21.2 Metal-Semiconductor Contacts
519
0
10
20
30
40
50
60
70
80
90
0
1
2
3
4
5
6
7
U
Cu
Ni
Co
Fe
Mn
Cr
V
Ti
Sc
Ca
K
Si
Al
Mg
C
B
Be
Sm
Nd
Ce
La
Ba
Cs
Sb
Sn
In
Cd
Te
Ag
Pd
Mo
Nb
Zr
Y
Li
Rb
Se
As
Ge
Ga
Zn
Rb
Ru
Th
Bi
Pb
Tl
Hg
Au
Pt
Ir
Os
Re
W
Ta
Hf
Lu
Tb
Gd
Eu
Sr
Fig. 21.2 Work function W m of various metals
The position of the Fermi level in the metal is given by the work function W m that is shown in
Fig. 21.2 for various metals (see also Table 21.2). A recent review of published values of metal work
functions is given in [1688]. The work function reflects the atomic shell structure; minima of the work
function exist for group-I elements. The work function is the energy difference between the vacuum
level (an electron is at rest in an infinite distance from the metal surface) and the metal Fermi level
(W m > 0). Since the electron density in the metal conduction band is very high, the position of the
metal Fermi level does not change considerably when charge is exchanged between the metal and the
semiconductor.
Since in a semiconductor the Fermi level depends strongly on the doping and temperature it is not
useful to characterize the material itself. For semiconductors the electron affinity χ sc = E vac − E C > 0
is defined as the energy difference between the vacuum level and the conduction-band edge (see
Fig. 12.21).
Ideal Band Diagram
When the metal and the semiconductor are not in contact (Fig. 21.3a), the metal is characterized by its
work function W m = E vac − E F and the semiconductor by its electron affinity χ sc . First, we assume
that W m > χ sc . For an n-type semiconductor, the energy difference between the Fermi level and the
conduction band is denoted as
− e V n = E C − E F
(21.1)
being negative, V n < 0, for nondegenerate semiconductors. Thus, the position of the semiconductor
Fermi level is given as
E F = E vac − χ sc + e V n .
(21.2)
If the metal and semiconductor are brought into contact the Fermi levels will equilibrate. For the case
of Fig. 21.3 (E F,sc > E F,m ) electrons will flow from the semiconductor to the metal. The negative
surface charge of the metal is compensated by a positive charge (due to D
+ ) in the semiconductor in
the vicinity of the surface. Eventually a (Schottky) barrier of height
4 F Bn
F Bn = W m − χ sc
(21.3)
4 We denote the energy barrier height with F B = −eφ B .
519
0
10
20
30
40
50
60
70
80
90
0
1
2
3
4
5
6
7
U
Cu
Ni
Co
Fe
Mn
Cr
V
Ti
Sc
Ca
K
Si
Al
Mg
C
B
Be
Sm
Nd
Ce
La
Ba
Cs
Sb
Sn
In
Cd
Te
Ag
Pd
Mo
Nb
Zr
Y
Li
Rb
Se
As
Ge
Ga
Zn
Rb
Ru
Th
Bi
Pb
Tl
Hg
Au
Pt
Ir
Os
Re
W
Ta
Hf
Lu
Tb
Gd
Eu
Sr
Fig. 21.2 Work function W m of various metals
The position of the Fermi level in the metal is given by the work function W m that is shown in
Fig. 21.2 for various metals (see also Table 21.2). A recent review of published values of metal work
functions is given in [1688]. The work function reflects the atomic shell structure; minima of the work
function exist for group-I elements. The work function is the energy difference between the vacuum
level (an electron is at rest in an infinite distance from the metal surface) and the metal Fermi level
(W m > 0). Since the electron density in the metal conduction band is very high, the position of the
metal Fermi level does not change considerably when charge is exchanged between the metal and the
semiconductor.
Since in a semiconductor the Fermi level depends strongly on the doping and temperature it is not
useful to characterize the material itself. For semiconductors the electron affinity χ sc = E vac − E C > 0
is defined as the energy difference between the vacuum level and the conduction-band edge (see
Fig. 12.21).
Ideal Band Diagram
When the metal and the semiconductor are not in contact (Fig. 21.3a), the metal is characterized by its
work function W m = E vac − E F and the semiconductor by its electron affinity χ sc . First, we assume
that W m > χ sc . For an n-type semiconductor, the energy difference between the Fermi level and the
conduction band is denoted as
− e V n = E C − E F
(21.1)
being negative, V n < 0, for nondegenerate semiconductors. Thus, the position of the semiconductor
Fermi level is given as
E F = E vac − χ sc + e V n .
(21.2)
If the metal and semiconductor are brought into contact the Fermi levels will equilibrate. For the case
of Fig. 21.3 (E F,sc > E F,m ) electrons will flow from the semiconductor to the metal. The negative
surface charge of the metal is compensated by a positive charge (due to D
+ ) in the semiconductor in
the vicinity of the surface. Eventually a (Schottky) barrier of height
4 F Bn
F Bn = W m − χ sc
(21.3)
4 We denote the energy barrier height with F B = −eφ B .