Download Effective Mass

Effective Mass
• The electrons in a crystal are not free, but instead interact with
the periodic potential of the lattice.
• In applying the usual equations of electrodynamics to charge
carriers in a solid, we must use altered values of particle mass.
We named it Effective Mass.
P.Ravindran, PHY02E Semiconductor Physics, 30 January 2013: Intrinsic and Extrinsic semiconductors
1
Effective Mass – an example
Find the (E,k) relationship for a free electron and relate it to the
electron mass.
E
k
The electron momentum is: p  mv  h k
1 2 1 p2 h 2 2
E  mv 

k
2
2 m 2m
d 2E h 2

2
dk
m
h2
m 2
d E
dk 2
P.Ravindran, PHY02E Semiconductor Physics, 30 January 2013: Intrinsic and Extrinsic semiconductors
Effective Mass
Most energy bands are close to parabolic at their minima (for
conduction bands) or maxima (for valence bands).
EC
m* 
h2
d 2E
EV
dk 2
Remember that in GaAs:
• The effective mass of an electron in a band with a given (E,k)
relationship is given by
E
L

m ()  m ( X
*
X
*
or
L)
1.43eV
k
3
Effective Mass
• At k=0, the (E,k) relationship near the minimum is usually
parabolic:
h2 2
E (k ) 
k  Eg
*
2m
2
d
In a parabolic band, E2 is constant. So, effective mass is
dk
constant.
In most semiconductors the effective mass is a tensor quantity.
m 
*
ij
h2
d 2 Eij
dkij2
P.Ravindran, PHY02E Semiconductor Physics, 30 January 2013: Intrinsic and Extrinsic semiconductors
Effective Mass
EV
d 2E
0
2
dk
m*  0
m* 
2
EC
d E
0
2
dk
h2
d 2E
dk 2
m 0
*
Table: Effective mass values for Ge, Si and GaAs.
Effective mass
*
m
m
n
*
p
Ge
Si
GaAs
0.55m0
1.1m0
0.067 m0
0.37 m0
0.56 m0
0.48m0
† m0 is the free electron rest mass.
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
►
P.Ravindran, PHY02E Semiconductor Physics, 30 January 2013: Intrinsic and Extrinsic semiconductors
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Intrinsic Semiconductor
k B  1.38 10 23 JK -1
a) Energy level diagrams showing the excitation of an electron from the valence band to
the conduction band. The resultant free electron can freely move under the
application of electric field.
b) Equal electron & hole concentrations in an intrinsic semiconductor created by the
thermal excitation of electrons across the band gap
P.Ravindran, PHY02E Semiconductor Physics, 30 January 2013: Intrinsic and Extrinsic semiconductors
 For T > 0K, some VB electrons get enough thermal energy to be excited (through
Eg) up to the CB.
 Consequently, the semiconductor material will have some electrons in the
previously empty CB and some unoccupied states in the previously full VB
P.Ravindran, PHY02E Semiconductor Physics, 30 January 2013: Intrinsic and Extrinsic semiconductors
Semiconductors: Doping
– When silicon is doped with phosphorous, it becomes a n-type
semiconductor, in which an electrical current is carried by
negatively charged electrons
– When silicon is doped with boron, it becomes a p-type
semiconductor, in which an electrical current is carried by
positively charged holes
P.Ravindran, PHY02E Semiconductor Physics, 30 January 2013: Intrinsic and Extrinsic semiconductors
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Holes
C
Eg
Energy band
Model
V
Vacancy
Bond
Model
P.Ravindran, PHY02E Semiconductor Physics, 30 January 2013: Intrinsic and Extrinsic semiconductors
Donors and acceptors
At 0K, the donor energy
level is filled with electrons
and too little thermical
energy is needed in order to
excite these electrons up to
the CB. So, between 50100K, electrons are virtually
“donated” to the CB.
Likewise, acceptor levels can be
thermically settled with VB
electrons, by there generating
holes.
P.Ravindran, PHY02E Semiconductor Physics, 30 January 2013: Intrinsic and Extrinsic semiconductors
Donor and Acceptor levels with temperature
P.Ravindran, PHY02E Semiconductor Physics, 30 January 2013: Intrinsic and Extrinsic semiconductors
Extrinsic Semiconductors (n-type)
Extrinsic semiconductor (p-type)
P.Ravindran, PHY02E Semiconductor Physics, 30 January 2013: Intrinsic and Extrinsic semiconductors
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Charge Carriers in doped Semiconductors
Extrinsic Material
•
Doping: the process to create carriers in semiconductors by purposely introducing impurities
into the crystal
-
•
•
There are two types of doped semiconductors, n-type and p-type.
Extrinsic semiconductors: the materials that have a characteristic of n0  p0  ni when they are
doped
n-type semiconductors:
- A “donor” impurity from column V (P, As, Sb; donor) introduces an donor energy level (Ed) near the
-
bottom of CB Ec( within the band gap)
At 50K, all of the electrons in Ed (filled with electrons at 0K) are “donated” to CB
n0  ni , p0 
(e-: majority carrier, h+: minority carrier)
n-type semiconductor
Energy Band Model
P.Ravindran, PHY02E Semiconductor Physics, 30 January 2013: Intrinsic and Extrinsic semiconductors
Charge Carriers in Semiconductors
Extrinsic Material (continued)
•
p-type semiconductor (e-: minority carrier, h+:majority carrier):
-
An “acceptor” impurity from column III (B, Al, Ga, In; acceptor) introduces an acceptor energy level (Ea) near the
top of VB Ev( within the band gap)
At 50K, all of the energy states in Ea (empty at 0K) “accept” electrons from the VB, leaving behind holes in the VB.
- p  n , n 
0
i
0
•
p-type
Energy Band Model
Bond Model:
-
An As atom (column V; 5 valence electrons) in Si lattice has 4
valence electrons to complete the covalent bonds with the 4
neighboring Si atoms, plus one extra electron; the fifth
valence electron is loosely bound to As atom  A small
amount of thermal E enables this extra electron to
overcome its Coulombic binding to the impurity atom 
extra electron is donated to the lattice
-
The column III impurity B has only 3 valence electrons to
contribute to the covalent bonding, leaving one bond
incomplete (hole)  With a small amount of thermal E, this
incomplete bond is transferred to other atoms as the
bonding electrons exchange positions (electron hopping)
Bond Model
P.Ravindran, PHY02E Semiconductor Physics, 30 January 2013: Intrinsic and Extrinsic semiconductors
5 valance dopant: donor (As, P, Sb)
conduction band
valance band
• Electron:
• Hole:
majority carrier
minority carrier
n-type semiconductor
P.Ravindran, PHY02E Semiconductor Physics, 30 January 2013: Intrinsic and Extrinsic semiconductors
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3 valance dopant: acceptor (B, Ga, In)
conduction band
valance band
• Electron:
• Hole:
minority carrier
majority carrier
p-type semiconductor
P.Ravindran, PHY02E Semiconductor Physics, 30 January 2013: Intrinsic and Extrinsic semiconductors
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n-Type Semiconductor
a) Donor level in an n-type semiconductor.
b) The ionization of donor impurities creates an increased electron
concentration distribution.
P.Ravindran, PHY02E Semiconductor Physics, 30 January 2013: Intrinsic and Extrinsic semiconductors
p-Type Semiconductor
a) Acceptor level in an p-type semiconductor.
b)
The ionization of acceptor impurities creates an increased hole
concentration distribution
P.Ravindran, PHY02E Semiconductor Physics, 30 January 2013: Intrinsic and Extrinsic semiconductors
Intrinsic & Extrinsic Materials
• Intrinsic material: A perfect material with no impurities.
n  p  ni  exp( 
Eg
2k BT
)
n,p & ni are the electron, hole, & intrinsic concentrations respectively. Eg is
the gap energy, T is temperature.
• Extrinsic material: donor or acceptor type semiconductors.
pn  ni
2
• Majority carriers: electrons in n-type or holes in p-type.
• Minority carriers: holes in n-type or electrons in p-type.
• The operation of semiconductor devices is essentially based on the injection
and extraction of minority carriers.
P.Ravindran, PHY02E Semiconductor Physics, 30 January 2013: Intrinsic and Extrinsic semiconductors
Calculation of carrier concentration
FD statistics:
f (W ) 
1
 W  WF 
1  exp

