Valence band structure calculations of strained Ge$_{1-x}$Sn$_x$ quantum well pFETs

TL;DR

This paper investigates how Sn content, strain, and substrate orientation affect the valence band structure of GeSn quantum wells in pFETs, using empirical pseudopotential and k·p models to reveal effects on hole populations and effective mass.

Contribution

It provides a detailed theoretical analysis of valence band structures in strained GeSn quantum wells, incorporating wave-function coupling effects influenced by cap thickness and band offsets.

Findings

Increased Sn content raises hole population in the quantum well.

Thinner caps and higher Sn content reduce the transport effective mass.

Valence band structure depends on substrate orientation and strain conditions.

Abstract

The dependence of valence band structures of Ge$_{1-x}$Sn$_x$ with 0 $\leq$ $x$ $\leq$ 0.2 on Sn content, biaxial strain, and substrate orientation is calculated using the nonlocal empirical pseudopotential method. The first valence subband structure in p-type Ge cap/fully strained Ge$_{1-x}$Sn$_x$ quantum well/Ge (001) and (111) inversion layers are theoretically studied using the 6$\times$6 k$\cdot$p model. A wave-function coupling of a Ge cap with respect to a strained Ge$_{1-x}$Sn$_x$ quantum well, which is influenced by the cap thickness, valence band offset, and confined effective mass, changes the energy dispersion relation in the two-dimensional $k$-space. The increase in Sn content and the decrease in cap thickness increase the hole population in the strained Ge$_{1-x}$Sn$_x$ quantum well to reduce the transport effective mass at the zone center in the Ge/strained Ge$_{1-x}$Sn$_x$/Ge inversion layers.

Figures

Tables

Table 1. Table 1: Fitted 6 × \times 6 k ⋅ ⋅ \cdot p parameters.

	Symbol	Ge	Sn
Luttinger parameters	${\gamma }_1$	9.4415	$-$6.8605
	${\gamma }_2$	2.7433	$-$5.8624
	${\gamma }_3$	3.9298	$-$4.3485
Deformation potentials (eV)	$b$	$-$2.06	$-$2.3
	$d$	$-$7.4608	$-$4.1
Additional internal strain parameter (eV)	${\xi }_b$	$-$0.2322	2.872

Table 2. Table 2: Fitted bowing factors.

Bowing factor	Symbol	GeSn
Luttinger parameters	${\theta }_{\gamma 1}$	$-$66.6075
	${\theta }_{\gamma 2}$	$-$32.6386
	${\theta }_{\gamma 3}$	$-$32.8118
Deformation potentials (eV)	${\theta }_b$	$-$0.5502
	${\theta }_d$	7.906

Equations

$$P^{\mathrm{GeSn}}(x)=(1-x)P^{\mathrm{Ge}}+x{P}^{\mathrm{Sn}}-x(1-x){\theta }_P,$$

$$b_{\mathrm{mod}}^{\mathrm{GeSn}}(x)=b^{\mathrm{GeSn}}(x)+(1-x){\xi }_b^{\mathrm{Ge}}+x{\xi }_b^{\mathrm{Sn}},$$

$$E(K,\varphi )=E_{1^{\mathrm{st}}}+\frac{{\hslash }^{2}K(\varphi {)}^2}{2m^{\ast }(K,\varphi )},$$

