2.14: El DOS 3-d- materiales a granel sin confinamiento
- Page ID
- 84366
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)En 3-d, no hay confinamiento de electrones. La única restricción en\(k_{x}\)\(k_{y}\), o\(k_{z}\) son las condiciones de contorno periódicas. Acabamos de demostrar que si el sistema tiene volumen\(L_{x}\times L_{y}\times L_{z}\) entonces cada valor permitido de k-espacio ocupa un volumen de\(2\pi/ L_{x}\times 2\pi/L_{y}\times2\pi/L_{z} = 8\pi^{3}/V\).
Para determinar el número de estados permitidos, integraremos sobre todo k -espacio. Es conveniente hacer esto en coordenadas esféricas. Si k es la magnitud del vector k, el número de modos dentro de una capa esférica de espesor dk es entonces
\[ n_{s}(k)dk=2\times \frac{1}{8\pi^{3}/V}\times 4\pi k^{2}dk \nonumber \]
donde\(V = L_{x}\times L_{y}\times L_{z}\), y el factor de dos cuentas para espín electrónico. Las funciones de onda no confinadas dentro de nuestra caja 3-d son ondas planas en todas las direcciones, es decir, la función de onda podría describirse mediante
\[ \psi(x,y,z) = \psi_{0}e^{ik_{x}x}e^{ik_{y}y}e^{ik_{z}z} \nonumber \]
Sustituir en la ecuación de Schrödinger da
\[ -\frac{\hbar^{2}}{2m} \left( \frac{d^{2}}{dx^{2}}\frac{d^{2}}{dy^{2}} \frac{d^{2}}{dx^{2}}\right) \psi = E \psi \nonumber \]
Lo que da
\[ \frac{\hbar^{2}}{2m} (k_{x}^{2} + k_{x}^{2}+ k_{x}^{2}) = E \nonumber \]
Reordenando:
\[ E = \frac{\hbar^{2}k^{2}}{2m} \nonumber \]
El uso de la ecuación (2.14.5) para relacionar E con k da:
\[ g(E)dE = \frac{V}{2\pi^{2}}\left( \frac{2m}{\hbar^{2}}\right)^{\frac{3}{2}} \sqrt{E}\ dE \nonumber \]
donde g (E) es la densidad de estados por unidad de energía.
En la Figura 2.14.2 se muestra una comparación de la densidad de estados en materiales 1-d, 2-d y 3-d.
