Budapest University of Technology and Economics
Department of Electron Devices
Microelectronics, BSc course
Basic semiconductor
physics
http://www.eet.bme.hu/~poppe/miel/en/03-smicond_phys.ppt
http://www.eet.bme.hu
Budapest University of Technology and Economics
Department of Electron Devices
Summary of the
physics required
► Charge
carriers in semiconductors
► Currents in semiconductors
► Generation, recombination; continuity
equation
11-09-2012
Microelectronics BSc course, Basic semiconductor physics © András Poppe & Vladimír Székely, BME-EET 2008-2011
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Budapest University of Technology and Economics
Department of Electron Devices
Energy bands
► Resulting
from principles of quantum physics
Discrete energy
levels:
More atoms – more
energy levels of
electrons:
There are so many of
them that they form
energy bands:
The discrete (allowed) energy levels of an atom become energy
bands in a crystal lattice
11-09-2012
Microelectronics BSc course, Basic semiconductor physics © András Poppe & Vladimír Székely, BME-EET 2008-2011
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Budapest University of Technology and Economics
Department of Electron Devices
Valance band, conductance band
conductance
band
valance band
v – valance band / uppermost full
c – conductance band / lowermost empty
►
Valance band – these electrons form the chemical bonds
 almost full
►
Conductance bend – electrons here can freely move
 almost empty
11-09-2012
Microelectronics BSc course, Basic semiconductor physics © András Poppe & Vladimír Székely, BME-EET 2008-2011
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Budapest University of Technology and Economics
Department of Electron Devices
Electrons and holes
conduction band
►
Generation: using the
average thermal energy
►
Electrons: in the bottom of
the conduction band
►
Holes: in the top of the
valance band
►
Both the electrons and the
holes form the electrical
current
recombination
generation
valance band
Electron:
negative charge, positive mass
Hole:
positive charge, positive mass
11-09-2012
Microelectronics BSc course, Basic semiconductor physics © András Poppe & Vladimír Székely, BME-EET 2008-2011
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Budapest University of Technology and Economics
Department of Electron Devices
Conductors and insulators
forbidden gap
band gap
semiconductor
For Si: Wg = 1.12 eV
insulator
for SiO2: Wg = 4.3 eV
1 eV = 0.16 aJ = 0.16 10
11-09-2012
metal
-18
J
Microelectronics BSc course, Basic semiconductor physics © András Poppe & Vladimír Székely, BME-EET 2008-2011
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Budapest University of Technology and Economics
Department of Electron Devices
Band structure of semiconductors
conduction band
indirect
direct
Valance band
W 
1
p
W 
2
F 
dt
11-09-2012
P 
P
2
2 m eff
2m
dP
1
h
2
k
GaAs: direct band  opto-electronic devices (LEDs)
Si: indirect band
Microelectronics BSc course, Basic semiconductor physics © András Poppe & Vladimír Székely, BME-EET 2008-2011
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Budapest University of Technology and Economics
Department of Electron Devices
Generation / recombination
Spontaneous process: thermal excitation – jump into
the conduction band / recombination  equilibrium
~~~~>  = Wg/h
Direct recombination
may result in light
emission (LEDs)
<~~~~ h > Wg
Light absorption results
in generation (solar
cells)
Experiment
11-09-2012
Microelectronics BSc course, Basic semiconductor physics © András Poppe & Vladimír Székely, BME-EET 2008-2011
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Budapest University of Technology and Economics
Department of Electron Devices
Crystalline structure of Si
►
Si N=14
4 bonds,
IV-th column of the periodic table
undoped or intrinsic semiconductor
real 3D
simplified 2D
►
►
Diamond lattice, lattice constant a=0.543 nm
Each atom has 4 nearest neighbor
11-09-2012
Microelectronics BSc course, Basic semiconductor physics © András Poppe & Vladimír Székely, BME-EET 2008-2011
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Budapest University of Technology and Economics
Department of Electron Devices
5 valance dopant: donor (As, P, Sb)
conduction band
valance band
►
Electron: majority carrier
► Hole:
minority carrier
11-09-2012
n-type semiconductor
Microelectronics BSc course, Basic semiconductor physics © András Poppe & Vladimír Székely, BME-EET 2008-2011
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Budapest University of Technology and Economics
Department of Electron Devices
