Thermoelectric Materials and Devices - Hierarchical Nanostructures for Energy Devices - page 113

where k B is the Boltzmann constant. As we can see in eqn (6.6), the Seebeck
coecient is proportional to the energy derivative of the differential con-
ductivity, s(E), at the Fermi energy E F . The differential conductivity is de-
fined as 2

d n 3 r 4 n g | 7

s ð E Þ¼ q 2 t ð E Þ ðð v x dk y dk z D

q 2 t ð E Þ v x ð E Þ D ð E Þ (6 : 7)

where t is the momentum dependent relaxation time, k is the wavevector,
v is the carrier velocity and D(E) is the density of state. Then electrical con-
ductivity can be written as

s ¼ ð s ð E Þ dE (6 : 8)

E

Here, n(E) is the energy-dependent electron density, i.e. n(E) ¼ D(E) f(E).
Further, D(E) is the density of states, f(E) is the Fermi-Dirac distribution, q is
the electron charge and m(E) is the carrier mobility. We can rewrite eqn (6.6)
using D(E) and f(E)as
S ¼ p 2
3

þ 1
m

k B
q k B T

1
n

D ð E Þ df ð E Þ
dE

þ f ð E Þ dD ð E Þ
dE

dm ð E Þ
dE

(6 : 9)

E ¼ E F

We can confirm from eqn (6.9) that the Seebeck coecient increases when
the energy derivative of the electron density of states (dD(E)/dE) at the Fermi
level increases. Low-dimensional materials have discrete, sharp densities of
states due to the so-called quantum confinement effect (see Figure 6.3(a)).
Thus, if we control the Fermi level so as to locate it near a sharp peak of the
density of states, we can enhance the Seebeck coecient dramatically without
reducing electrical conductivity too much. In order to achieve this quantum
confinement effect in 1D or 2D materials, however, a very small confinement
direction size, typically less than 20 nm, and a very low temperature are re-
quired. Careful doping control is also necessary to locate the Fermi level near
a sharp DOS peak, which could be realized by gate-induced charge density
modulation. Wu et al. 27 demonstrated large power factor enhancement in
relatively thick InAs nanowires with 50-70 nm diameters at temperatures
below 20 K, which they attributed to the resonance effect by quantum dot
states in the nanowires, not to the 1D quantum confinement effect.

.

6.2.2.2 Electron Filtering
Several strategies can be used to enhance a material's thermoelectric prop-
erties, including the energy filtering effect. 28-31 An energy barrier appears at
a hetero-junction between different semiconductors, and between a semi-
conductor and a metal (Figure 6.4(a)). 32 This barrier acts as a filter for charge
carriers. While high-energy electrons can cross this barrier, electrons with
energies lower than the barrier height cannot pass. The high-energy electron
transport is required to be a thermionic or quasi-thermionic emission in