Chapter

SEMICONDUCTOR
BANDSTRUCTURE

In this chapter we present gures discussing bandstructures of important semiconductors and the
concept of doping and mobile carriers.
 SEMICONDUCTOR BANDSTRUCTURE
In semiconductors we are pimarily interested in the
valence band and conduction band. Moreover, for
most applications we are interested in what happens
near the top of the valence band and the bottom of
the conduction band. These states originate from the
atomic levels of the valence shell in the elements
making up the semiconductor.
IV Semiconductors

C 1s22s22p2

Si 1s22s22p63s23p2

Ge 1s22s22p63s23p63d104s24p2

III-V Semiconductors

Ga 1s22s22p63s23p63d104s24p1

As 1s22s22p63s23p63d104s24p3

Outermost atomic levels are either s-type or p-type.

© Prof. Jasprit Singh www.eecs.umich.edu/~singh

 BANDSTRUCTURE OF SEMICONDUCTORS
The k-vector for the electrons in a crystal is limited to a
space called the Brillouin zone. The figure shows the
Brillouin zone for the fcc lattice relevant for most
semiconductors. The values and notations of certain
important k-points are also shown. Most semiconductors
have bandedges of allowed bands at one of these points.

IMPORTANT HIGH SYMMETRY POINTS

Γ point: kx = 0 = ky = kz L

X point: kx = 2π ; ky = kz = 0 Γ
a X
L point: kx = ky = kz = π
a
a = lattice constant (cube edge)

A TYPICAL BANDSTRUCTURE: Si
6
L3 Γ2
4 Γ2
L1
2 Γ15 Γ15
Γ25 Γ25
0 L Χ1
3
ENERGY (eV)

–2 Χ4
–4
–6
–8 L1
L
–10 2
Γ1 Γ1
–12
L Λ Γ ∆ Χ UK Σ Γ

© Prof. Jasprit Singh www.eecs.umich.edu/~singh

 BANDSTRUCTURE NEAR BANDEDGES

Behavior of electrons near the bandedges determines most device

properties. Near the bandedges the electrons can be described by
simple effective mass pictures, i.e., the electrons behave as if they
are in free space except their masses are m*.

DIRECT

AAA AAA
CONDUCTION
BAND

AAA
AAA AAA
AA
AAA
AA
AAA
AA
AAA
A
A
A
AA INDIRECT

AA
AAA
AAA
Direct CONDUCTION
bandgap BAND

Indirect
bandgap

∆ I
II
VALENCE I Heavy Hole Band
BAND
II Light Hole Band
III
III Split-Off Band

∆ = Split-Off Energy
k=0
k

Schematic of the valence band, direct bandgap, and indirect bandgap

conduction bands. The conduction band of the direct gap
semiconductor is shown in the solid line, while the conduction band
of the indirect semiconductor is shown in the dashed line. The
curves I, II, and III in the valence band are called heavy hole, light
hole, and split-off hole states, respectively.

© Prof. Jasprit Singh www.eecs.umich.edu/~singh

 EFFECTIVE MASS DESCRIPTION

CONDUCTION BAND: Direct bandgap material

h2k2
Ec(k) = Ec(0) + 2m*c
with 2 1
2pcv
1
m*c
= 1 +
m m 2 3 E
2 +
gΓ E
1
gΓ + ∆
( )
The smaller the bandgap, the smaller the effective mass.

SPLIT-OFF BAND: h2k2

Eso = –∆ – 2m*so
1 2
2pcv
= –1 +
m*so m (3m2EgΓ + ∆)

HEAVY HOLE; LIGHT HOLE:

In a simple approximation the heavy hole and light hole bands can
also be represented by masses m*hh and m* h. However, the real
picture is more complex.

CdS
0.2 AlAs
AlSb
( mmoc )

ZnTe GaP
*

ZnSe
CdSe
EFFECTIVE MASS

CdTe
0.1 InP
GaAs
GaSb
InAs
InSb
0
0 1.0 2.0 3.0
BANDGAP Eg (eV)

© Prof. Jasprit Singh www.eecs.umich.edu/~singh

 BANDSTRUCTURE: SILICON
Although the bandstructure of Si is far from ideal, having an indicrect bandgap, hig hhole masses,
and small spin-orbit splitting, processing related advantages make Si the premier semiconductor
for consumer electronics. On the right we show constant energy ellipsoids for Si conduction band.
There are six equivalent valleys in Si at the bandedge.

