Electronic structure of semiconductor nanowires

We compute the subband structure of several group IV and III-V $⟨001⟩$-, $⟨110⟩$-, and $⟨111⟩$-oriented nanowires using $s{p}^3$ and $s{p}^3{d}^5{s}^{*}$ tight-binding models. In particular, we provide the band gap energy of the nanowires as a function of their radius $R$ in the range $R=1–20\mathrm{nm}$. We then discuss the self-energy corrections to the tight-binding subband structure, that arise from the dielectric mismatch between the nanowires (with dielectric constant ${\epsilon }_{\mathrm{in}}$) and their environment (with dielectric constant ${\epsilon }_{\mathrm{out}}$). These self-energy corrections substantially open the band gap of the nanowires when ${\epsilon }_{\mathrm{in}}>{\epsilon }_{\mathrm{out}}$, and decrease slower $(\propto 1∕R)$ than quantum confinement with increasing $R$. They are thus far from negligible in most experimental setups. We introduce a semi-analytical model for practical use. This semianalytical model is found in very good agreement with tight-binding calculations when ${\epsilon }_{\mathrm{in}}>{\epsilon }_{\mathrm{out}}$.

Figure 1

Band structure of a $⟨111⟩$-oriented Si nanowire with radius $R=3.75\mathrm{nm}$ ($s{p}^3{d}^5{s}^{*}$ TBB model).

Figure 2

(a) Electron and (b) hole confinement energies in $⟨111⟩$-oriented Si nanowires, in the $s{p}^3{d}^5{s}^{*}$ TBB and JMJ models, and in the $s{p}^3$ YMN model. ${\epsilon }_{\mathrm{c}}(\infty )$ and ${\epsilon }_{\mathrm{v}}(\infty )$ are the bulk band edges. The solid lines are fits to the TB data according to Eq. (2) and Table .

Figure 3

Electron (lowest conduction subband) and hole (highest valence subband) effective masses in $⟨111⟩$-oriented Si nanowires, in the $s{p}^3{d}^5{s}^{*}$ TBB model and in the $s{p}^3$ YMN model.

Figure 4

The splitting $\Delta$ between the conduction band minimum at $k=0$ and the conduction band minimum at $k\simeq \pm 0.4\pi ∕\ell$ ($⟨001⟩$-oriented Si nanowires) or at $k\simeq \pm 0.8\pi ∕\ell$ ($⟨110⟩$-oriented Si nanowires). The solid and dashed lines are fits to the TB data according to Eq. (2) and Table (TBB and YMN models).

Figure 5

Band structure of a $⟨111⟩$-oriented Ge nanowire with radius $R=3.75\mathrm{nm}$ ($s{p}^3{d}^5{s}^{*}$ TBB model).

Figure 6

Electron confinement energies and masses (inset) in $⟨111⟩$-oriented $\mathrm{InAs}$ nanowires, in the $s{p}^3{d}^5{s}^{*}$ TBB model, and in the $s{p}^3$ YMN model. The solid line is a fit to ${\epsilon }_{\mathrm{c}}\left(R\right)-{\epsilon }_{\mathrm{c}}(\infty )$ according to Eq. (2) and Table . The bulk conduction band effective mass in $\mathrm{InAs}$ is $m^{*}=0.023m_0$ (horizontal dash-dotted line).

Figure 7

Comparison between the TB band gap energies ($s{p}^3{d}^5{s}^{*}$ TBB model) and (i) the local density approximation (LDA) for $⟨001⟩$-oriented (Ref. 32) and $⟨110⟩$-oriented Si nanowires; (ii) the scanning tunneling spectroscopy (STS) data of Ref. 74 for $⟨110⟩$- and $⟨112⟩$-oriented Si nanowires. Spin-orbit coupling was not taken into account in the LDA and TB calculations, which only affects the band gap energies by a few meV.

Figure 8

The semiclassical self-energy potential $\Sigma \left(r\right)$ in a Si nanowire with radius $R=3.75\mathrm{nm}$ $({\epsilon }_{\mathrm{in}}=11.7)$, for ${\epsilon }_{\mathrm{out}}=1$ and ${\epsilon }_{\mathrm{out}}=50$.

Figure 9

The prefactor $\alpha$ of the self-energy correction $⟨\Sigma ⟩\left(R\right)=\alpha ∕R$ [Eq. (10)] as a function of ${\epsilon }_{\mathrm{out}}$ for ${\epsilon }_{\mathrm{in}}=11.7$, and as a function of ${\epsilon }_{\mathrm{in}}$ for ${\epsilon }_{\mathrm{out}}=1$. The vertical line is the dielectric constant of bulk Si, ${\epsilon }_{\mathrm{in}}=11.7$.

Figure 10

The conduction (a) and valence (b) band edge energies $E_{\mathrm{c}}\left(R\right)$ and $E_{\mathrm{v}}\left(R\right)$, as well as the self-energy corrections $⟨{\Sigma }_{\mathrm{c}}⟩\left(R\right)$ and $⟨{\Sigma }_{\mathrm{v}}⟩\left(R\right)$ in $⟨111⟩$-oriented Si nanowires (${\epsilon }_{\mathrm{out}}=1$, $s{p}^3{d}^5{s}^{*}$ TBB model). The results from a “full' calculation [including $\pm \Sigma \left(r\right)$ in the Hamiltonian] and from first-order perturbation theory (pert) are shown. The solid and dashed lines are the results from the fits of Sec. 2B [Eq. (2) and Table ], and from the semianalytical model [Eqs. (10, 11)].

Figure 11

The self-energy correction ${\Sigma }_{nk}=\pm ⟨{\phi }_{nk}\mid \Sigma \mid {\phi }_{nk}⟩$ as a function of the bare quasiparticle energy ${\epsilon }_{nk}=⟨{\phi }_{nk}\mid H_{\mathrm{TB}}\mid {\phi }_{nk}⟩$ for a $⟨111⟩$-oriented Si nanowire with radius $R=3.75\mathrm{nm}$ (${\epsilon }_{\mathrm{out}}=1$, full calculation, TBB model). The 48 lowest conduction subbands and the 48 highest valence subbands at $57k$ points in $[0,\pi ∕\ell [$ are represented. The vertical lines are the bulk conduction and valence band edges, while the horizontal lines are the self-energy corrections computed from Eqs. (10, 11).

Figure 12

The conduction (a) and valence (b) band edge energies $E_{\mathrm{c}}\left(R\right)$ and $E_{\mathrm{v}}\left(R\right)$, as well as the self-energy corrections $⟨{\Sigma }_{\mathrm{c}}⟩\left(R\right)$ and $⟨{\Sigma }_{\mathrm{v}}⟩\left(R\right)$ (insets) in $⟨111⟩$-oriented Si nanowires (${\epsilon }_{\mathrm{out}}=50$, $s{p}^3{d}^5{s}^{*}$ TBB model). The results from a “full' calculation [including $\pm \Sigma \left(r\right)$ in the Hamiltonian] and from first-order perturbation theory (pert) are shown. The solid and dashed lines are the results from the fits of Sec. 2B [Eq. (2) and Table ], and from the semianalytical model [Eqs. (10, 11)].