of Density States: g Definition: g d = number of states between and d (per unit volume) For a single band k , g d
1 V
1 8 d
3 k ,s d
d
2
3
k
1 4 3
dS dk
d
dk
dS = area element on constant energy surface k dk = change in k, along k k (perpendicular to dS), corresponding to d
dS
d
k k
dk k dk
1
k k
dk k k dk v g dk
1 4
3
v g = group velocity
With dk
k k
d ,
g
1 4
3
k
k
k k
gn
n
dS
For multiple bands:
g
n
n
k n k
dS
Another formula for single band: g
1 V
k 4
1
3 k ,s
BZ
d 3 k k
Pf:
2
1
1 1 k d V V ks
1 k 2
k ,s
1
QED
van Hove singularities in g arise when k n k 0 (denominator = 0); these occur at critical points k c . Consider a single band in repeated zone near k c . 2 1 k smooth k c k k c k c k k c kc 2 k c quadratic terms in k k c
Make a principal axis transformation (diagonalize a 3 3 symmetric matrix) 2 2 k k c qi q k kc qi referred to principal axes. i 2mi mi = effective mass Class (# of negative mi's) 0 1 2 3 Name
m1,2,3 0 m1,2 0, m3 0, etc. m1 0, m2,3 0, etc. m1,23 0
M0 (minimum; parabolic) M1 (saddle point) M2 (saddle point) M3 (maximum; parabolic)
van Hove's theorem: a periodic function of variables (e.g. k in repeated-zone) ! must have at least Cn critical points of type Mn in each -dimensional n ! n ! primitive cell. 2D 3 n0 c0 1 Mo 3 Max 1 c1 3 M1 Example: in 3D; 3 Min 3 2 c2 3 M2 3
3 c3 1
M3
In 3D, there are at least one minimum, one maximum, and three saddle points of each type. This follows from topological considerations. Consider the region near a M0 of a single band. 1 1 2 2 g d 3 k k d 3k c 4 qi 3 BZ 4 3 i 2mi qi q k k c , define pi 2mi
g 1 4
3 3
o
8m1m2 m3
3
d 3 p c p2
8m 1 m2 m3 4
3 3
o
4 p2 c p2 dp
8m 1 m2 m3 4
3
2
p2 c p 2 dp2
2m1m2 m 3 g 2 3 0,
c ,
c c
g
So, it is a weak singularity. However, g' = at c .
c
Many bands: the critical points for each band are generally at different energies. So, each singularity may sit atop a smooth "background" of density of states from other bands. Mo , g c1 c2 c c3 c1 c2 c
c , c 0 , c
, c
M1 ,
g
c1 c2 c c1 c2 c c3
c1 c2 c c1 c2 c c3
c , c
, c
M2 ,
g
c , c
M3 ,
g
c1 c2 c c3 c1 c2 c
c , c , c
Example of 3d g (schematic): g
Experimentally, these critical point features can be seen by high-resolution XPS (x-ray photoelectron spectroscopy). XPS: an incident monochromatic x-ray beam ejects electrons from the sample; the energy distribution of the photoelectrons shows the valence density of states (modulated by cross section variation).
XPS spectrum of GaP [L. Ley, R.A. Pollack, F.R. McFeely, G. P. Kowalczyk, and D.A. Shirley, Phys. Rev. B 9, 600 (1974)]
Top: XPS spectrum with background removed 2nd: DOS broadened by lifetime & instrumental resolution 3rd: DOS (note the singularities) 4th: band structure of GaP
Density of states for a free-electron-like material
0D: a set of delta functions at n (atomic levels, or particle in a box) 2 n 2 k z 1D: a set of subbands n ; at each band edge g 2m n 2D: a set of subbands n
2 2 2 k x k y
3D: a free-electron-like band Wikipedia:
2m 2 2 2 2 k x k y k z
; at each band edge g
n
2m
; at band bottom g
0D
2D 1D
3D