Full text

• March 6 2017

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1 Introduction

Sn incorporating into Ge potentially makes Ge1-xSnx alloys as a direct-gap semiconductor [1, 2, 3, 4, 5], which make them suitable for use in n-channel field effect transistors (nFETs) owing to the small effective mass of the $\Gamma$ valley [6]. Several studies on GeSn pFETs show high hole mobilities of strained GeSn grown on Ge (001) [7, 8], (110) [9], and (111) [10, 11] substrates. The reduction in effective mass along the channel direction ($m_{\mathrm{chan}}$) due to biaxial compressive strain is responsible for the increased hole mobility [8, 12]. In GeSn/Ge (001), the biaxial compressive strain lifts the light hole band along the [110] direction [8]. Increasing the Sn content can further reduce the light hole effective mass [4]. In addition, the Ge cap on the GeSn channel can prevent holes from scatterings at the Ge/oxide interface [8]. Nevertheless, knowledge of valence subband structures in GeSn inversion layers is very limited owing to the unknown 6$\times$6 k$\cdot$p Luttinger parameters and deformation potential parameters of GeSn alloys. Note that linearly interpolated 6$\times$6 k$\cdot$p parameters based on the linear virtual crystal approximation (VCA) have been widely used in calculations related to valence subband structures in the SiGe inversion layers [13, 14]. Low’s pioneering approach [4] fitted multiple sets of nonlinear Luttinger parameters ($\gamma_{1}$, $\gamma_{2}$, and $\gamma_{3}$) of relaxed GeSn alloys from full-band structures calculated using the nonlocal empirical pseudopotential method (EPM) for discrete Sn content. However, this approach increases the computational complexity, as it requires taking different strains and proportions of Sn content into account. In this work, one set of 6$\times$6 k$\cdot$p parameters for Ge1-xSnx alloys, where 0 $\leq$ $x$ $\leq$ 0.2 with in-plane biaxial strain of $-$3% $\leq$ ${\varepsilon}_{||}$ $\leq$ 1%, are fitted based on the calculated valence band structures and effective masses of strained GeSn (001), (110), and (111) wafer orientations using EPM. The fitted 6$\times$6 k$\cdot$p parameters are used to calculate the valence subband structures in inversion layers of Ge cap/10 nm GeSn quantum well (QW)/Ge (001) and (111) pFETs. Most of the holes in these cases are located in the first subband, of which favorable for carrier transport is implied by a small $m_{\mathrm{chan}}$. The dependence of $m_{\mathrm{chan}}$ of the first subband at the zone center on the Sn content, cap thickness, and wafer orientation is theoretically studied.

2 Theoretical Framework

The calculation flowchart in this work is summarized in figure 1. To study valence subband structures in the inversion layer of Ge cap/GeSn QW/Ge pFETs on (001) and (111) Ge substrates, the wafer orientation and the corresponding transferred strain tensor in the crystal coordinate system [15] are determined first. Effective masses (m*∗) and valence band structures near the $\Gamma$ point ($k$ = [$-$0.05 : 0.05] in 2$\pi$/$a_{0}$ unit) along [100], [001], [110], [$-$110], and [111] directions, and band energies at the $\Gamma$ point of Ge1-xSnx* with 0 $\leq$ $x$ $\leq$ 0.2 under $-$3, $-$2, $-$1, 0, and 1% in-plane biaxial strain on (001), (110), and (111) wafer orientations are calculated using EPM and used to fit 6$\times$6 k$\cdot$p parameters through the 6$\times$6 k$\cdot$p model [13, 16]. The fitted 6$\times$6 k$\cdot$p parameters are obtained by solving nonlinear least squares with the above values calculated by EPM. Note that our EPM [17] has been reported in elsewhere and the calculated bandgaps agree with the experimental data. Linear interpolation of elastic constants (C11, C12, and C44) [18], and a bowing factor of 0.047 Å [19] for the lattice constant ($a_{0}$) of GeSn alloys are used in the calculations.

The fitted Luttinger parameters ($\gamma_{1}$, $\gamma_{2}$, and $\gamma_{3}$) and deformation potential parameters ($b$ and $d$) of Ge and Sn are listed in table 1. The fitted Luttinger parameters of Ge are similar with the previous reported results [20]. To take into account GeSn alloys, the 6$\times$6 k$\cdot$p parameters are expressed as a quadratic polynomial