3 valance dopant: acceptor (B, Ga, In)
conduction band
valance band
►
Electron: minority carrier
► Hole:
majority carrier
11-09-2012
p-type semiconductor
Microelectronics BSc course, Basic semiconductor physics © András Poppe & Vladimír Székely, BME-EET 2008-2011
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Budapest University of Technology and Economics
Department of Electron Devices
Calculation of carrier concentration
FD statistics:
f (W ) 
1
 W  WF 
1  exp 

 kT

occupation
probability
possible
energy states
electrons
concentrations
holes

n
g
Wv
c
Wc
11-09-2012
(W ) f (W ) dW
p 
g
v
(W ) 1  f (W )  dW
0
Microelectronics BSc course, Basic semiconductor physics © András Poppe & Vladimír Székely, BME-EET 2008-2011
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Budapest University of Technology and Economics
Department of Electron Devices
Calculation of carrier concentration
► The
► If
results is:
n  const T
3/2
p  const T
3/2
 Wc  WF 
exp  

kT


 WF  Wv 
exp  

kT


there is no doping: n = p = ni
 it is called intrinsic material
Wc  W F  W F  Wv
WF 
11-09-2012
Wc  Wv
= Wi
WF: Fermi-level
2
Microelectronics BSc course, Basic semiconductor physics © András Poppe & Vladimír Székely, BME-EET 2008-2011
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Budapest University of Technology and Economics
Department of Electron Devices
The Fermi-level
►
Formal definition of the Fermi-level: the energy level where
1
the probability of occupancy is 0.5:
f (W ) 
 0 .5
 W  WF 
1  exp 

 kT

►
In case of intrinsic semiconductor this is in the middle of the
band gap:
electrons
WF 
Wc  Wv
2
holes
►
This is the intrinsic Fermi-level Wi
11-09-2012
Microelectronics BSc course, Basic semiconductor physics © András Poppe & Vladimír Székely, BME-EET 2008-2011
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Budapest University of Technology and Economics
Department of Electron Devices
Carrier concentrations
n  const T
3/2
 Wc  WF 
exp  

kT


n  p  const  T
p  const T
3
3/2
 WF  Wv 
exp  

kT


exp  W g / kT

► Depends
on temperature only, does not depend
on doping concentration
n p  n
2
i
Mass action law
11-09-2012
For silicon at 300 K absolute
temperature
ni = 1010 /cm3
(10 electrons in a cube of size 0.01
mm x 0.01 mm x 0.01 mm)
Microelectronics BSc course, Basic semiconductor physics © András Poppe & Vladimír Székely, BME-EET 2008-2011
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Budapest University of Technology and Economics
Department of Electron Devices
Carrier concentrations
► Si,
Problem
T = 300 K, donor concentration ND = 10
► What
17
/cm
3
are the hole and electron concentrations?
 Donor doping 
n  ND = 1017 /cm3
 Hole concentration: p = ni2/n = 1020/1017 = 103/cm3
► What
is the relative density of the dopants?
 1 cm3 Si contains 51022 atoms
 thus, 1017/ 51022 = 210-6
 The purity of doped Si is 0.999998
11-09-2012
Microelectronics BSc course, Basic semiconductor physics © András Poppe & Vladimír Székely, BME-EET 2008-2011
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Budapest University of Technology and Economics
Department of Electron Devices
Carrier concentrations
n  const T
3/2
n i  const T
3/2
 Wc  WF 
exp  

kT


 Wc  Wi 
exp  

kT


 W F  Wi 
 exp 

ni
kT


n
 W F  Wi 
n  n i exp 

kT


 W F  Wi 
p  n i exp  

kT


11-09-2012
►
Just re-order the equations
kT = 1.3810-23 VAs/K  300
K =4,14 10-21 J = 0.026 eV
= 26 meV
Thermal energy
In doped
semiconductors
the Fermi-level is
shifted wrt the
intrinsic Fermilevel
Microelectronics BSc course, Basic semiconductor physics © András Poppe & Vladimír Székely, BME-EET 2008-2011
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Budapest University of Technology and Economics
Department of Electron Devices
Temperature dependence
ni2
n  p  const  T
=
Wg
2
2 3
n i  n i  
2
dT
T
kT

d
► How
strong is it for Si?
2
d ni
2
ni




3
exp  W g / kT
2
d ni
2
ni
Wg  d T


  3 
 T
kT


Problem
1,12  d T

 3 
 0 . 15 d T

0 , 026  300

11-09-2012

 15% / oC
Microelectronics BSc course, Basic semiconductor physics © András Poppe & Vladimír Székely, BME-EET 2008-2011
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Budapest University of Technology and Economics
Department of Electron Devices
Temperature dependence of carrier
Problem
concentrations
Si, T = 300 K, donor dopant concentration ND = 1017 /cm3
n  ND = 1017 /cm3
2
 n  p  ni
p = ni2 / n = 1020/1017 = 103/cm3
How do n and p change, if T is increased by 25 degrees?
n  ND = 1017 /cm3 – unchnaged
ni2 = 1020 1.1525 = 331020
 p = ni2 / n = 331020/1017 = 3.3104/cm3
Only the minority carrier concentration increased!
