4.1 Electron Confinement at the Semiconductor-Oxide Interface

In order to calculate the subband structure in the inversion layer the Schrödinger equation and the Poisson equation have to be considered as a coupled system of differential equations, which has to be solved self-consistently by numerical methods [Vasileska00]. The energy levels $E$ and envelope wavefunctions $\psi$ are determined by a solution of the effective Schrödinger equation

$$[T-e\Phi (z)]\psi = E\psi\ ,$$ (4.1)

where $\Phi (z)$ is the electrostatic potential and $T$ is the operator for the kinetic energy. The electrostatic potential determining the shape of the potential well is the solution of the Poisson equation

$$\nabla^2 \Phi = - {\frac{e}{\ensuremath{\kappa_\mathrm{si}}{}}} [N_{\mathrm{dop}}(z) + p(z) - n(z)]\ .$$ (4.2)

Here, $N_{\mathrm{dop}}(z)$ is the doping profile in the semiconductor, and $p(z)$ and $n(z)$ denote the hole and electron concentration, respectively. The boundary conditions for the potential are $\lim_{z \rightarrow \infty} \Phi(z) = 0$ for the bulk interface and

$$\ensuremath{\kappa_\mathrm{diel}}\frac{\mathrm{d}\Phi}{\mathrm{d}... ...} = \kappa_\mathrm{sc} \frac{\mathrm{d}\Phi}{\mathrm{d}z}{\Big\vert}_{z=0^+}\ ,$$ (4.3)

for the Si-SiO$_2$ interface. In (4.3) $\kappa_\mathrm{diel}$ denotes the dielectric permittivity of the insulating layer and $\kappa_\mathrm{sc}$ that of the semiconductor. Assuming the effective mass approximation, the kinetic energy operator $T$ in (4.1) can be written as

$$T=\frac{1}{2} \sum_{i,j} \nu_{ij} p_ip_j\ ,\qquad i,j = x,y,z\ .$$ (4.4)

where $p_j=-i\hbar (\partial/\partial x_j)$ denotes the momentum operator, and $\nu_{ij}$ is the reciprocal effective mass tensor. A coordinate system is chosen such that the $z$ axis is normal to the semiconductor-insulator interface.

Figure 4.1: Angles $\phi$ and $\theta$ with respect to the coordinate system which diagonalizes the inverse effective mass tensor.

The conduction band valleys of semiconductors are typically oriented along high symmetry lines of the first Brillouin zone. To calculate the subband structure for any substrate orientation, it is necessary to introduce a unitary transformation from the crystallographic system $x',y',z'$ to the interface coordinate system. The momentum operator and the reciprocal effective mass tensor in the interface coordinate system are transformed as

$$p_j = \sum_{k} u_{jk}{p_k}'\ ,$$ (4.5)

$$\nu_{ij} = \sum_{k} u_{ik} u_{jk} {\nu_{kk}}'\ .$$ (4.6)

Here, $u_{jk}$ are the elements of a unitary matrix, ${\nu_{kk}}'=1/{m_k}'$, where ${m_k}'$ denote the principal effective masses of the constant-energy ellipsoid in the semiconductor. The unitary transformation matrix from the crystallographic coordinate system to the interface coordinate system is given by

$$\ensuremath{{\underaccent{\bar}{U}}}=\begin{pmatrix}\cos \phi \co... ...ta& \cos\phi & \sin\phi\sin\theta \\ -\sin\theta& 0 & \cos\theta \end{pmatrix},$$ (4.7)

and involves a rotation of $\phi$ about the $z'$ axis followed by a subsequent rotation of $\theta$ about the new $y$ axis (compare Figure 4.1). The direction of the new $z$ axis after the two rotations is thus given by the last column of ${\underaccent{\bar}{U}}$,

$${\ensuremath{\mathitbf{e}}}_z = (\cos\phi\sin\theta, \sin\phi\sin\theta, \cos\theta)\ .$$ (4.8)

Since the potential is assumed to be a function of $z$ only, it is possible to the separate the trial solution of (4.1) into a $z$-dependant factor $\xi(z)$, and a plane wave factor representing free motion in the $xy$ plane [Stern67]

$$\psi(x,y,z) = \xi(z) \exp(ik_1x+ik_2y)\ .$$ (4.9)