[TABLE]

where $P$ can be $\gamma_{1}$, $\gamma_{2}$, $\gamma_{3}$, $b$ or $d$, and the coefficient $\theta_{P}$ is a corresponding bowing factor listed in table 2. A linearly interpolated hydrostatic deformation potential for the valence band of GeSn alloys is obtained from reference [21] ($a_{\mathrm{v}}^{Ge}$ = 2.23 eV and $a_{\mathrm{v}}^{Sn}$ = 1.58 eV). Moreover, for strained Ge1-xSnx (110) and (111) substrates with in-plane biaxial strain, there is a relative displacement of two atoms in the unit cell due to generated shear strains [22, 23]. An additional internal strain parameter ($\xi$) was used in the EPM calculation to take into account the internal strain effects [24]. The used $\xi$ of Ge1-xSnx in EPM is obtained by the linear interpolation of the reported $\xi$ of Ge [22] and Sn [25]. In 6$\times$6 k$\cdot$p model, the internal strain effect was usually ignored due to few information for parameter correction [14, 26]. To consider the effect in 6$\times$6 k$\cdot$p model, we use an additional parameter ($\xi_{b}$) to modify the fitted deformation potential parameter ($b$) of Ge1-xSnx as

[TABLE]

where the $b$ of Ge1-xSnx is obtained from equation (1), and the fitted $\xi_{b}$ of Ge and Sn are listed in table 1.

The calculated valence band structures of Ge1-xSnx along [$-$110] and [100] directions with varying Sn content ($x$ = 0, 0.1, and 0.2) and in-plane biaxial compressive strain (${\varepsilon}_{||}$ = 0 and $-$1%) using EPM (solid line) and 6$\times$6 k$\cdot$p (dashed line) are shown in figure 2. The agreements in band energies between the two methods at the $\Gamma$ point and two uppermost band structures near the $\Gamma$ point are presented. The split-off band structure fitting is ignored to obtain best fit of the other two valence bands.

The calculated effective masses of the heavy hole bands ($m_{\mathrm{HH}}$) along $\langle 111\rangle$, $\langle 110\rangle$, and $\langle 100\rangle$ directions, and light hole bands ($m_{\mathrm{LH}}$) along $\langle 110\rangle$ directions of relaxed Ge1-xSnx (r-Ge1-xSnx) alloys using EPM and 6$\times$6 k$\cdot$p are shown in figure 3(a). The rapid decrease of $m_{\mathrm{LH}}$ along $\langle 110\rangle$ direction with increasing $x$ for r-Ge1-xSnx is compared against the slow decrease of $m_{\mathrm{HH}}$ along $\langle 111\rangle$, $\langle 110\rangle$, and $\langle 100\rangle$ directions. In strained Ge1-xSnx (s-Ge1-xSnx) (001) with ${\varepsilon}_{||}$ = $-$1% (figure 3(b)), the effective masses along [110] and [$-$110] directions are the same due to symmetry of the band structures. For s-Ge1-xSnx (110) and (111) with ${\varepsilon}_{||}$ = $-$1%, the symmetry is broken by the strain and the top valence band ($m_{\mathrm{top}}$) along [$-$110] direction is smaller than that along [110] (figure 3(c) and (d)). The slow decrease of $m_{\mathrm{top}}$ along the out-of-plane direction in s-Ge1-xSnx (001), (110), and (111) is compared against the rapid decrease of $m_{\mathrm{top}}$ along the other directions. Note that in-plane $\langle 110\rangle$ and out-of-plane directions are usually defined as channel and confined directions, respectively. The $m_{\mathrm{top}}$ calculated by 6$\times$6 k$\cdot$p based on VCA (dashed lines) along [110], [$-$110], and [$-$110] directions for s-Ge1-xSnx (001) (figure 3(b)), (110) (figure 3(c)), and (111) (figure 3(d)) with ${\varepsilon}_{||}$ = $-$1%, respectively, are also shown for comparison. The used linearly interpolated 6$\times$6 k$\cdot$p parameters are adopted from literatures [20, 27, 28].

