5.4 Carrier Mobility

Lead telluride attracts attention due to its extraordinarily high carrier mobilities at low temperatures. Values of $800,000\,\ensuremath{\mathrm{cm^2/Vs}}$ and $256,000\,\ensuremath{\mathrm{cm^2/Vs}}$ have been reported for electrons and holes, respectively at $4.2\,\ensuremath{\mathrm{K}}$ , which reduce to $1,900\,\ensuremath{\mathrm{cm^2/Vs}}$ and $900\,\ensuremath{\mathrm{cm^2/Vs}}$ at room temperature [270]. The mobilities for electrons and holes, $\ensuremath{\ensuremath{\mu}_{\ensuremath{n}}}$ and $\ensuremath{\ensuremath{\mu}_{\ensuremath{p}}}$ are limited by carrier scattering within the semiconductor. The electron mobility model is based on experimental and Monte-Carlo simulation data [271], while the hole mobility model relies on collected measurement data.

The Monte-Carlo technique serves as a powerful link between measurement data and models for device simulation in hierarchical device simulation [79]. Macroscopic average quantities such as carrier mobility and energy relaxation times are derived from the microscopic behavior of single electrons with statistical methods. Bulk Monte-Carlo simulations have been carried out using the Vienna Monte Carlo simulator (VMC) [204] applying a single particle Monte Carlo technique. For the analysis, the two lowest conduction band valleys at L and W points, respectively are incorporated. Several relevant stochastic mechanisms are considered, which are phonon scattering in the acoustic and optical branch, polar optical phonon scattering, optical deformation potential scattering, L-L intravalley scattering, and scattering by ionized impurities. The band structure is described by a non-parabolic approximation of the valleys using Kane's formula [77]

$$\gamma(\ensuremath{\mathcal{E}}(k)) = \ensuremath{\mathcal{E}}( 1 + \alpha \ensuremath{\mathcal{E}}) = \frac{\hbar^2 k^2}{2 m^*} \,.$$ (5.41)

Compared to widely used materials such as silicon or germanium, the material parameters of lead telluride show higher uncertainties, especially at higher temperatures. The influences of the according single material parameters on the mobility are assessed and some parameters are adjusted in an iterative process to account for available measurement data for certain doping and temperature values. This calibrated set of models finally serves as a basis for the extraction of bulk mobility data in order to further calibrate the according mobility models in the device simulator MINIMOS-NT [268].

Only a few Monte-Carlo simulations are currently documented in literature for lead telluride. The negative differential mobility at "high-field" conditions in lead telluride has been investigated in [272] at $77\,\ensuremath{\mathrm{K}}$ . This work has been extended to selected lead-tin telluride alloys in [273]. A comprehensive investigation including both measurements and Monte-Carlo simulation results for the hot-electron behavior in lead telluride as well as lead-tin telluride alloys is presented in [274]. However, all these studies are limited to a temperature of $77\,\ensuremath{\mathrm{K}}$ and focus on the influence of the W-valley. In recent work, Palankovski et al. [275] provided results for the electron mobility as a function of temperature up to $500\,\ensuremath{\mathrm{K}}$ , carrier concentration, and electric field.

In contrast to semiconductors with wider band gaps such as silicon, the temperature dependence of several parameters becomes more pronounced. Thus, these parameters are modeled accordingly by introducing temperature dependent expressions. Model parameters applied in the simulations are collected in Table 5.9.

