic energy band diagram :
E
Conduction band
conduction electron
Band gap
hole
Valence band
Elemental Semiconductors
In the periodic table,
Carrier density : Cu (metal) ~ 10 23 cm -3
Ge (semiconductor) ~ 10 13 cm -3
Semimetal : conduction and valence bands are slightly overｌaped.
As (semimetal) ~ 10 20 cm -3
Sb (semimetal) ~ 10 19 cm -3
C (semimetal) ~ 10 18 cm -3
Bi (semimetal) ~ 10 17 cm -3
Fabrication of a Si-Based Integrated Circuit
Czochralski method :
Si purity (99.999999999 %)
* http://www.wikipedia.org/
Compound Semiconductors
In the periodic table,
III-V compounds : GaAs, InAs, InSb, AlP, BP, ...
II-VI compounds : ZnO, CdS, CdTe, ...
IV-IV compounds : SiC, GeSi
IV-VI compounds : PbSe, PbTe, SnTe, ...
Shockley Model
Contributions for electrical transport :
E
Conduction band
Band gap
conduction electron
(number density : n)
positive hole
(number density : p)
Valence band
 Ambipolar conduction
 Intrinsic semiconductor
   e   h  nq e  pq h  ni q e   h 
n i  n  p
* http://www.wikipedia.org/
Carrier Number Density of an Intrinsic Semiconductor
Carrier number density is defined as
n
 f EgEdE
Here, the Fermi distribution function is f E 

1
exp E  EF  kBT 1
For the carriers like free electrons with m*, the density of states is
* 3 2

1
2m
gE  2
E


2
2
2    
For electrons with effective mass me*, g(E) in the conduction band is written

with respect to the energy level EC,
* 3 2

1
2me
g C E  2
E  EC


2  2 

2  
For holes with effective mass mp*,
g(E) in the valence band is written

with respect to the energy level EV = 0,
2m * 3 2
1  p 
g V E  2
E
2  2 
2  

* H. Ibach and H. Lüth, Solid-State Physics (Springer, Berlin, 2003).
Carrier Number Density of an Intrinsic Semiconductor (Cont'd)
fp(E) for holes equals to the numbers of unoccupied states by electrons :
f p E  1 f e E
n is an integral in the conduction band from the bottom EC to top Ect :

n

E Ct
EC
f e Ege EdE 

E Ct
EC
* 3 2

1 2me
1
E

E
dE


C
2  2 
exp E  EF  kBT 1
2 

p is an integral in the valence band from the bottom -EVb to top 0 :

p



0
E Vb
0
E vb
f p Eg p EdE 

0
E vb
2m * 3 2




1  p 
1
E
1

dE
2  2 

2 
 exp E  EF  kBT 1 


2m * 3 2
1  p 
1
E
dE
2  2 
exp  E  EF  kBT 1
2 

Here, EC (= Eg = EC - EV) >> kBT  E - EF  EC /2 for EC  E  Ect (EF ~ EC /2)


f e E  exp E  EF  kBT 
Similarly, EC >> kBT  -(E - EF)  EC /2 for EVb  E  0
f p E  expE  EF  kBT 
Carrier Number Density of an Intrinsic Semiconductor (Cont'd)
For E - EF > 3kBT,
f e EF  3kBT   0.05 and hence ECt  ∞
Similarly, f p EF  3kBT   0.05 and hence EVb  -∞





* 3 2

1 2me
n  2 
 2 

2 


2m * 3 2
1  p 
p
2 
2 2 




EC
0

E  EC exp E  EF  kBT dE
E exp E  EF  kBT dE
As a result,
n  N C expEC  EF  kBT   N C f e EC
N  N T 3 2
C
Ce


2m * k 3 2

e B

N Ce  2


2

h



N  N T 3 2
Vp
 V

2m * k 3 2
n  N V exp EF kBT   N V f p 0 
p B
 N  2

Vp
2


h




Fermi Level of an Intrinsic Semiconductor
For an intrinsic semiconductor, n  p  n i
N C expEC  EF  kBT  N V expEF kBT 


m * 
1  3
p
EF  EC  kBT ln * 
m 
2
4
 e 
Assuming, me* = mp* = m*
EF 
1
1
EC  Eg
2
2
np product is calculated to be

np  n i  N CN V exp EC
2
2kBT 3
kBT   4 2  me* mp*
 h 


32
exp  EC kBT 
 constant for small ni

 can be applied for an extrinsic (impurity) semiconductor
Extrinsic Semiconductors
Doping of an impurity into an intrinsic semiconductor :
n-type extrinsic semiconductor :
e.g., Si (+ P, As, Sb : donor)
conduction electron
As neutral donor
As positive donor
p-type extrinsic semiconductor :
e.g., Si (+ Ga, Al, B : acceptor)
conduction band
B neutral acceptor
holes
B negative acceptor
* M. Sakata, Solid State Physics (Baifukan, Tokyo, 1989).
Carrier Number Density of an Extrinsic Semiconductor
Numbers of holes in the valence band EV should equal to
the sum of those of electrons in the conduction band EC and in the acceptor level EA :
p  n  nA
n
Similar to the intrinsic case,

 E 
p  N V f p 0  N V exp  F 
 kBT 
 Eg  EF 
n  N C f e Eg  N C exp 

 kBT 
 


Assuming numbers of neutral acceptors are NA,
nA 
NA
E  E 
F
1 + 2exp  A

 k BT 
For EA - EF > kBT,

 E
N V exp  F
 kBT

nA  N A

 Eg  EF 
 N C exp 
 N A

 kBT 
nA
p
valence band
EC (Eg)
EF
EA
EV (0)
Carrier Number Density of an Extrinsic Semiconductor (Cont'd)
At low temperature, one can assume p >> n,
p  nA
As nA is very small, EA > EF
 E  E 
NA
F
nA 
exp  A
 
