1 \part{Theory of the solid state}
3 \chapter{Atomic structure}
5 \chapter{Reciprocal lattice}
7 Example of primitive cell ...
9 \chapter{Electronic structure}
11 \section{Noninteracting electrons}
13 \subsection{Bloch's theorem}
15 \section{Nearly free and tightly bound electrons}
17 \subsection{Tight binding model}
19 \section{Interacting electrons}
21 \subsection{Density functional theory}
23 \subsubsection{Hohenberg-Kohn theorem}
25 The Hamiltonian of a many-electron problem has the form
31 T & = & \langle\Psi|\sum_{i=1}^N\frac{-\hbar^2}{2m}\nabla_i^2|\Psi\rangle\\
32 & = & \frac{-\hbar^2}{2m} \sum_{i=1}^N \int d\vec{r} d\vec{r}' \,
33 \langle \Psi | \vec{r} \rangle \langle \vec{r} |
35 | \vec{r}' \rangle \langle \vec{r}' | \Psi \rangle\\
36 & = & \frac{-\hbar^2}{2m} \sum_{i=1}^N \int d\vec{r} d\vec{r}' \,
37 \langle \Psi | \vec{r} \rangle \nabla_{\vec{r}_i}
38 \langle \vec{r} | \vec{r}' \rangle
39 \nabla_{\vec{r}'_i} \langle \vec{r}' | \Psi \rangle\\
40 & = & \frac{-\hbar^2}{2m} \sum_{i=1}^N \int d\vec{r} d\vec{r}' \,
41 \nabla_{\vec{r}_i} \langle \Psi | \vec{r} \rangle
42 \delta_{\vec{r}\vec{r}'}
43 \nabla_{\vec{r}'_i} \langle \vec{r}' | \Psi \rangle\\
44 & = & \frac{-\hbar^2}{2m} \sum_{i=1}^N \int d\vec{r} \,
45 \nabla_{\vec{r}_i} \Psi^*(\vec{r}) \nabla_{\vec{r}_i} \Psi(\vec{r})
47 V & = & \int V(\vec{r})\Psi^*(\vec{r})\Psi(\vec{r})d\vec{r} \text{ ,} \\
48 U & = & \frac{1}{2}\int\frac{1}{\left|\vec{r}-\vec{r}'\right|}
49 \Psi^*(\vec{r})\Psi^*(\vec{r}')\Psi(\vec{r}')\Psi(\vec{r})
52 represent the kinetic energy, the energy due to the external potential and the energy due to the mutual Coulomb repulsion.
55 As can be seen from the above, two many-electron systems can only differ in the external potential and the number of electrons.
56 The number of electrons is determined by the electron density.
58 N=\int n(\vec{r})d\vec{r}
60 Now, if the external potential is additionally determined by the electron density, the density completely determines the many-body problem.
63 Considering a system with a nondegenerate ground state, there is obviously only one ground-state charge density $n_0(\vec{r})$ that corresponds to a given potential $V(\vec{r})$.
65 n_0(\vec{r})=\int \Psi_0^*(\vec{r},\vec{r}_2,\vec{r}_3,\ldots,\vec{r}_N)
66 \Psi_0(\vec{r},\vec{r}_2,\vec{r}_3,\ldots,\vec{r}_N)
67 d\vec{r}_2d\vec{r}_3\ldots d\vec{r}_N
69 In 1964, Hohenberg and Kohn showed the opposite and far less obvious result \cite{hohenberg64}.
71 \begin{theorem}[Hohenberg / Kohn]
72 For a nondegenerate ground state, the ground-state charge density uniquely determines the external potential in which the electrons reside.
76 The proof presented by Hohenberg and Kohn proceeds by {\em reductio ad absurdum}.
77 Suppose two potentials $V_1$ and $V_2$ exist, which yield the same electron density $n(\vec{r})$.
78 The corresponding Hamiltonians are denoted $H_1$ and $H_2$ with the respective ground-state wavefunctions $\Psi_1$ and $\Psi_2$ and eigenvalues $E_1$ and $E_2$.
79 Then, due to the variational principle (see \ref{sec:var_meth}), one can write
81 E_1=\langle \Psi_1 | H_1 | \Psi_1 \rangle <
82 \langle \Psi_2 | H_1 | \Psi_2 \rangle \text{ .}
85 Expressing $H_1$ by $H_2+H_1-H_2$, the last part of \eqref{subsub:hk01} can be rewritten:
87 \langle \Psi_2 | H_1 | \Psi_2 \rangle =
88 \langle \Psi_2 | H_2 | \Psi_2 \rangle +
89 \langle \Psi_2 | H_1 -H_2 | \Psi_2 \rangle
91 The two Hamiltonians, which describe the same number of electrons, differ only in the potential
93 H_1-H_2=V_1(\vec{r})-V_2(\vec{r})
97 E_1<E2+\int n(\vec{r}) \left( V_1(\vec{r})-V_2(\vec{r}) \right) d\vec{r}
101 By switching the indices of \eqref{subsub:hk02} and adding the resulting equation to \eqref{subsub:hk02}, the contradiction
103 E_1 + E_2 < E_2 + E_1 +
105 \int n(\vec{r}) \left( V_1(\vec{r})-V_2(\vec{r}) \right) d\vec{r} +
106 \int n(\vec{r}) \left( V_2(\vec{r})-V_1(\vec{r}) \right) d\vec{r}
109 is revealed, which proofs the Hohenberg Kohn theorem.% \qed
112 \section{Electron-ion interaction}
114 \subsection{Pseudopotential theory}
116 The basic idea of pseudopotential theory is to only describe the valence electrons, which are responsible for the chemical bonding as well as the electronic properties for the most part.