 kT 
occupation
probability
possible energy
states
electrons
concentrations
holes

n   g c (W ) f (W ) dW
Wc
p
Wv
 g (W ) 1  f (W ) dW
v
0
P.Ravindran, PHY02E Semiconductor Physics, 30 January 2013: Intrinsic and Extrinsic semiconductors
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Temperature Effects
As the crystal temperature rises, the crystal expands and the gap
energy gets lower.
 By submitting to pressure, the crystal is compressed and the gap
energy rises.

Varshni
Equation
T 2
Eg (T )  Eg (0) 
T 
Bandgaps of silicon, germanium and gallium arsenide
S.M.Sze, “Physics of Semiconductor Devices”, 2nd Edition, John
Wiley&Sons, 1981
P.Ravindran, PHY02E Semiconductor Physics, 30 January 2013: Intrinsic and Extrinsic semiconductors
dopant in a rather simple way
* In the case where the dopant is a DONOR t
a core with a net POSITIVE charge
 The donor is thus essentially similar
energy of the surplus electron can b
IONIZATION ENERGY of the hydroge
 Intrinsic Semiconductor
 Extrinsic Semiconductor
n  p  ni
• THE IO
(SUBJE
Doping of Semic
Si:As
FOR A
• A similar argument to that
above may also be•BY
mad
+
INC
THE DIE
the pseudo hydrogen atom
consists
of
a
POSITIVE
e
 Donor impurities – provide extra electrons
* This gives a SIMILAR
estimate for the binding
to conduction
A GROUP V DOPANT IN A SILICON LATTICE
CAN BE VIEWED AS A PSEUDO HYDROGEN ATOM!
(type n)
 Acceptor impurities – provide exceeding
holes to conduction
(type p)
• A GROUP I
BE VIEWED
Si:B
• IN THIS CA
NEGATIVEL
B
e+
* The ACTUAL donor and acceptor binding ene
reasonably WELL with these SIMPLE estimat
P.Ravindran, PHY02E Semiconductor Physics, 30 January 2013: Intrinsic and
ExtrinsicENERGY
semiconductors
BINDING
FOR DONORS (eV)
Concept of Fermi-Dirac distribution function
Distribution of electrons over a range of allowed energy levels
at thermal equilibrium is given by :
f(E) =
1
1 + e(E – EF) / kT
K – Boltzman’s constant = 8.62 x 10-5 eV / K= 1.38 x 10 -23 J / K
f(E) gives the probability of findiing a electron in a particular energy E.
P.Ravindran, PHY02E Semiconductor Physics, 30 January 2013: Intrinsic and Extrinsic semiconductors