Silicon
6
Eg = 1.1eV
5
at 300K
4 kz

3 (001)
ENERGY (eV)

2 3.4
Six equivalent
1.1
(010) valleys at
1 (100)
Eg conduction
bandedge
0
–1 kx (100) (010) ky
(001)
–2

–3 4.37 x 10–4 T2
Eg = 1.17 – (eV) T = Temperature in K
–4 T – 636
k
(b)
(a)

• Indirect gap material weak optical transitions, cannot be used to produce lasers.
2π 2π
• Valleys along the x-axis and –x-axis: k0x = a (0.85,0,0) and k0x = a (–0.85,0,0):

E(k) = Ec + h
2 (kx – k0x)2
kx2 + kz2
; ml = 0.98 m0; mt = 0.19 m0
+
2 m*l m*t
similar E-k relations for other 4 valleys.

• Density of states mass = 1.08 m0 (6 valleys included).

• Heavy hole mass: 0.49 m0 ; light hole mass: 0.16 m0.

• Intrinsic carrier concentration at 300 K: 1.5 x 1010 cm–3.

© Prof. Jasprit Singh www.eecs.umich.edu/~singh

 BANDSTRUCTURE: GaAs
The bandgap at 0 K is 1.51 eV and at 300 K it is 1.43 eV. The bottom of the conductionband
is at k = (0,0,0), i.e., the G-point. The upper conduction band valleys are at the L-point.

Gallium Arsenide
6
Eg = 1.43eV
CONDUCTION
5 at 300 K BAND
4

3
ENERGY (eV)

2 0.58
0.3
1 Eg 5.4 x 10–4 T2
Eg = 1.519 – (eV)
T + 204
0
T = Temperature in K
–1
–2

–3
–4
[111] [100]
k

CONDUCTION BAND: • Electron mass is light. m* = 0.067 m0

• Upper valley mass is large. m* = 0.25 m0 results in negative
differential resistance at higher fields.
• Material is direct bandgap and has strong optical transistions
can be used for light emission.

VALENCE BAND: • Heavy hole mass: 0.45 m0; light hole mass = 0.08 m0.
Intrinsic carrier concentration at 300 = 1.84 x 106 cm–3.

© Prof. Jasprit Singh www.eecs.umich.edu/~singh

 BANDSTRUCTURE: Ge AlAs, InAs, InP

Germanium Aluminum Arsenide

6 6
Eg = 0.66eV Eg = 2.15eV
5 5 at 300K
at 300K
4 4
3 3
2 2
ENERGY

ENERGY
0.3
Eg
(eV)

(eV)
1 1 2.75
Eg 0.90
0 0
–1 –1
–2 –2
–3 –3
–4 –4
(a) L [111] Γ [100]X (b) L [111] Γ [100]X

Indium Arsenide Indium Phosphide

6 6
Eg = 0.35eV Eg = 1.34eV
5 5
at 300K at 300K
4 4
3 3
ENERGY

ENERGY

2 2
(eV)

0.9
(eV)

0.6
9 4
1 1 Eg
E
0 g 0
–1 –1
–2 –2
–3 –3
–4 –4
L [111] Γ [100]X L [111] Γ [100]X
(c) (d) k
k

(a) Bandstructure of Ge. (b) Bandstructure of AlAs. (c) Bandstructure of InAs.

Since no adequate substitute matches InAs directly, it is often used as an alloy
(InGaAs, InAlAs, etc.,) for devices. (d) Bandstructure of InP. InP is a very
important material for high speed devices as well as a substrate and barrier layer
material for semiconductor lasers.

© Prof. Jasprit Singh www.eecs.umich.edu/~singh

 ELECTRONIC PROPERTIES OF SOME SEMICONDUCTORS

Material Bandgap Relative Material Electron Hole

(eV) Dielectric Mass Mass
Constant (m0) (m0)

C 5.5, I 5.57 AlAs 0.1

Si 1.124, I 11.9 AlSb 0.12 * = 0.98
mdos
Ge 0.664, I 16.2 GaN 0.19 * = 0.60
mdos
SiC 2.416, I 9.72 * = 0.60
GaP 0.82 mdos
GaAs 1.424, D 13.18
GaAs 0.067 m*lh = 0.082
AlAs 2.153, I 10.06 m*hh = 0.45

InAs 0.354, D 15.15 GaSb 0.042 * = 0.40

mdos

GaP 2.272, I 11.11 Ge ml = 1.64 * = 0.044

mlh
mt = 0.082 m*hh = 0.28
InP 1.344, D 12.56
mdos= 0.56
InSb 0.230, D 16.8
InP 0.073 * = 0.64
mdos
CdTe 1.475, D 10.2
InAs 0.027 * = 0.4
mdos
AlN 6.2, D 9.14
InSb 0.13 m*dos = 0.4
GaN 3.44, D 10.0
Si ml = 0.98 m*lh = 0.16
ZnSe 2.822, D 9.1 * = 0.49
mt = 0.19 mhh
ZnTe 2.394, D 8.7 mdos= 1.08

Properties of some semiconductors. D and I stand for direct and indirect gap,
respectively. The data are at 300 K. Note that Si has six conducton band
valleys, while Ge has four.

