energy bands
DESCRIPTION
energy bandTRANSCRIPT
Semiconductor DevicesRevision energy band diagram
Dr. K. FobeletsRoom 714
De Broglie“all particles can behave like waves”
Thus electrons in a semiconductor lattice have an energy associated to their electron wave.
Particle-wave duality↓
Quantum mechanics
Wave-particle duality - lightNewton Huygens
1650
T. Young1800
particles waves
Wave-particle duality - lightPlanck Einstein
1900
fhE
Waves can behave as particlesWorkfunction = minimum energy to release electron from material
Wave-particle duality - matterDe Broglie Einstein
1924
Matter (particles) can behave as waves.
hp
420
22 cmcpE
Classical description (1D): EVmpPEKE 2
2
Re-written by Schrödinger (1D): )()()()(2 2
22
xExxVdxxd
m
Schrodinger 1925
Schrödinger equationWave equation for particles
2)(xBack to particles: probability of finding the particle at x
mExVx)()( wave function of particle
electrostatic potential seen by particletotal energy of particlemass of particlereduced Planck constant
Solving the Schrödinger equationWhat is V(x) in a periodic lattice?
Electrostatic potential in lattice: -V(x)
a
V(x): electrostatic potential = work done/q = PE/q
xkqQPE potential energy related to the work done by bringing an e- (q<0) towards
an atom (Q>0)
+e-
Simplifying problemStep 1 – electron in a box (quantum well – QW):
PE
∞∞
-a/2
)()(2 2
22
xEdxxd
m
0)2/,2/( aax
mkkE
evennank
kxa
x
2)(
)sin(2)(
22
a/20
mkkE
oddnank
kxa
x
2)(
)cos(2)(
22
∞∞
x
Simplifying problemStep 1 – relevance
,2,1 nank
Discrete energy levels En within QW!
22
222
22n
mah
mkEn
∞∞
-a/2 a/2
∞∞
x
Ener
gy
E1
E2
E3
Simplifying problemStep 2 – electrons in different boxes
V(x)∞∞ ∞
electrons in different boxes: Wells not ∞ highThus e- will influence each otherPauli’s exclusion principleThus small changes to energy values for each e-
↓Splitting of levels in to bands of closely spacedenergy levels.
Ener
gyE1
E2
E3
~1022 cm-3 of Si atoms
each
with
14 e
-
Distribution of e-
Ener
gy
E1
E2
E3
Pauli’s exclusion principle~1022 cm-3 of Si atoms each with 14 e-
Schrödinger in periodic lattice: e- on discrete energy levels in bands
→ Bands of discrete energy levels
T=0K
How will all these e- be distributed?
• There are a discreet number of energy levels: STATEs• Two electrons per state (Pauli exclusion principle). • Quantum mechanics shows that available states occur in bands with gaps where no
states exist → bandgap• Electrons bound closest to nucleus will sit on the lowest state.• Valence electrons will fill the upper levels.
Completely filled with e- at T=0K
Ener
gy
Eg
Ec and Ev – band edgesEn
ergy
T=0K
→ Valence band
→ Conduction band
→ EV
→ EC→ EG → Forbidden band → bandgap
• Shine light with hn>Eg• or with sufficient thermal energy
Ener
gy
Eg
free electronsfree holes
CONDUCTION BAND
VALENCE BAND
• With doping: n-type
Ener
gy
Eg
More free electronsfree holes
• With doping: p-type
Ener
gy
Eg
free electronsMore free holes
What can energy bands teach us?
empty
filled
empty
filled
partially filled
filled
or
overlap
Ec
EvEg
Ec
Ev
Ec
Ev
Ec
Ev
Eg
metal
Small bandgapConduction band partially filled
No bandgap
What can energy bands teach us?
empty
filled
empty
filled
partially filled
filled
or
overlap
Ec
EvEg
Ec
Ev
Ec
Ev
Ec
Ev
Eg
metal insulator
Very large bandgapConduction band empty
What can energy bands teach us?
empty
filled
empty
filled
partially filled
filled
or
overlap
Ec
EvEg
Ec
Ev
Ec
Ev
Ec
Ev
Eg
metal semiconductor insulator
Bandgap ~ kT/qSome free electrons and holes at room temperature
Result of simplification
A silicon lattice is 3D V(x,y,z) much more complex.Thus energy band diagram much more complex.
We lost the information on the mass of the carriers (electronsand holes). More accurate calculations show:1. The existence of 2 types of carriers
electrons in conduction bandholes in valence band
2. Mass of electrons and holes is differentmcarrier for Si: mn < mp
Carrier statistics
• How many carriers are available and how are they distributed in energy?– Density of states– Occupation probability of states
Density of states
Solution of the 1-D Schrödinger equation: discrete energy levels
n
mkkE
nank
2)(
,2,122
EmEg23
22
22
1)(
Density of states = number of statesper unit volume, per unit energy
Each integer value of n = called state
Occupation probability of states
Electrons are fermions, they must obey Pauli’s exclusion principle: no two of them may occupy the same state.Statistical mechanics → Fermi-Dirac distribution function f(E).Gives the probability of finding an electron at a certain energy E.
