Thermoelectric power factor of nanoporous semiconductors

L. D.

 

Hicks

and

M. S.

 

Dresselhaus

,  , ().

A. R.

 

Abramson

,

W. C.

 

Kim

,

S. T.

 

Huxtable

,

H.

 

Yan

,

Y.

 

Wu

,

A.

 

Majumdar

,

C. -L.

 

Tien

, and

P.

 

Yang

,  , ().

J. R.

 

Lim

,

J. F.

 

Whitacre

,

J. -P.

 

Fleurial

,

C. -K.

 

Huang

,

M. A.

 

Ryan

, and

N. V.

 

Myung

,  , ().

Since many semiconductor nanowires are naturally covered with a layer of electrically insulating oxide that is several nanometers thick, electron transport inside the nanowires and in the matrix material can be decoupled.

G. D.

 

Mahan

and

J. O.

 

Sofo

,  , ().

A. N.

 

Tikhonov

and

A. A.

 

Samarsky

, (, , ).

D. A.

 

Broido

and

N.

 

Mingo

,  , ().

N. W.

 

Ashcroft

and

N. D.

 

Mermin

, (, , ), p. .

These relations are directly obtained from the Schrödinger equation with infinite barrier boundary conditions. They are strictly valid only if the confining potential is infinite outside the material. A proof goes as follows: inside the material, the wave function corresponding to the set of quantum numbers $α$⁠, with $kz=0$ satisfies $EαΨα=(ℏ2∕2m⊥)(∂2∕∂x2+∂2∕∂y2)Ψα$⁠, subject to the condition $Ψ=0$ at the pore boundaries. Trivially, for a material with different effective mass, $m⊥′$ and the same pore size and aspect ratio, the energy of the corresponding state $α$ changes as $E′=m⊥∕m⊥′E$⁠. Substituting this in the definition of $ρ2D$⁠, Eq. (6), one obtains Eq. (10). A change in the pore separation, $d$⁠, keeping the aspect ratio $d∕δ$ constant, is analogous to a change in length units. Let $Ψ$ be the wave function corresponding to the case with pore separation $d$⁠, and $Ψ′$ be the one with pore separation $d′$⁠. At any point inside the material (excepting the pores) the wave function satisfies $Eα=(ℏ2∕2m⊥)[(∂2∕lx2+∂2∕∂y2)Ψα]∕Ψα$⁠. Since the wave functions in the $d$ and $d′$ cases are related by $Ψα(x,y)∝Ψ′α(xd′∕d,yd′∕d)$⁠, this implies that their energies are related as $Eα′=(d∕d′)2Eα$⁠. Substituting this into Eq. (6) and taking into account the change in the Brillouin zone size, one obtains Eq. (9).

J. O.

 

Sofo

and

G. D.

 

Mahan

,  , ().

D. A.

 

Broido

and

T. L.

 

Reinecke

,  , ().

R.

 

Prasher

,  , ().

W. A.

 

Harrison

, (, , ).

N.

 

Mingo

and

D. A.

 

Broido

,  , ().