Quantum kinetics approach to calculation of the low field mobility in the hole inversion layers of silicon MOSFET’s

Abstract

Analytic expressions for low field mobility have been obtained in the quantized p-type inversion layers. The confining potential is approximated by a triangular quantum well. Main attention is paid to study the dependence of the hole mobility on transverse effective field at different temperatures and concentrations of the ionized impurities. Acoustic and optical phonons, charged impurities, and surface roughness have been adopted as scattering system. Theoretical considerations are based on the quantum kinetic equation and special form of the non-equilibrium distribution function (shifted Fermi distribution). Calculations show that the acoustic phonon limited mobility does not depend on the transverse effective electrical field $E_\mathrm{eff}$ and has a temperature dependence closer to experiment than known expression for the universal mobility. At the same time, the mobility limited by scattering with optical phonons and surface roughness is proportional to $E_\mathrm{eff} ^{-1/3}$ and $E_\mathrm{eff} ^{-2}$, respectively. The mobility limited by scattering by ionized impurities is a weak function of the transverse effective field. Results of the calculations are compared with known experimental data.

Appendix

The Airy functions A(z) allow the exact analytical solution of the Schrödinger equation in triangular quantum well [30, 31].

$$\begin{aligned} Ai\left( z \right) =\frac{1}{\pi }\int \limits _0^\infty {\cos \left( {\frac{t^{3}}{3}+zt} \right) } \mathrm{d}t. \end{aligned}$$

(30)

In this case the envelope functions can be written as

$$\begin{aligned}&\xi _{n(m)} \left( z \right) =\frac{1}{\sqrt{I_{n(m)} L_E^{(v)} }}Ai\left( {\frac{z}{L_E^{(v)} }+a_{n(m)} } \right) , \end{aligned}$$

(31)

$$\begin{aligned}&I_{n(m)} =\int \limits _0^\infty {Ai\left( {y+a_{n(m)} } \right) ^{2}\mathrm{d}y} , \end{aligned}$$

(32)

here $n,m=1,2,3, \ldots$ are the quantum numbers.

At scattering by acoustic and optic phonons integration (11) gives the overlap integrals in the following forms

$$\begin{aligned} A_{nm}^{(A)}= & {} \frac{1}{I_n I_m }\left[ {\int \limits _0^\infty {Ai\left( {y+a_n } \right) Ai\left( {y+a_m } \right) \mathrm{d}y} } \right] ^{2}\simeq \delta _{nm} ,\nonumber \\ \end{aligned}$$

(33)

$$\begin{aligned} A_{v,n,m}^{(O)}= & {} \frac{A_{nm} }{L_E^{(v)} },\nonumber \\ A_{nm}= & {} \frac{1}{2\pi I_n I_m } \int \limits _{-\infty }^\infty \left| \int \limits _0^\infty \exp \left( {-ixy} \right) Ai\left( {y+a_n } \right) \right. \nonumber \\&\left. Ai\left( {y+a_m } \right) \mathrm{d}y \right| ^{2}\mathrm{d}x , \end{aligned}$$

(34)

Here $A_{11} =0.417$, $A_{12} =0.158$, $A_{22} =0.291$, $A_{23} =0.135$, $A_{33} =0.236$.

At scattering by charged scattering centers the overlap integral is a function of the transfer momentum [30, 44]

$$\begin{aligned} A_{n,m}^{(I)} \left( {q_\bot } \right) =\int \limits _0^\infty {\exp \left( {-\left| {q_\bot } \right| z} \right) \xi _n \left( z \right) \xi _m \left( z \right) } \mathrm{d}z. \end{aligned}$$

(35)

Using expression (31) and (32) we can write integral (35) in the following form

$$\begin{aligned} A_{v,n,m}^{(I)} (q_\bot )= & {} \frac{1}{I_n I_m }\int \limits _0^\infty \exp \left( {-\left| {\vec {q}_\bot } \right| L_E^{(v)} y} \right) \nonumber \\&Ai\left( {y+a_n } \right) Ai\left( {y+a_m } \right) \mathrm{d}y. \end{aligned}$$

(36)

At scattering by surface roughness the overlap integral is [17, 30, 45]

$$\begin{aligned} A_{nm}^{(\mathrm{SR})}= & {} \left[ {\left. {\frac{\hbar ^{2}}{2m_{zz}^{(v)} }\frac{\mathrm{d}\xi _n }{\mathrm{d}z}\frac{\mathrm{d}\xi _m }{\mathrm{d}z}} \right| _{z=0} } \right] ^{2}\nonumber \\= & {} \left[ {\left. {eE_\mathrm{eff} L_E^{(v)3}\frac{\mathrm{d}\xi _n }{\mathrm{d}z}\frac{\mathrm{d}\xi _m }{\mathrm{d}z}} \right| _{z=0} } \right] ^{2}=e^{2}E_\mathrm{eff} ^{2}. \end{aligned}$$

(37)