EEF
f(E)
1T=0K
T1
T2>T1
kTEE
EfFexp1
1)(
1/2
Probability function and band diagramRoom temperature
Energy
EF
f(E) 1 1/2 0
EG
empty
full
insula
torsem
icond
uctor
metal w
ith
overl
appin
g ban
dsGrey area is full at T=0
Density of carriers • Carriers can only occupy the available states (energy
levels)• Thus no carriers in the bandgap• Density of states:
EEmh
Eg
EEmh
Eg
vpv
cnc
323
323
28)(
28)(
• The Fermi-Dirac distribution function f(E) will give the probability a state is filled with an electron
• Integrate gc(E)f(E) over the band gives the number of free electrons
• Integrate gv(E)[1-f(E)] over the band gives the number of free holes
Density of carriers
v
c
E
v
Ec
dEEfEgp
dEEfEgn
)(1)(
)()(
kTEENp
kTEENn
vFV
FcC
exp
exp
Density of free electrons and holes
23
2
*
23
2
*
22
22
hkTm
N
hkTm
N
pV
nC
Effective density of states
For kTEEkTEE vFFc &
Density of carriers
Fc EE EcEF
Ev
n
kTEENn Fc
C exp
vF EE
Ec
EFEv
p
kTEENp vF
V exp
Density of carriers
kTEE
np
kTEE
nn
Fii
iFi
exp
expEi is the position of the Fermi level when the material is intrinsic. Lies ± midgap.
Density of free electrons and holes
EcEF
Ev
Ei
How conductive is the semiconductor?
• Fermi level represents character of semiconductor Intrinsic n-type p-type
EcEF
Ev
Ec EF
Ev
Ec
EFEv
Position of Fermi level: EF
n=p=ni n>p n<p
Material dependent
kTEENn Fc
C exp
kTEENp vF
V exp
Carriers: graphically (see B. Streetman)
EF
E
1/2 F(E)1
E E
Ec
Ev
g(E) carrierconcentration
a) intrinsic
electrons
holes
EF
E
1/2 F(E)1
E E
Ec
Ev
g(E) carrierconcentration
b) n-type
electrons
holes
EF
E
1/2 F(E)1
E E
Ec
Ev
g(E) carrierconcentration
c) p-type
electrons
holes
Extrinsicsemiconductor
Intrinsicsemiconductor
Density of free carriers and position of the Fermi-level are related.
g(E) Density of states
f(E) Fermi-Dirac distribution function
T>0K
f(E)
f(E)
f(E)
Linking potential energy in the energy band diagram E(x) with electrostatic potential V(x) and electric field .
)(xEPE
exExV
)()(
dxxdVx )()( E
)(xE
Energy band diagram.
What does an energy band diagram describe?It gives the potential energy of free carriers
above Ec for free electrons below Ev for free holes
In between is bandgap in which no carriers can reside
EcEF
Ev
EiEG
Influence of an electric field on the energy band diagram
Ec
Ev
EG
Represents the potential energy of the carriers
x
When an electric field, E is applied, carriers gain potential energy (represented by E) in the direction of the field.
Ei
If Ethen Higher potential
energy E here.
Since electrons drift in opposite direction as electric field.
E
Influence of an electric field on the energy band diagram
Represents the energy of the carriers
When an electric field is applied, carriers gain potential energy (represented by E) in the direction of the field.
E(x)Thus:Ec(x)
Ev(x)EG
Ei(x)
Influence of an electric field on carrier movement in the energy band diagram
• Energy band diagram of a semiconductor under an electric field
EEc
EvEG
e- opposite direction to electric field.
h+ in direction of electric field.
Influence of an electric field on the energy band diagram: maths for e-
E Ec
EvEG
Electric field E points “uphill” for electrons!
exExV
)()(
dxxdVx )()( E
dxxdE
ex )(1)( E
dxxdE
ex c )(1)( E
)(xEPE
Excess carriers Internal electric field
• & Vext = 0V ; I = 0A
• Current densities:Jn = emnnE + eDndn/dxJp = emppE - eDpdp/dx
• Dn dn/dx = -mnn E Dp dp/dx = mpp E
• E = -(Dn/mnn) dn/dx E = (Dp/mpp) dp/dx
n(x)
x
If
ThenE(x)
Ec
EvEg
EF
xTVq
kTD
m
e- diffusion
e- drift
=0=0
All transport equations
qxExV
dxxdVx
xNxNxpxnexdxxd
dxxnd
dxxpd
dxxdpeDxxpexJ
dxxdneDxxnexJ
AD
p
n
ppp
nnn
)()(
)()(
)()()()()()(
0)(
0)(
)()()()(
)()()()(
2
2
2
2
E
E
E
E
m
m
)()(
)(
)()(
)()(
)(
)()(
2
2
xpxN
nxp
xNxp
xnxN
nxn
xNxn
nD
in
Ap
pA
ip
An
Steady state (DC) in short layers Low carrier injection at 300K
Conclusions
• Two types of carriers in a semiconductor– Electrons negative charge and electron mass mn
– Holes positive charge and hole mass mp
• Two types of current in semiconductors– Drift current, caused by external electric field– Diffusion current, caused by excess carrier
concentration.
+
• Free electrons reside in the conduction band, have a negative charge and an effective mass mn
• Free holes reside in the valence band, have a positive charge and a different effective mass mp
• The simplified energy band gives the potential energy of the carriers
• No carriers can reside in the bandgap, EG
• The position of the Fermi level, EF defines the type and density of free carriers.
• An electric field (internal or external) causes a tilt in the band diagram.
For the exams you do not have to be able to:
• Solve the Schrodinger equation• Derive the density of states• Derive the expression for the electron and hole concentration.
The solutions for these are derived in the study group questions, but the purpose is solely to satisfy your curiosity.The equations will be given in a formulae sheets and you have to be able to apply them, but not derive them.