118 \subsubsection{Orthogonalized planewave method}
120 Due to the orthogonality of the core and valence wavefunctions, the latter exhibit strong oscillations within the core region of the atom.
121 This requires a large amount of planewaves $\ket{k}$ to adequatley model the valence electrons.
123 In a very general approach, the orthogonalized planewave (OPW) method introduces a new basis set
125 \ket{k}_{\text{OPW}} = \ket{k} - \sum_t \ket{t}\bra{t}k\rangle \text{ ,}
127 with $\ket{t}$ being the eigenstates of the core electrons.
128 The new basis is orthogonal to the core states $\ket{t}$.
130 \braket{t}{k}_{\text{OPW}} =
131 \braket{t}{k} - \sum_{t'} \braket{t}{t'}\braket{t'}{k} =
132 \braket{t}{k} - \braket{t}{k}=0
134 The number of planewaves required for reasonably converged electronic structure calculations is tremendously reduced by utilizing the OPW basis set.
136 \subsubsection{Pseudopotential method}
138 Following the idea of orthogonalized planewaves leads to the pseudopotential idea, which --- in describing only the valence electrons --- effectively removes an undesriable subspace from the investigated problem.
140 Let $\ket{\Psi_\text{V}}$ be the wavefunction of a valence electron with the Schr\"odinger equation
142 H \ket{\Psi_\text{V}} = \left(\frac{1}{2m}p^2+V\right)\ket{\Psi_\text{V}}=
143 E\ket{\Psi_\text{V}} \text{ .}
145 \ldots projection operatore $P_\text{C}$ \ldots
147 \subsubsection{Semilocal form of the pseudopotential}
149 Ionic potentials, which are spherically symmteric, suggest to treat each angular momentum $l,m$ separately leading to $l$-dependent non-local (NL) model potentials $V_l(r)$ and a total potential
151 V=\sum_{l,m}\ket{lm}V_l(\vec{r})\bra{lm} \text{ .}
153 In fact, applied to a function, the potential turns out to be non-local in the angular coordinates but local in the radial variable, which suggests to call it asemilocal (SL) potential.
155 Problem of semilocal potantials become valid once matrix elements need to be computed.
156 Integral with respect to the radial component needs to be evaluated for each planewave combination, i.e.\ $N(N-1)/2$ integrals.
158 \bra{k+G}V\ket{k+G'} = \ldots
161 A local potential can always be separated from the potential \ldots
163 V=\ldots=V_{\text{local}}(\vec{r})+\ldots
166 \subsubsection{Norm conserving pseudopotentials}
170 \subsubsection{Fully separable form of the pseudopotential}
172 KB transformation \ldots
174 \subsection{Spin-orbit interaction}
176 Relativistic effects can be incorporated in the normconserving pseudopotential method up to but not including order $\alpha^2$ with $\alpha$ being the fine structure constant.
177 This is advantageous since \ldots
178 With the solutions of the all-electron Dirac equations, the new pseudopotential reads
180 V(\vec{r})=\sum_{l,m}\left[
181 \ket{l+\frac{1}{2},m+{\frac{1}{2}}}V_{l,l+\frac{1}{2}}(\vec{r})
182 \bra{l+\frac{1}{2},m+{\frac{1}{2}}} +
183 \ket{l-\frac{1}{2},m-{\frac{1}{2}}}V_{l,l-\frac{1}{2}}(\vec{r})
184 \bra{l-\frac{1}{2},m-{\frac{1}{2}}}
187 By defining an averaged potential weighted by the different $j$ degeneracies of the $\ket{l\pm\frac{1}{2}}$ states
189 \bar{V}_l(r)=\frac{1}{2l+1}\left(
190 l V_{l,l-\frac{1}{2}}(\vec{r})+(l+1)V_{l,l+\frac{1}{2}}(\vec{r})\right)
192 and a potential describing the difference in the potential with respect to the spin
194 V^{\text{SO}}_l(\vec{r})=\frac{2}{2l+1}\left(
195 V_{l,l+\frac{1}{2}}(\vec{r})-V_{l,l-\frac{1}{2}}(\vec{r})\right)
197 the total potential can be expressed as
200 \ket{l}\left[\bar{V}_l(\vec{r})+V^{\text{SO}}_l(\vec{r})LS\right]\bra{l}
203 where the first term correpsonds to the mass velocity and Darwin relativistic corrections and the latter is associated with the spin-orbit (SO) coupling.
