Blochs Theorem

The Bloch theorem is one of the tools that helps us to mathematically deal with solids [5,6], The mathematical condition behind the Bloch theorem is the fact that the equations which governs the excitations of the crystalline structure such as lattice vibrations, electron states and spin waves are periodic. Then, to jsolve the Schrodinger equation for a crystalline solid where the potential is periodic, [V(r + R) = V(r), this theorem is applied [5,6], 

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If V( r ) is the potential seen by an electron belonging to the solid, then the one electron wave function, /(r ), satisfies the Schrodinger equation  

In the case of lattice waves and spin waves, the procedure is different but the principle is the same. The periodic potential is represented with the help of a Fourier series 

the wave function /(r) and the wave function /(r + R) must differ only in a constant, then 

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For a periodic lattice, it can be shown (Bloch theorem) that the solutions to the one-electron Schrodinger equation are of the... 

In the bulk of a perfect crystal, the electronic states follow the Bloch theorem, which have the form... 

The concept of surface states was proposed by Tamm (1932) using a one-dimensional analytic model. We start with reviewing the proof of the Bloch theorem for a one-dimensional periodic potential U x) with periodicity a (Kittel, 1986) ... 

Stimulated by a variety of commercial applications in fields such as xerography, solar energy conversion, thin-film active devices, and so forth, international interest in this subject area has increased dramatically since these early reports. The absence of long-range order invalidates the use of simplifying concepts such as the Bloch theorem, the counterpart of which has proved elusive for disordered systems. After more than a decade of concentrated research, there remains no example of an amorphous solid for the energy band structure, and the mode of electronic transport is still a subject for continued controversy. 

Again, performing the trace with the aid of the Bloch theorem, and passing as from Eqs. (68) to (69), gives... 

Next, perform the trace over the slow mode with the help of the Bloch theorem. Then, with Eqs. (H.l) and (H.4), the ACF (123) becomes... 

To calculate this thermal average, it is suitable to use the following theorem, the demonstration of which is given in the book by Louisell [54] which, deals with the quantum theory of light. Another possibility is to use the Bloch theorem... 

Have the correct form to be a solution of Equation 1.8. As a result, the Bloch theorem affirms that the solution to the Schrodinger equation may be a plane wave multiplied by a periodic function, that is [5,6],... 

It is necessary to state now that the rigorous fulfillment of the Bloch theorem needs an infinity lattice. In order to calculate the number of states in a finite crystal, a mathematical requirement named the Bom-Karman cyclic boundary condition is introduced. That is, if we consider that a crystal with dimensions Nxa, N2b, /V3c is cyclic in three dimensions, then [5]... 

In absence of the Bloch theorem,155 the zero-order disorder effect—termed the minimum, or diagonal, disorder156 —enters the coulombic interaction hamiltonian in the following way ... 

Simply put, the Bloch theorem guarantees that, if the correct wavefunction is found for the zeroth cell, the wavefunctions outside the cell are a repetition of that wavefunction, multiplied by the factor exp (ik R). Among many choices, the wavefunctions uk(r) can be Wannier45 functions, which are defined to be mutually orthogonal, (uk(r) uk (r)) — kk while for atomic or molecular wavefunctions this orthogonality does not necessarily hold. 

In the last decade an abundant literature has focused more and more on the properties of low-symmetry systems having large unit cells which render unwieldy the traditional description in terms of the Bloch theorem. Low-symmetry systems include compUcated ternary or quaternary compounds, man-made superlattices, intercalated materials, etc. The k-space picture becomes totally useless for higher degrees of disorder as exhibited by amorphous materials, microcrystallites, random alloys, phonon-induced disorder, surfaces, adsorbed atoms, chemisorption effects, and so on. 

It is a basic consequence of the translational symmetry of a solid that its Kohn-Sham eigenfunctions can be uniquely labeled by four quantum numbers, the band index n and a wavevector k, as in xj/ y.. The diagram s n,k) that represents the n, k dependence of the corresponding eigenenergies is called the band structure. The Bloch theorem asserts that the t>e written in the form of a Fourier... 

When treating periodic systems, the orbital expansion given so far is incomplete in principle. Although, the charge density is necessarly periodic, there can be for the wavefunction itself a phase factor from one periodic image to the other. This is the essence of the Bloch theorem stipulating that orbitals can be written as... 

The presence of the periodic potential f/(r) has important consequences with regard to the solutions of the time-independent Schrodinger equation associated with the Hamiltonian (4.71). In particular, a fundamental property of eigenfunctions of such a Hamiltonian is expressed by the Bloch theorem. 

The Bloch theorem states that the eigenfunctions of the Hamiltonian (4.71), (4.72) are products of a wave of the form (4.73) and a function that is periodic on the lattice, that is. 

For future reference we cite here without proof a useful identity that involves the harmonic oscillator Hamiltonian H = p /2m + (1 /2)ma> q and an operator of the general formH = explaip + a2q with constant parameters ai and that is, the exponential of a linear combination of the momentum and coordinate operators. The identity, known as the Bloch theorem, states that the thermal average A )t (under the hannonic oscillator Hamiltonian) is related to the thennal average ((aip + Q 2 )2)t according to... 

The band structure of crystalline solids is usually obtained by solving the Schrodinger equation of an approximate one-electron problem. In the case of non-metallic materials, such as semiconductors and insulators, there are essentially no free electrons. This problem is taken care of by the Bloch theorem. This important theorem assumes a potential energy profile V(r) being periodic with the periodicity of the lattice. In this case the Schrodinger equation is given by... 

Usual crystal orbitals extend all over the system concerned and are obtained so as to fulfill the Bloch theorem (Bloch, 1928). However, it is known to be rather useful to convert the wave function of the system into the localized function for the purpose of the discussion of the local nature of the system, such as the exciton. One such function is the Wannier function ap derived by the Fourier transformation of the crystal orbital... 

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