11-09-2012
T=16.5 oC 10
Microelectronics BSc course, Basic semiconductor physics © András Poppe & Vladimír Székely, BME-EET 2008-2011
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Budapest University of Technology and Economics
Department of Electron Devices
Currents in
semiconductors
► Drift
current
► Diffusion current
Not discussed:




11-09-2012
currents due to temperature gradients
currents induced by magnetic fields
energy transport besides carrier transport
combined transport phenomena
Microelectronics BSc course, Basic semiconductor physics © András Poppe & Vladimír Székely, BME-EET 2008-2011
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Budapest University of Technology and Economics
Department of Electron Devices
Drift current
Thermal movement of electrons
No electrical field
vs   E
11-09-2012
There is E electrical field
 = mobility
m2/Vs
Microelectronics BSc course, Basic semiconductor physics © András Poppe & Vladimír Székely, BME-EET 2008-2011
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Budapest University of Technology and Economics
Department of Electron Devices
Drift current
J   v
 charge density
v velocity (average)
Jn  q n n E
Jp  q pp E
J  q n  n  p  p  E
J e E
e 
Differencial
Ohm's law
 e  q n  n  p  p 
1
e
Specific resistance
11-09-2012
vs   E
Specific electrical conductivity of the
semiconductor
Microelectronics BSc course, Basic semiconductor physics © András Poppe & Vladimír Székely, BME-EET 2008-2011
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Budapest University of Technology and Economics
Department of Electron Devices
Drift velocity [cm/s]
Carrier mobilities
Si
electrons
holes
n= 1500 cm2/Vs
p= 350 cm2/Vs
Electric field [V/cm]
11-09-2012
Microelectronics BSc course, Basic semiconductor physics © András Poppe & Vladimír Székely, BME-EET 2008-2011
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Budapest University of Technology and Economics
Department of Electron Devices
Carrier mobilities
►
►
Mobility decreases with
increasing doping
concentration
300 K
At around room
temperature mobilities
decrease as temperature
increases
 ~ T -3/2
Si, holes
11-09-2012
Microelectronics BSc course, Basic semiconductor physics © András Poppe & Vladimír Székely, BME-EET 2008-2011
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Budapest University of Technology and Economics
Department of Electron Devices
Diffusion current
►
Reasons:
 concentration difference
(gradient)
 thermal movement
►
►
Proportional to the gradient
D: diffusion constant [m2/s]
J n  q D n grad n
J p   q D p grad p
11-09-2012
Microelectronics BSc course, Basic semiconductor physics © András Poppe & Vladimír Székely, BME-EET 2008-2011
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Budapest University of Technology and Economics
Department of Electron Devices
Total currents
J n  qn  n E  q D n grad n
J p  q p  p E  q D p grad p
D 
kT

Einstein's relationship
q
UT 
kT
q

T  300 K
1 . 38  10
 23
[VAs/K]  300 [K]
1 . 6  10
 19
 0 . 026 V  26 mV
[As]
Thermal voltage
11-09-2012
Microelectronics BSc course, Basic semiconductor physics © András Poppe & Vladimír Székely, BME-EET 2008-2011
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Budapest University of Technology and Economics
Department of Electron Devices
Generation, recombination
►
conduction band
Life-time: average time an
electron spends in the
conduction band
 n,  p
recombination
1 ns … 1 s
generation
valance band
rn 
n
n
11-09-2012
rp 
p
p
►
Generation rate: g [1/m3s]
►
Recombination rate: r [1/m3s]
g n  rn
equilibriu m

n0
n
Microelectronics BSc course, Basic semiconductor physics © András Poppe & Vladimír Székely, BME-EET 2008-2011
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Budapest University of Technology and Economics
Department of Electron Devices
Continuity equation
d
 N 
dt
1
q

A
d N
V volume
A surface
N electron
dt  V
dn
dt
11-09-2012
Jn d A  gn V 


1
1
q V
1
q

n
n
V
Jn d A  gn 
A
 
div J n  g n 
Microelectronics BSc course, Basic semiconductor physics © András Poppe & Vladimír Székely, BME-EET 2008-2011
n
n
n
n
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Budapest University of Technology and Economics
Department of Electron Devices
Diffusion equation
dn
1
n
 div  J   g 
J  qn 
dt
n
q
dn
dt
dp
dt
11-09-2012
n
n
n
n
E  q D n grad n
 
  n div n E  D n divgrad n  g n 
 
   p div p E  D p divgrad
p gp 
n
n
p
p
Microelectronics BSc course, Basic semiconductor physics © András Poppe & Vladimír Székely, BME-EET 2008-2011
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Budapest University of Technology and Economics
Department of Electron Devices
Example for the solition of the diff. eq.:
dn
dt
 
  n div n E  D n divgrad n  g n 
2
0  Dn
d n
0  Dn
d n
dx
2
 gn 
2
dx
2

n0
n

11-09-2012
D n n
n
n
n
p
n
n
n ( x )  n 0  ( n e  n 0 ) exp(  x /
Ln 
n
D n n )
diffusion length
Microelectronics BSc course, Basic semiconductor physics © András Poppe & Vladimír Székely, BME-EET 2008-2011
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