The functions $\xi$ must satisfy the equation

$$\frac{\hbar^2}{2}\nu_{33} \frac{\mathrm{d}^2 \xi}{\mathrm{d}z^2} ... ...u_{23}k_2)\frac{\mathrm{d} \xi}{\mathrm{d}z} +(e\phi(z) + \hat{E})\xi(z) = 0\ ,$$ (4.10)

where

$$\hat E=E-\frac{\hbar^2}{2}(\nu_{11}k_1^2 + 2\nu_{12}k_1k_2+\nu_{22}k_2^2)\ .$$ (4.11)

Following Stern and Howard [Stern67], the first derivative in the above equation can be eliminated using the substitution^4.1

$$\xi(z) = \zeta(z)\exp\left (-\frac{iz}{\nu_{33}}(\nu_{13}k_1 + \nu_{23} k_2)\right ).$$ (4.12)

The differential equation for $\zeta(z)$ takes the form

$$\frac{\mathrm{d}^2\zeta_i(z)}{\mathrm{d}z^2} + \frac{2m_\perp}{\h... ...(E_i + e\phi(z))\zeta_i(z)=0\ ,\quad\mathrm{using}\quad m_\perp = 1/\nu_{33}\ .$$ (4.13)

The eigenfunctions being subject to the boundary conditions $\lim_{z \rightarrow \infty} \zeta_i(z) = \lim_{z \rightarrow 0} \zeta_i(z) = 0$ and the eigenvalues of (4.13) are labeled by a subscript $i$. The energy spectrum is given by

$$E(k_1,k_2)= E_i + \frac{\hbar^2}{2}\left [ \left ( \nu_{11} - \fr... ...k_1k_2 + \left ( \nu_{22} - \frac{\nu_{23}^2}{\nu_{33}}\right ) k_2^2 \right ].$$ (4.14)

and represents constant-energy ellipses above the minimum energy $E_i$. The fact that $E_i$ is independent of $k_1$ and $k_2$ is a result of the boundary condition $\zeta(0)=0$. The energy levels $E_i$ for a given value of $m_\perp$ generate a set of subband minima called subband ladder. Since the value of the quantization mass depends on the substrate orientation, so do the number and the degeneracy of the subband ladders. Obviously, if conduction band valleys have the same orientation with respect to the surface, these valleys belong to the same subband ladder. Because of the kinetic energy term (4.4) in the Schrödinger equation the valleys with the largest quantization mass $m_\perp$ have the lowest energy. Following a widely used convention, the subbands belonging to the ladder lowest in energy are labeled $0,1,2,\ldots$, those of the second ladder $0',1',2',\ldots$, the third ladder , and so on [Ando82].

The principal effective masses $m_{\shortparallel ,1}$ and $m_{\shortparallel ,2}$, associated with motion parallel to the surface can be deduced from (4.14). This equation represents an ellipse whose principal axes are not parallel to $k_1$ and $k_2$. Introducing the matrix

$$\ensuremath{{\underaccent{\bar}{M}}} = \begin{pmatrix}\nu_{11}-{\... ...nu_{12}-\nu_{13}\nu_{23}/\nu_{33} & \nu_{22}-\nu_{23}^2/\nu_{33} \end{pmatrix},$$ (4.15)

equation (4.14) can be written as

$$E(k_1,k_2)= E_i + \frac{\hbar^2}{2} {\ensuremath{\mathitbf{k}}}^T \ensuremath{{\underaccent{\bar}{M}}} {\ensuremath{\mathitbf{k}}}\ .$$ (4.16)

The inverse effective masses $m_{\shortparallel,1}^{-1}$ and $m_{\shortparallel,2}^{-1}$ are the eigenvalues of $\ensuremath{{\underaccent{\bar}{M}}}$ and can be calculated by solving the secular equation

$$\det\left (\ensuremath{{\underaccent{\bar}{M}}} - m_{\shortparallel}^{-1} \mathbb{1}\right )=0\ ,$$ (4.17)

where $\mathbb{1}$ denotes the two dimensional unity matrix.

Subsections

• 4.1.1 The Si-SiO$_2$ Interface
• 4.1.2 Substrate Orientation (001)
• 4.1.3 Substrate Orientation (110)
• 4.1.4 Substrate Orientation (111)

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