Table 5.9: Parameters for several scattering models incorporated in Monte-Carlo simulations.

quantity	symbol	value	unit
valley separation energy	$\Delta \ensuremath{\mathcal{E}}_\ensuremath{\mathrm{WL}}$	$0.15 + 0.04 \ensuremath{\left(\ensuremath{\frac{\ensuremath{T_{\mathrm{L}}}}}{300\,\ensuremath{\mathrm{K}}}\right)}$	eV
effective masses	$m^*_{\ensuremath{\mathrm{L,l}}}$	$0.25 + 0.11 \ensuremath{\left(\ensuremath{\frac{\ensuremath{T_{\mathrm{L}}}}}{... ...ath{\frac{\ensuremath{T_{\mathrm{L}}}}}{300\,\ensuremath{\mathrm{K}}}\right)}^2$
	$m^*_{\ensuremath{\mathrm{L,t}}}$	$0.024 + 0.0112 \ensuremath{\left(\ensuremath{\frac{\ensuremath{T_{\mathrm{L}}}... ...ath{\frac{\ensuremath{T_{\mathrm{L}}}}}{300\,\ensuremath{\mathrm{K}}}\right)}^2$
	$m^*_{\ensuremath{\mathrm{W,l}}}$	$0.5$
	$m^*_{\ensuremath{\mathrm{W,t}}}$	$0.5$
sound velocities	$\ensuremath{v_{\rm {sl}}}$	$3297 - 170\ensuremath{\left(\ensuremath{\frac{\ensuremath{T_{\mathrm{L}}}}}{30... ...ath{\frac{\ensuremath{T_{\mathrm{L}}}}}{300\,\ensuremath{\mathrm{K}}}\right)}^2$	m/s
	$\ensuremath{v_{\rm {st}}}$	$2016 - 121 \ensuremath{\left(\ensuremath{\frac{\ensuremath{T_{\mathrm{L}}}}}{3... ...ath{\frac{\ensuremath{T_{\mathrm{L}}}}}{300\,\ensuremath{\mathrm{K}}}\right)}^2$	m/s
lattice constant	$a$	$6.462$	$\mathrm{\AA}$
mass density	$\rho$	$8241$	kg/m$^3$
non-parabolicity constants	$\alpha_\ensuremath{\mathrm{L}}$	$3$	1/eV
	$\alpha_\ensuremath{\mathrm{W}}$	$3$	1/eV
relative permittivities	$\ensuremath {\ensuremath {\epsilon }_{\ensuremath {\mathrm {s}}}}$	$428 - 48 \ensuremath{\left(\ensuremath{\frac{\ensuremath{T_{\mathrm{L}}}}}{300\,\ensuremath{\mathrm{K}}}\right)}$
	$\ensuremath{\ensuremath{\epsilon}_{\infty}}$	$38 - 5 \ensuremath{\left(\ensuremath{\frac{\ensuremath{T_{\mathrm{L}}}}}{300\,\ensuremath{\mathrm{K}}}\right)}$
acoustical deformation potential	$\ensuremath{\mathcal{E}}_{\ensuremath{\mathrm{ADP}}}$	$10$	eV
optical phonon energy	$\hbar \omega_{\ensuremath{\mathrm{LO}}}$	$13.6$	meV
intervalley phonon energy	$\hbar \omega_{\ensuremath{\mathrm{ij}}}$	$10.5$	meV
intervalley coupling constant	$D_{ij}$	$1.6 \times 10^8$	eV/cm
ODP coupling constant	$D_\ensuremath{\mathrm{o}}$	$1.2 \times 10^9$	eV/cm

Due to the low driving forces far below saturation effects within thermoelectric applications, mobility model parameters are obtained for the low-field case. Low-field mobilities for electrons and holes are modeled by a two-stage model [268]. The temperature dependent mobilities for intrinsic and low doped samples, where lattice scattering is the dominant scattering mechanism, are expressed by a power law

$$\ensuremath{\ensuremath{\mu}_\nu^{\ensuremath{\mathrm{L}}}}= \ens... ...,\ensuremath{\mathrm{K}}}\right)}^{\ensuremath{\gamma_{0,\ensuremath{\nu}}}}\,.$$ (5.42)

Figure 5.11: Temperature dependence of the electron mobility in lead telluride for different dopings.

Figure 5.12: Doping dependent electron mobility degradation in lead telluride at room temperature.