© Prof. Jasprit Singh www.eecs.umich.edu/~singh

 SOME IMPORTANT PROPERTIES OF Si AND GaAs

PROPERTY SI GAAS

Electron m*l = 0.98 m* = 0.067

effective mass m*t = 0.19
(m0) m*dos = 1.08
m*σ = 0.26

Hole m*hh = 0.49 m*hh = 0.45

effective mass m*lh = 0.16 m*lh = 0.08
(m0) m*dos = 0.55 m*dos = 0.47

Bandgap 1.17 – 4.37 x 10–4 T2 1.519 – 5.4 x 10–4 T2

(eV) T + 636 T + 204

Electron affinity 4.01 4.07

(eV)

For Si: m*dos: To be used in calculating density of states, position of Fermi level
m*σ: To be used in calculating response to electric field, e.g., in mobility

© Prof. Jasprit Singh www.eecs.umich.edu/~singh

 HOLES IN SEMICONDUCTORS: WHAT ARE HOLES?

In a filled band (valence band) no current can flow, since electrons are
normally Fermi particles and obey the Pauli exclusion principle. The
electrons can “move” if there is an empty state available. The empty
states in the valence band are called holes.

E
Conduction
band

Wavevector associated
kh with the missing electron
ke
k

Electron
removed
+ VALENCE
BAND

Missing electron = hole

Diagram illustrating the wavevector of the missing electron ke. The

wavevector of the system with the missing electron is –ke, which is
associated with the hole.

© Prof. Jasprit Singh www.eecs.umich.edu/~singh

 HOLES IN SEMICONDUCTORS: HOW DO HOLES MOVE?

Holes behave as if they carry a positive charge.

t2 > t1>0

t=0 t = t1 t = t2
Ε Ε Ε
Fx Fx Fx
F F F
kx kx kx
E G E G E G
D H D H D H
C I C I C I
B J B J B J
A K A K A K

(a) (b) (c)

e
ve je

vh
h
jh

(d)

The movement of an empty electron state, i,e,. a hole under an electric

field. The electrons move in the direction opposite to the electric field
so that the hole moves in the direction of the electric field thus behaving
as if it were positively charged, as shown in (a), (b), and (c). (d) The
velocities and currents due to electrons and holes. The current flow is in
the same direction, even though the electron and holes have opposite
velocities. The electron effective mass in the valence band is negative,
but the hole behaves as if it has a positive mass.

© Prof. Jasprit Singh www.eecs.umich.edu/~singh

FREE CARRIERS IN SEMICONDUCTORS: INTRINSIC CARRIERS

In semiconductors, at finite temperatures, there are electrons in the conduction band

and holes in the valence band.

METALS
Evac

EF
Electrons in the

ENERGY
Ec
conduction band
can carry current

(a)

SEMICONDUCTORS
Evac
Electrons in the conduction band
– – (density, n) carry current
Ec
ENERGY

Mobile carrier density = n + p

+ + + EV
Holes in the valence band (density, p)
Valence band carry current
(b)
(a) A schematic showing allowed energy bands in electrons in a metal. The
electrons occupying the highest partially occupied band are capable of
carrying current. (b) A schematic showing the valence band and conduction
band in a typical semiconductor. In semiconductors only electrons in the
conduction band holes in the valence band can carry current.