2
 k BT 
 E  N
 E  E 
F
A
F
N V exp 
exp  A


k
T
2
k
T
 B 


B
 E 
 E 
NA
F
exp 
exp  A 

2N V
 kBT 
 2kBT 
k T 2N V EA
EF  B ln

2
NA
2




n
By substituting N V  N VpT
nA
p
valence band
32
k BT 2N Vp 3 2  EA
EF 
ln
T 
2
 N A
 2
E
For T ~0, EF  A
2
At high temperature, one can assume n >> nA, p  n

Similar to the intrinsic case, for me = mp
*

*,
EF 
Eg
2
EC (Eg)
EA
EF
EV (0)
Temperature Dependence of an Extrinsic Semiconductor

Eg
2kB


EA
2k B

* H. Ibach and H. Lüth, Solid-State Physics (Springer, Berlin, 2003).
Semiconductor Junctions
Work function  :
Current density of thermoelectrons :
vacuum level
  
J  AT exp 

k
T
 B 
2

3s
2p
 Richardson-Dushman equation
A : Richardson constant (~120 Acm 2/K)

2s
Metal - metal junction :
EFA
A
barrier
A
+
- +
+
-
B
vacuum level
 A -  B : contact potential
EFB
=FAE=
- FB
EFB
E
FA E
B
EFB
EFA
A
A - B
B
1s
Na 11+
EF
Metal - Semiconductor Junction - n-Type
Metal - n-type semiconductor junction :
vacuum level
S
M
 S : electron affinity
EC : conduction band
ED : donor level
EFS
EFM
M
n-S
EV : valence band
depletion layer
qVd =  M -  S : Schottky barrier height
M - S
EFM
M
- ++
++
-
EC
ED
EFS
n-S
EV
* http://www.dpg-physik.de/
Metal - Semiconductor Junction - p-Type
Metal - p-type semiconductor junction :
vacuum level
 S : electron affinity
M
S
EFM
M
EC
p-S
EFS
EA : acceptor level
EV
depletion layer
EC
S - M
EFM
M
+
+
+
+
qVd =  S -  M
p-S
EFS
EA
EV
Einstein Relationship
At the equillibrium state,
Numbers of electrons diffuses towards -x direction are
De
dn
dx
EC
ED
EF
(-x direction)
(n : electron number density, De : diffusion coefficient)

Drift velocity of electrons with mobility e under E is
v d   e E
Numbers of electrons travel towards +x direction under E are


nvd   e nE
(+x direction)
As E is generated by the gradient of EC, E is along -x and vd is +x.
 enE  De
dn
0
dx
(equillibrium state)
Assuming EV = 0, electron number density is defined as

 E  E 
F
n  N e exp  C

 k BT 
EV
x
Einstein Relationship (Cont'd)
Now, an electric field E produces voltage VCF = VC - VF
EC  EF  qVCF  qVC  VF 
dVCF 1 d EC  EF 
E  

dx
q
dx


Accordingly,
d EC  EF 
dn
dn
1



n  qE
dx d EC  EF 
dx
kBT
  e nE  De




De   e
nqE
kBT
kBT
q
 Einstein relationship
Therefore, a current density Jn can be calculated as
 qn dV

dn 
dn 
x
J n  q e nE  qDe


 qDe 



dx
 kBT dx dx 
  qV  V 
 qV 
d
J n  Bexp 
 exp  d 
kBT 

 kBT 
 

Rectification in a Schottky Junction
By applying a bias voltage V onto a metal - n-type semiconductor junction :
M - S
q(Vd - V)
forward bias
J Foward
  qV  V  
 qV 
d
 J MS  JSM  Bexp 
 exp  d 
k BT 

 kBT 
 

  qV
 J 0exp 
 k BT
M - S
q(Vd - V)
 
 qV 
1 J 0 exp 

k
T
 
 B 
J

J Foward
J Reverse  J 0
V : large
 qV 
 J 0 exp 

k
T
 B 
V

  qV  V  
 qV 
d
J Reverse  J MS  J SM  Bexp 
 exp  d 

 k T 

 k BT 
B
 

  qV  
 J 0exp 
 1 J 0 V : large 
  k BT  
reverse bias
* H. Ibach and H. Lüth, Solid-State Physics (Springer, Berlin, 2003).
pn Junction
Fabrication method :
• Annealing method :
n-type : Spread P2O5 onto a Si substrate and anneal in forming gas.
p-type : Spread B2O3 onto a Si substrate and anneal in forming gas.
• Epitaxy method (“epi” = on + “taxy” = arrangement) :
Oriented overgrowth
n-type : thermal deformation of SiH4 (+ PCl3) on a Si substrate
p-type : thermal deformation of SiH4 (+ BBr3) on a Si substrate
pn Junction Interface
By connecting p- and n-type semiconductors,
p : Most of accepters become - ions
 Holes are excited in EV.
n : Most of donors become + ions
 Electrons are excited in EC.
Fermi level EF needs to be connected.
qVd
 Built-in potential : qVd = Efn - EFp
Electron currents balances
pn=np
Majority carriers :
p : holes, n : electrons
Minority carriers :
p : electrons, p : holes
* H. Ibach and H. Lüth, Solid-State Physics (Springer, Berlin, 2003).
Rectification in a pn Junction
Under an electrical field E,
hole
electron
drift current
forward bias
hole
Current rectification :
very small drift current
reverse bias
* M. Sakata, Solid State Physics (Baifukan, Tokyo, 1989).
** H. Ibach and H. Lüth, Solid-State Physics (Springer, Berlin, 2003).