Figure 5.13: Temperature and doping dependent hole mobility in lead telluride.

Doping dependent mobility degradation due to ionized impurity scattering is introduced using a simplified Caughey-Thomas equation [276]

$$\ensuremath{\ensuremath{\mu}_\nu^{\ensuremath{\mathrm{LI}}}}= \fr... ...ath{\mathrm{tot}}}}{N_\nu^{\ensuremath{\mathrm{ref}}}} \right)^{\alpha_\nu}}\,,$$ (5.43)

where $\ensuremath{\ensuremath{\mu}_\nu^{\ensuremath{\mathrm{L}}}}$ depicts the temperature dependent mobilities for undoped samples (5.42) and $\ensuremath{N_\ensuremath{\mathrm{tot}}}$ stands for the total impurity concentration. The mobility degradation with increasing impurity concentration is determined by $N_\nu^{\ensuremath{\mathrm{ref}}}$ and $\alpha_\nu$ . While $N_\nu^{\ensuremath{\mathrm{ref}}}$ depicts the impurity concentration, where the according mobility becomes $\ensuremath{\ensuremath{\mu}_\nu^{\ensuremath{\mathrm{L}}}}/2$

$$N_\nu^{\ensuremath{\mathrm{ref}}} = N_{\nu,300}^{\ensuremath{\mat... ...,\ensuremath{\mathrm{K}}}\right)}^{\ensuremath{\gamma_{1,\ensuremath{\nu}}}}\,,$$ (5.44)

the exponent $\alpha_\nu$ models the gradient of the mobility degradation with increasing impurity concentration

$$\alpha_\nu = \alpha_{\nu,300} \ensuremath{\left(\ensuremath{\frac... ...,\ensuremath{\mathrm{K}}}\right)}^{\ensuremath{\gamma_{2,\ensuremath{\nu}}}}\,.$$ (5.45)

The parameters for lead telluride are collected in Table 5.10 for electrons and holes, respectively.

Table 5.10: Parameters for lead telluride mobility models.

	Electrons	Holes
$\ensuremath{\ensuremath{\mu}_{\nu,300}^{\ensuremath{\mathrm{L}}}}$	$1900\,\ensuremath{\mathrm{cm^2/Vs}}$	$900\,\ensuremath{\mathrm{cm^2/Vs}}$
${\ensuremath{\gamma_{0,\ensuremath{\nu}}}}$	$-2.4$	$-2.45$
$N_{\nu,300}^{\ensuremath{\mathrm{ref}}}$	$4.4 \times 10^{19}\,\ensuremath{\mathrm{cm^{-3}}}$	$8.2 \times 10^{19}\,\ensuremath{\mathrm{cm^{-3}}}$
${\ensuremath{\gamma_{1,\ensuremath{\nu}}}}$	$0.56$	$2.05$
$\alpha_{\nu,300}$	$0.96$	$1.38$
${\ensuremath{\gamma_{2,\ensuremath{\nu}}}}$	$-0.32$	$0.3$

The validity of the model for electrons is illustrated in Figures 5.11 and 5.12. Fig. 5.11 depicts the temperature dependent electron mobility for doping concentrations of $3 \times 10^{18}\,\ensuremath{\mathrm{cm^{-3}}}$ and $5.25 \times 10^{19}\,\ensuremath{\mathrm{cm^{-3}}}$ , respectively. The temperature dependence follows a power law, where the exponent's value of $-2.4$ for low doped samples reduces for higher dopings. Fig. 5.12 illustrates the mobility degradation with increasing dopant concentrations at room temperature. Ueta's data are based on epitaxial layers grown on $\ensuremath{\mathrm{BaF_2}}$ and are thus somewhat lower than bulk values due to additional surface scattering and lattice mismatch. An overview of hole mobility data is given in Fig. 5.13, where measurement data from [270,277,194,278,279,280,281] has been used as a basis for the hole mobility model.

M. Wagner: Simulation of Thermoelectric Devices