For small electron (n), hole (p) densities we can use Boltzmann approximation:

n = Nc exp [(EF– Ec)/kBT]

m*ekBT 3/2
where
Nc = 2 ( )
2πh2

m*hkBT 3/2
p=2 ( )
2πh2
exp [(Ev– EF)/kBT]

= Nv exp [(Ev– EF)/kBT]

kBT 3/2
Intrinsic case: ni = pi = 2 ( )
2πh2
(m*em*h)3/4 exp (–Eg/2kBT)

© Prof. Jasprit Singh www.eecs.umich.edu/~singh

 INTRINSIC CARRIER DENSITIES FOR SOME SEMICONDUCTORS
T(°C)
1000 500 200 100 27 0 –20
1019
1018
INTRINSIC CARRIER DENSITY ni (cm–3)

1017

1016 Ge

1015
Temperature dependence of
1014 Si
ni, pi in Si, Ge, GaAs
1013

1012
11
10
10
10
109
GaAs
10 8

10 7

106
0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0
1000/T (K–1)

CONDUCTION BAND VALENCE BAND INTRINSIC CARRIER

MATERIAL EFFECTIVE DENSITY (NC ) EFFECTIVE DENSITY (NV ) CONCENTRATION (ni = pi)

Si (300 K) 2.78 x 1019 cm–3 9.84 x 1018 cm–3 1.5 x 1010 cm–3

Ge (300 K) 1.04 x 1019 cm–3 6.0 x 1018 cm–3 2.33 x 1013 cm–3

GaAs (300 K) 4.45 x 1017 cm–3 7.72 x 1018 cm–3 1.84 x 106 cm–3

Effective densities and intrinsic carrier concentrations of Si, Ge and GaAs. The numbers for
intrinsic carrier densities are the accepted values even though they are smaller than the values
obtained by using the equations derived in the text.

© Prof. Jasprit Singh www.eecs.umich.edu/~singh

 DOPING OF SEMICONDUCTORS: DONORS AND ACCEPTORS

If an impurity atom replaces a host semiconductor atom in a crystal it could donate (donor) an
extra electron to the conduction band or it could accept (acceptor) an electron from the valence
band producing a hole.

All 4 outer EC
electrons go into
the valence band
EV
Silicon
host atom

= + +
Electron-ion
Pentavalent Coulombic
donor impurity = Silicon-like + attraction

A schematic showing the approach one takes to understand donors in semiconductors. The
donor problem is treated as the host atom problem together with a Coulombic interaction term.
The silicon atom has four “free” electrons per atom. All four electrons are contributed to the
valence band at 0 K. The dopant has five electrons out of which four are contribted to the
valence band, while the fifth one can be used for increasing electrons in the conducton band.

E Conduction
band
Si Si Si
Ec
+ Excess Ed
Si As positive Donor
( )( εεso )2 eV
m*

AAAA
Ed = Ec –13.6 m o
charge level
–
Si Si Si

AAAA
Excess electron Ev Valence

AAAA
n-type
band
silicon
Arsenic (As) atom donates one
electron to the conduction band to
produce an n-type silicon

Charges associated with an arsenic impurity atom in silicon. Arsenic has five valence electrons,
but silicon has only four valence electrons. Thus four electrons on arsenic form tetrahedral
covalent bonds similar to silicon, and the fifth electron is available for conduction. The arsenic
atom is called a donor because when ionized it donates an electron to the conduction band.

© Prof. Jasprit Singh www.eecs.umich.edu/~singh

 FREE CARRIERS IN DOPED SEMICONDUCTORS
If electron (hole) density is measured as a function of temperature in a doped semiconductor, one
observes three regimes:
Freezeout: Temperature is too small to ionize the donors (acceptors), i.e.,
kBT < EC – ED (kBT< ED – EV).
Saturation: Most of the donors (acceptors) are ionzed.
Intrinsic: Temperature is so high that ni > doping density.

TEMPERATURE (K)
500
1000 300 200 100 75 50
10 17
Si
Nd = 1015cm–3
Intrinsic range
ELECTRON DENSITY n (cm–3)

10 16

Saturation range
10 15
Freeze-out
range

14
10 ni

10 13
0 4 8 12 16 20
1000/T (K–1)
Electron density as a function of temperature for a Si sample with donor impurity
concentration of 1015 cm–3. It is preferable to operate devices in the saturation region
where the free carrier density is approximately equal to the dopant density.

It is not possible to operate devices in the intrinsic regime, since the devices always have a high
carrier density that cannot be controlled by electric fields.
every semiconductor has an upper temperature beyond which it cannot be used in devices.
The larger the bangap, the higher the upper limit.

