Bloch function

Representation of an isosurface of the absolute square of a Bloch wave function in silicon

The Bloch function or Bloch wave (after Felix Bloch ) is a general form for the solution of the stationary Schrödinger equation for a particle in a periodic potential , e.g. B. the wave function of an electron in a crystalline solid .

The shape of these wave functions is determined by the Bloch theorem , which is a special case of the Floquet theorem : ${\displaystyle \psi }$

Theorem: Let there be a periodic potential with the periodicity : ${\displaystyle V({\vec {r}})}$ ${\displaystyle {\vec {R}}}$ ${\displaystyle V({\vec {r}})=V({\vec {r}}+{\vec {R}})}$ Then there exists a basis of solutions to the stationary Schrödinger equation of the form ${\displaystyle \psi _{\vec {k}}({\vec {r}})=e^{\mathrm {i} {\vec {k}}\cdot {\vec {r}}}\cdot u_{\vec {k}}({\vec {r}})}$ With the Euler's number ${\displaystyle e}$ the imaginary unit ${\displaystyle \mathrm {i} }$ any vector ${\displaystyle {\vec {k}}}$ a periodic function with period that is dependent on the parameter : ${\displaystyle {\vec {k}}}$ ${\displaystyle u_{\vec {k}}({\vec {r}})}$ ${\displaystyle {\vec {R}}}$ ${\displaystyle u_{\vec {k}}({\vec {r}})=u_{\vec {k}}({\vec {r}}+{\vec {R}})}$

The periodicity of the potential is transferred to and thus to the probability of the observed particle being in the potential. For an electron in such an intrinsic energy state, the probability of being in each unit cell is therefore the same and shows the same spatial course. In a crystalline solid, the periodicity is given by the crystal lattice , which is a lattice vector . If the potential is independent of time, it can be assumed to be real. ${\displaystyle V({\vec {r}})}$ ${\displaystyle u_{k}({\vec {r}})}$ ${\displaystyle |\psi ({\vec {r}})|^{2}=|\psi ({\vec {r}}+{\vec {R}})|^{2}}$ ${\displaystyle {\vec {R}}}$ ${\displaystyle u_{k}({\vec {r}})}$

statement

According to Bloch's theorem, the single-particle energy eigenstates are parameterized via wave vectors , whereby all real numbers can pass through. The wave vectors of the first Brillouin zone ( etc.) are sufficient for a complete parameterization . Because a Bloch function remains unchanged if it is replaced by , with any vector of the reciprocal lattice , and at the same time the function is replaced by . It is true , because by definition is , and thus the function is periodic like . When describing wave functions and lattice energies, this enables the transition from the expanded zone scheme to the reduced zone scheme. ${\displaystyle {\vec {k}}}$ ${\displaystyle {\vec {k}}=\left(k_{x},k_{y},k_{z}\right)}$ ${\displaystyle {\vec {k}}}$ ${\displaystyle |k_{x}|\leq \pi /a_{x}}$ ${\displaystyle \psi ({\vec {r}})=e^{\mathrm {i} {\vec {k}}\cdot {\vec {r}}}\cdot u_{\vec {k}}({\vec {r}})}$ ${\displaystyle {\vec {k}}}$ ${\displaystyle {\vec {k}}'={\vec {k}}+{\vec {R_{G}}}}$ ${\displaystyle {\vec {R_{G}}}=\sum _{i=1}^{3}n_{i}\cdot {\frac {2\pi }{a_{i}}}}$ ${\displaystyle u_{\vec {k}}({\vec {r}})}$ ${\displaystyle u_{{\vec {k}}'}({\vec {r}})=e^{-\mathrm {i} {\vec {r}}_{}\cdot {\vec {R_{G}}}}u_{\vec {k}}({\vec {r}})}$ ${\displaystyle \psi ({\vec {r}})=e^{\mathrm {i} {\vec {k}}\cdot {\vec {r}}}\cdot u_{\vec {k}}({\vec {r}})\equiv e^{\mathrm {i} {\vec {k}}'\cdot {\vec {r}}}\cdot u_{{\vec {k}}'}({\vec {r}})}$ ${\displaystyle e^{\mathrm {i} {\vec {R_{}}}\cdot {\vec {R_{G}}}}=1}$ ${\displaystyle u_{{\vec {k}}'}({\vec {r}})}$ ${\displaystyle u_{\vec {k}}({\vec {r}})}$

