2.10: The 1-D DOS - Quantum Wires Confined in 2-D
- Page ID
- 50140
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)The density of states, \(g(E)\) is defined as the number of allowed states within energy range \(dE\), i.e. the total number of states within the energy range \(-\infty < E < E_{F}\) is
\[ n_{s}(E_{F})=\int^{E_{F}}_{-\infty} g(E)dE \nonumber \]
To determine \(g(E)\) we will count \(k\) states and then use the relation between \(E\) and \(k\) (known as the dispersion relation) to change variables from \(k\) to \(E\).
We showed above that the energy of electrons in a quantum wire is
\[ E_{n_{x}, n_{y}}=\frac{\hbar^{2} \pi^{2}}{2 m}\left(\frac{n_{x}^{2}}{L_{x}^{2}}+\frac{n_{y}^{2}}{L_{y}^{2}}\right)+\frac{\hbar^{2} k_{z}^{2}}{2 m}, \quad n_{x}, n_{y}=1,2, \ldots \nonumber \]
Thus, counting the x-y modes is straight forward, since in the confined potential they are discrete. But to count the modes in the z-direction we impose periodic boundary conditions.
This should be OK if the wire is sufficiently long since the boundaries of the wire are then less significant. Periodic boundary conditions cause \(k_{z}\) to be quantized, and each allowed value of k-space occupies a length \(2\pi /L_{z}\).
For convenience, we will integrate with respect to the magnitude of \(k_{z}\). Since we are integrating \(|k_{z}|\) from 0 to \(\infty\), not \(-\infty < k_{z}<\infty\), there is an extra factor of two to account for modes with negative \(k_{z}\), and an additional factor of two to account for the two possible electron spins per k state.
\[ n_{s}\left( |k_{z}| \right) = 2\times 2\times \int^{|k_{z}|}_{0} \frac{1}{2 \pi /L_{z}}dk \nonumber \]
Next we need to change variables in Equation 2.10.3, i.e. we need g(E) where
\[ n_{s}\left( E_{F} \right) = \int^{E_{F}}_{-\infty} g(E)dE \nonumber \]
Now \(|k_{z}|\) is related to the energy by
\[ E-E_{n_{x},n_{y}}= \frac{\hbar^{2}|k_{z}|^{2}}{2m}, E \geq E_{n_{x},n_{y}} \nonumber \]
Using the dispersion relation of Equation 2.10.5 in Equation 2.10.3 gives,
\[ g(E) d E=\frac{2 L}{\pi} \sqrt{\frac{m}{2 \hbar^{2}}} \sum_{n_{x}, n_{y}} \frac{u\left(E-E_{n_{x}, n_{y}}\right)}{\sqrt{E-E_{n_{x}, n_{y}}}} d E , \nonumber \]
where u is the unit step function. The DOS is plotted in Figure 2.10.2. Note that the flat region in the dispersion relation as k → 0 yields infinite peaks in the DOS at the bottom of each band. The peaks have finite area, however, since the wire contains a finite number of states.