After fitting 6$\times$6 k$\cdot$p parameters, the self-consistently calculation between 6$N_{z}\times 6N_{z}$ k$\cdot$p Schrödinger and Poisson equations [29, 30] is performed to calculate the two-dimensional hole gas system (2DHG) as shown in figure 1. The $N_{z}$ denotes mesh points along the confined $z$ direction. A uniform mesh of 2 Å size is used. Linear interpolation of dielectric constants of GeSn alloys are used [18]. A valence band offset ($\Delta E_{\mathrm{v}}$) at a heterointerface calculated by the model-solid theory (MST) [31, 32] are used to determine the valence band lineup potential $V(z)$ in the 6$N_{z}\times 6N_{z}$ k$\cdot$p Schrödinger equation. Note that a proposed correction term on the average valence band offset of GeSn alloys [33] is used in MST instead of the linearly interpolated average valence band offset [34]. Our reported calculated \Delta$$E_{\mathrm{v}} of $\sim$80 meV in s-Ge0.95Sn0.05/r-Ge by MST [31] is within the reported measurement accuracy range of $\pm$ 50 meV for $x$ $\leq$ 0.08 [35]. A duplicate valence band edge (VB) shift in the $V(z)$ potential by strain in the 6$N_{z}\times 6N_{z}$ k$\cdot$p matrix is removed. The treatment of the 6$N_{z}\times 6N_{z}$ k$\cdot$p Hamiltonian matrix at a heterointerface is referred to reference [36]. A 2D density of state (DOS) of each subband is calculated from a tabulation of each subband structure and used to determine the hole density in the inversion layer [37]. The effective mass Schrödinger equations are also performed to calculate the electron density in conduction band heterostructures [38]. A conduction band offset (\Delta$$E_{\mathrm{c}}) at a heterointerface is determined by the corresponding bandgaps and \Delta$$E_{\mathrm{v}}. Uniform doping density $N_{\mathrm{D}}$ of 1$\times 10^{14}$ cm*-3* is used in the Poisson equation [30]. The Anderson mixing method is used to accelerate the convergence of iterative solution [39].

3 Results and Discussion

At the total inversion hole density $N_{\mathrm{inv}}$ = 5$\times 10^{12}$ cm*-2*, the calculated inversion layers in 3 nm and 1 nm Ge cap/10 nm s-Ge1-xSnx QW on Ge (001) substrate with the first subband hole density distribution are shown in figure 4(a) ($x$ = 0.05) and 4(c) ($x$ = 0.1). The Fermi level energy is fixed at 0 eV in the valence band. The quantized energy of the first subband ($E_{1^{\mathrm{st}}}$) crosses the Ge cap and s-Ge1-xSnx QW ($x$ = 0.05 and 0.1) layers. For $x$ = 0.05 (figure 4(a)), there is a substantial hole density of $E_{1^{\mathrm{st}}}$ in the 3 nm cap as compared to a low hole density of $E_{1^{\mathrm{st}}}$ in the 1 nm cap. These hole distributions influence the first subband constant 2D energy contour of the Ge/s-Ge0.95Sn0.05/Ge (001) inversion layer as shown in figure 4(b). Note that the first subband energy dispersion can be expressed as

[TABLE]

where $K$ is the in-plane vector, $\phi$ is the polar angle in the range [0, $\pi$/2] from $k_{x}$, and $m^{\ast}$ is indirectly proportional to $\partial^{2}E(K,\phi)$/$\partial K(\phi)^{2}$ [30, 37]. The energy dispersion is hybridized by the Ge cap and s-Ge0.95Sn0.05 QW. The rapid change in energy (in meV) with increasing $K$ for the 1 nm cap is compared against that for the 3 nm cap (figure 4(b)). For $x$ = 0.1 (figure 4(c)), the hole density of $E_{1^{\mathrm{st}}}$ in the QW layer for the 3 nm and 1 nm cap are similar, which is due to the increase in \Delta$$E_{\mathrm{v}} with increasing $x$. As a result, there is a similar change in energy with increase in $K$ for the first subband constant 2D energy contours of the 3 nm and 1 nm cap/s-Ge0.9Sn0.1/Ge (001) inversion layers (figure 4(d)).