To describe all single-particle wave functions of the crystal , in particular the lattice-periodic function , in the reduced zone scheme, the contributions of the entire reciprocal lattice d. H. of all equivalent reciprocal lattice vectors, so that a further index n has to be introduced. This conveys the contribution of the nth Brillouin zone to the energy spectrum and to the wave function via the reciprocal lattice vector . But since it is discrete, a discrete energy spectrum develops for each one, which changes continuously as a function of within the first Brillouin zone. Energy bands are thus formed: ${\displaystyle \psi _{\vec {k}}({\vec {r}})}$ ${\displaystyle u_{\vec {k}}({\vec {r}})}$ ${\displaystyle {\vec {R_{G}}}}$ ${\displaystyle {\vec {R_{G}}}}$ ${\displaystyle {\vec {k}}}$ ${\displaystyle {\vec {k}}}$

${\displaystyle E({\vec {k}})\rightarrow E_{n}({\vec {k}},R_{G})}$or . ${\displaystyle u_{\vec {k}}({\vec {r}})\rightarrow u_{n{\vec {k}}}({\vec {r}},R_{G})\Rightarrow u_{\vec {k}}({\vec {r}})=\sum _{n}u_{n{\vec {k}}}({\vec {r}},R_{G})}$

This is the basis of the representation of the band structure in the band model, which is widespread in solid state physics .

Wave vector and n , called the band index, are therefore suitable indices for designating the single-particle energy eigenstates and single-particle wave function of the lattice. The wave vector is also known as a quasi-pulse or a crystal pulse . The name is based on the fact that in the case of a weakly variable function the momentum of the particle is approximately given by , so that the crystal momentum still has approximately the properties of the momentum, e.g. B. in the conservation of momentum during collisions or emission and absorption of photons. If so , that applies exactly. ${\displaystyle {\vec {k}}}$ ${\displaystyle u_{\vec {k}}({\vec {r}})\approx const}$ ${\displaystyle \langle {\vec {p}}\rangle \approx \hbar {\vec {k}}}$ ${\displaystyle u_{\vec {k}}({\vec {r}})=const}$

Simplified derivation

Since the potential is invariant to a translation around a vector (there is a lattice vector in a crystal ), it is also the Hamiltonian of the particle. An eigenfunction that is shifted by the distance is therefore definitely an eigenfunction for the same energy. If there is no degeneracy, it describes the same state as before the translation and can therefore only differ from the unshifted function by a fixed phase factor . ${\displaystyle V({\vec {r}})}$ ${\displaystyle {\vec {R}}}$ ${\displaystyle {\vec {R}}}$ ${\displaystyle {\hat {H}}={\frac {{\hat {p}}^{2}}{2m}}+V({\vec {r}})}$ ${\displaystyle {\vec {R}}}$ ${\displaystyle f}$

${\displaystyle \psi ({\vec {r}}+{\vec {R}})=f\ \psi ({\vec {r}})}$

If the translation is repeated n times, the phase factors ( ) multiply , while the distances add ( ). ${\displaystyle f^{n}}$ ${\displaystyle n{\vec {R}}}$

But since the particle density should be preserved:

${\displaystyle \left|\psi ({\vec {r}}+{\vec {R}})\right|^{2}=\left|\psi ({\vec {r}})\right|^{2}}$,

must be given in general by ${\displaystyle f}$

${\displaystyle f({\vec {R}})=e^{\mathrm {i} {\vec {k}}\cdot {\vec {R}}}}$

with an appropriate fixed vector . A simple periodicity then follows for a trained function . So is ${\displaystyle {\vec {k}}}$ ${\displaystyle \psi ({\vec {r}})}$ ${\displaystyle u({\vec {r}})=e^{-\mathrm {i} {\vec {k}}\cdot {\vec {r}}}\psi ({\vec {r}})}$ ${\displaystyle u({\vec {r}}+{\vec {R}})=u({\vec {r}})}$ ${\displaystyle \psi ({\vec {r}})=e^{\mathrm {i} {\vec {k}}\cdot {\vec {r}}}u({\vec {r}})}$

literature

• Felix Bloch: About the quantum mechanics of electrons in crystal lattices . In: Zeitschrift für Physik A , 52, 1929, pp. 555-600, doi: 10.1007 / BF01339455 .
• Hartmut Haug, Stephan Koch: Quantum Theory of the Optical and Electronic Properties of Semiconductors . Fourth Edition. World Scientific, Singapore / River Edge / London, p. 29 ff.
• Claude Cohen-Tannoudji, Bernard Diu, Franck Laloë: Quantum Mechanics 1 & 2 . 2nd Edition. Walter de Gruyter, Berlin / New York 1999.
• Charles Kittel: Introduction to Solid State Physics . 14th edition. Oldenbourg-Verlag, Munich 2006, p. 187 f.
• Harald Ibach, Hans Lüth: Solid State Physics . 5th edition. Springer-Verlag, Berlin / Heidelberg 1999, p. 160 ff.