© Prof. Jasprit Singh www.eecs.umich.edu/~singh

 Zinc Blende and Wurtzite
CRYSTAL STATIC LATTICE
MATERIAL STRUCTURE BANDGAP DIELECTRIC CONSTANT DENSITY
(EV) CONSTANT (Å) (gm-cm–3)

C DI 5.50, I 5.570 3.56683 3.51525

Si DI 1.1242, I 11.9 5.431073 2.329002
SiC ZB 2.416, I 9.72 4.3596 3.166
Ge DI 0.664, I 16.2 5.6579060 5.3234
BN HEX 5.2, I ε|| = 5.06 a = 6.6612 2.18
ε = 6.85 c = 2.5040
BN ZB 6.4, I 7.1 3.6157 3.4870
BP ZB 2.4, I 11. 4.5383 2.97
BAs ZB — — 4.777 5.22
AlN W 6.2,D ε = 9.14 a = 3.111 3.255
c = 4.981
AlP ZB 2.45,I 9.8 5.4635 2.401
AlAS ZB 2.153,I 10.06 5.660 3.760
AlSb ZB 1.615,I 12.04 6.1355 4.26
GaN W 3.44,D ε||=10.4 a = 3.175 6.095
ε = 9.5 c = 5.158
GaP ZB 2.272,I 11.11 5.4505 4.138
GaAs ZB 1.4241,D 13.18 5.65325 5.3176
GaSb ZB 0.75,D 15.69 6.09593 5.6137
InN W 1.89,D a = 3.5446 6.81
c = 8.7034
InP ZB 1.344,D 12.56 5.8687 4.81
InAs ZB 0.354,D 15.15 6.0583 5.667
InSb ZB 0.230,D 16.8 6.47937 5.7747
ZnO W 3.44,D ε||= 8.75 a = 3.253 5.67526
ε = 7.8 c = 5.213
ZnS ZB 3.68,D 8.9 5.4102 4.079
ZnS W 3.9107,D ε = 9.6 a = 3.8226 4.084
c = 6.2605
ZnSe ZB 2.8215,D 9.1 5.6676 5.266
ZnTe ZB 2.3941,D 8.7 6.1037 5.636
CdO R 0.84,I 21.9 4.689 8.15
CdS W 2.501,D ε = 9.83 a = 4.1362 4.82
c = 6.714
CdS ZB 2.50,D — 5.818 —
CdSe W 1.751,D ε||=10.16 a = 4.2999 5.81
ε = 9.29 c = 7.0109
CdSe ZB — — 6.052 —
CdTe ZB 1.475,D 10.2 6.482 5.87
PbS R 0.41,D* 169. 5.936 7.597
PbSe R 0.278,D* 210. 6.117 8.26
PbTe R 0.310,D* 414. 6.462 8.219
Data are given at room temperature values (300 K).
Key: DI: diamond; HEX: hexagonal; R: rocksalt; W: wurtzite; ZB: zinc blende;
*: gap at L point; D: direct; I: indirect ε||: parallel to c-axis; ε : perpendicular to c-axis

© Prof. Jasprit Singh www.eecs.umich.edu/~singh

 Bandgaps (in eV) of some semiconductors
TETRAHEDRALLY BONDED MATERIALS

V C Si Ge α-Sn
C 5.5i,D
Si 2.6i,Z/W 1.1i,D 0.7-1.1i
Ge 0.7-1.1 0.74i,D
α-Sn 0.09,D
i: Indirect gap
III-V N P As Sb D: Diamond
B 3.8,W 2.0i,Z 1.5i,Z Z: Zinc Blende
Al 5.9,W 2.5,Z 2.2,Z 1.7,Z
Ga 3.5,W 2.4i,Z 1.5,Z 0.81,Z W: Wurtzite
In 2.4,W 1.4,Z 0.41,Z 0.24,Z R: Rocksalt
II-VI O S Se Te O: Orthorhombic
Zn 3.4,W 3.6,Z/W 2.8,Z/W 2.4,Z Rh: Rhombohedral
Cd 1.3i,R 2.5,Z/W 1.8,Z/W 1.6,Z T: Trigonal
Hg 2.2,O/Rh 2.3,T –.06,Z –.3,Z
OR: Orthorhombic
I-VII F Cl Br I distorted rocksalt
Cu 3.4,Z 3.1,Z 3.1,Z M: Monoclinic
Ag 2.8i,R 3.2i,R 2.7i,R 3.0,W

NON-TETRAHEDRAL BONDED MATERIALS

IV-VI compounds
IV-VI O S Se Te
Ge 1.7,OR 1.1,OR 0.15,R
Sn 1.1,OR 0.9,OR 2.1,R
Pb 2.0,i 0.29,R 0.15,R 0.19,R
Group VI elements
VI S Se Te
3.6,O 1.9i,T 0.33,T
2.5,M
Group V elements
V P As Sb Bi
.33,O .17,Rh .10 .015
© Prof. Jasprit Singh www.eecs.umich.edu/~singh