206 \subsubsection{Excursus: real space representation within an iterative treatment}
208 In the following, the spin-orbit part is evaluated in real space.
209 Since spin is treated in another subspace, it can be treated separately.
210 The matrix elements of the orbital angular momentum part of the potential in KB form for orbital angular momentum $l$ read
212 \bra{\vec{r'}}(r\times p)\ket{\chi_{lm}}E^{\text{SO,KB}}_l\braket{\chi_{lm}}{\vec{r''}}
217 \bra{\vec{r'}}p\ket{\chi_{lm}} & = & -i\hbar\nabla_{\vec{r'}} \braket{\vec{r'}}{\chi_{lm}}
218 =-i\hbar\nabla_{\vec{r'}}\,\chi_{lm}(\vec{r'}) \\
219 r\ket{\vec{r'}} & = & r'\ket{\vec{r'}}
223 -i\hbar(r'\times \nabla_{\vec{r'}})\braket{\vec{r'}}{\chi_{lm}}
224 E^{\text{SO,KB}}_l\braket{\chi_{lm}}{\vec{r''}}
226 \label{eq:solid:so_me}
228 To further evaluate this expression, the KB projectors
230 \chi_{lm}=\frac{\ket{\delta V_l^{\text{SO}}\Phi_{lm}}}
231 {\braket{\delta V_l^{\text{SO}}\Phi_{lm}}
232 {\Phi_{lm}\delta V_l^{\text{SO}}}^{1/2}}
234 must be known in real space (with respect to $\vec{r'}$).
236 \braket{\vec{r'}}{\chi_{lm}}=
237 \frac{\braket{\vec{r'}}{\delta V_l^{\text{SO}}\Phi_{lm}}}{
238 \braket{\delta V_l^{\text{SO}}\Phi_{lm}}{\Phi_{lm}\delta V_l^{\text{SO}}}
243 \braket{\vec{r'}}{\delta V_l^{\text{SO}}\Phi_{lm}}=
244 \delta V_l^{\text{SO}}(r')\frac{u_l(r')}{r'}Y_{lm}(\Omega_{r'})
246 \label{eq:solid:so_r1}
248 In this expression, only the spherical harmonics are complex functions.
249 Thus, the complex conjugate with respect to $\vec{r''}$ is given by
251 \braket{\Phi_{lm}\delta V_l^{\text{SO}}}{\vec{r''}}=
252 \delta V_l^{\text{SO}}(r'')\frac{u_l(r'')}{r''}Y^*_{lm}(\Omega_{r''})
254 \label{eq:solid:so_r2}
256 Using the orthonormality property
258 \int Y^*_{l'm'}(\Omega_r)Y_{lm}(\Omega_r) d\Omega_r = \delta_{ll'}\delta_{mm'}
259 \label{eq:solid:y_ortho}
261 of the spherical harmonics, the squared norm of the $\chi_{lm}$ reduces to
263 \braket{\delta V_l^{\text{SO}}\Phi_{lm}}{\Phi_{lm}\delta V_l^{\text{SO}}} &=&
264 \int \braket{\delta V_l^{\text{SO}}\Phi_{lm}}{\vec{r'}}
265 \braket{\vec{r'}}{\Phi_{lm}\delta V_l^{\text{SO}}} d\vec{r'}\\
267 {\delta V_l^{\text{SO}}}^2(r')\frac{u_l^2(r')}{r'^2}Y^*_{lm}(\Omega_{r'})
269 r'^2 dr' d\Omega_{r'} \\
271 {\delta V_l^{\text{SO}}}^2(r') u_l^2(r') dr'
272 \int_{\Omega_{r'}}Y^*_{lm}(\Omega_{r'})Y_{lm}(\Omega_{r'}) d\Omega_{r'}\\
273 &=&\int_{r'}{\delta V_l^{\text{SO}}}^2(r') u_l^2(r') dr' \text{ .}
275 To obtain a final expression for the matrix elements \eqref{eq:solid:so_me}, the product of \eqref{eq:solid:so_r1} and \eqref{eq:solid:so_r2} must be further evaluated.
277 \braket{\vec{r'}}{\delta V_l^{\text{SO}}\Phi_{lm}}
278 \braket{\Phi_{lm}\delta V_l^{\text{SO}}}{\vec{r''}}&=&
279 \delta V_l^{\text{SO}}(r')\frac{u_l(r')}{r'}Y_{lm}(\Omega_{r'})
280 \delta V_l^{\text{SO}}(r'')\frac{u_l(r'')}{r''}Y^*_{lm}(\Omega_{r''})\nonumber\\
282 \delta V_l^{\text{SO}}(r')\frac{u_l(r')}{r'}
283 \delta V_l^{\text{SO}}(r'')\frac{u_l(r'')}{r''}
284 Y^*_{lm}(\Omega_{r''})Y_{lm}(\Omega_{r'})
286 All megnetic states $m=-l,-l+1,\ldots,l-1,l$ contribute to the potential for angular momentum $l$.
287 Finally, to evaluate $V^{\text{SO}}_l\ket{\Psi}$, the integral \ldots