For different orientations, the calculated inversion layers of the 4 nm Ge cap/10 nm s-Ge0.9Sn0.1 QW on Ge (001) and (111) substrates at $N_{\mathrm{inv}}$ = 5$\times 10^{12}$ cm*-2* with the corresponding first subband hole density distribution are shown in figure 5(a). The hole density of $E_{1^{\mathrm{st}}}$ in the cap layer for (111) substrate is lower compared to that for (001) in spite of the similar \Delta$$E_{\mathrm{v}}. The $E_{1^{\mathrm{st}}}$ of (111) is more confined towards the inside than that of (001), which results in low hole density in the cap layer owing to the larger confined mass of (111) (similar to the out-of-plane values of $m_{\mathrm{top}}$ in figure 3(b) and (d)). The first subband constant 2D energy contours of the Ge/s-Ge0.9Sn0.1/Ge (001) and (111) inversion layers are shown in figure 5(b) for comparison. The $k_{x}/k_{y}$ in the device coordinate system [15] are [100]$/$[010] and [11$-$2]$/$[$-$110] for (001) and (111), respectively. To study the changes in $m_{\mathrm{chan}}$, $m^{\ast}$ at the zone center is calculated using equation (3) with $\phi$ = $\pi$/4 and $\pi$/2 for the channel direction/wafer orientation of [110]/(001) and [$-$110]/(111), respectively.

The calculated $m_{\mathrm{chan}}$ and hole population in QW as a function of cap thickness of the first subband in the Ge cap/s-Ge1-xSnx ($x$ = 0.05 and 0.1)/Ge (001) and (111) inversion layers at $N_{\mathrm{inv}}$ = 5$\times 10^{12}$ cm*-2* are shown in figure 6. The $m_{\mathrm{chan}}$ in the (111) inversion layer (figure 6(c)) is smaller than that in the (001) inversion layer (figure 6(a)) owing to quantum confinement. This is different from the trend seen in bulk effective masses of s-Ge1-xSnx [110]/(001) (figure 3(b)) and [$-$110]/(111) (figure 3(d)). Note that the quantum confinement also causes drastic warped valence subband structures in the case of (110) [40, 41], which is not discussed here. The $m_{\mathrm{chan}}$ is reduced with increasing Sn content or decreasing cap thickness. This trend is contrary to the increased $m_{\mathrm{chan}}$ calculated by 6$\times$6 k$\cdot$p (VCA). In the case of 0 nm cap, there is $\sim$20% reduction of $m_{\mathrm{chan}}$ with an increase of 5% in Sn. The increase in $m_{\mathrm{chan}}$ with increasing cap thickness is approximately similar to the increase in the average of $m_{\mathrm{chan}}$ in Ge cap and s-Ge1-xSnx QW weighted by its corresponding hole population.

4 Summary

One set of 6$\times$6 k$\cdot$p Luttinger and deformation potential parameters for valence band structures and effective masses of Ge1-xSnx (001), (110), and (111) wafer orientations, where 0 $\leq$ $x$ $\leq$ 0.2 with in-plane biaxial strain of $-$3% $\leq$ ${\varepsilon}_{||}$ $\leq$ 1% are fitted based on the calculated valence band structures by EPM. The internal strain effects by shear strain were taken into account in the EPM and 6$\times$6 k$\cdot$p calculations. The first valence subband structures in the p-type Ge cap/10 nm s-Ge1-xSnx QW/Ge (001) and (111) inversion layers were theoretically studied using the self-consistent calculation of 6$N_{z}\times 6N_{z}$ k$\cdot$p Schrödinger and Poisson equations. The valence band lineups at the heterointerfaces were theoretically determined using the model-solid theory. In the QW structures, the (111) has a lower $m_{\mathrm{chan}}$ in the first subband than (001) owing to quantum confinement. The Sn content controls the effective mass and \Delta$$E_{\mathrm{v}}, and the \Delta$$E_{\mathrm{v}} and cap thickness control the hole population in the QW to affect the $m_{\mathrm{chan}}$ of the first subband. Using the fitted 6$\times$6 k$\cdot$p parameters, the increase in Sn content and the decrease in cap thickness reduce the $m_{\mathrm{chan}}$ as compared to the incorrect increased $m_{\mathrm{chan}}$ using the reported linearly interpolated 6$\times$6 k$\cdot$p parameters of Ge and Sn.

This work was supported by Ministry of Science and Technology, Taiwan, R.O.C under Grant Nos. 105-2622-8-002-001-, 105-2911-I-009-301, and 103-2221-E-002-232-MY3. The support of high-performance computing facilities by the Computer and Information Networking Center, National Taiwan University, is also highly appreciated.