Kronig–Penney Model
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In
quantum mechanics Quantum mechanics is the fundamental physical Scientific theory, theory that describes the behavior of matter and of light; its unusual characteristics typically occur at and below the scale of atoms. Reprinted, Addison-Wesley, 1989, It is ...
, the particle in a one-dimensional lattice is a problem that occurs in the model of a periodic
crystal lattice In crystallography, crystal structure is a description of ordered arrangement of atoms, ions, or molecules in a crystal, crystalline material. Ordered structures occur from intrinsic nature of constituent particles to form symmetric patterns that ...
. The potential is caused by
ion An ion () is an atom or molecule with a net electrical charge. The charge of an electron is considered to be negative by convention and this charge is equal and opposite to the charge of a proton, which is considered to be positive by convent ...
s in the periodic structure of the crystal creating an
electromagnetic field An electromagnetic field (also EM field) is a physical field, varying in space and time, that represents the electric and magnetic influences generated by and acting upon electric charges. The field at any point in space and time can be regarde ...
so electrons are subject to a regular potential inside the lattice. It is a generalization of the
free electron model In solid-state physics, the free electron model is a quantum mechanical model for the behaviour of charge carriers in a metallic solid. It was developed in 1927, principally by Arnold Sommerfeld, who combined the classical Drude model with quan ...
, which assumes zero potential inside the lattice.


Problem definition

When talking about solid materials, the discussion is mainly around crystals – periodic lattices. Here we will discuss a 1D lattice of positive ions. Assuming the spacing between two ions is , the potential in the lattice will look something like this: The mathematical representation of the potential is a periodic function with a period . According to
Bloch's theorem In condensed matter physics, Bloch's theorem states that solutions to the Schrödinger equation in a periodic potential can be expressed as plane waves modulated by periodic functions. The theorem is named after the Swiss physicist Felix Bloch, ...
, the wavefunction solution of the
Schrödinger equation The Schrödinger equation is a partial differential equation that governs the wave function of a non-relativistic quantum-mechanical system. Its discovery was a significant landmark in the development of quantum mechanics. It is named after E ...
when the potential is periodic, can be written as: \psi (x) = e^ u(x), where is a
periodic function A periodic function, also called a periodic waveform (or simply periodic wave), is a function that repeats its values at regular intervals or periods. The repeatable part of the function or waveform is called a ''cycle''. For example, the t ...
which satisfies . It is the Bloch factor with Floquet exponent k which gives rise to the band structure of the energy spectrum of the Schrödinger equation with a periodic potential like the Kronig–Penney potential or a cosine function as it was shown in 1928 by Strutt. The solutions can be given with the help of the
Mathieu functions In mathematics, Mathieu functions, sometimes called angular Mathieu functions, are solutions of Mathieu's differential equation : \frac + (a - 2q\cos(2x))y = 0, where are real-valued parameters. Since we may add to to change the sign of , i ...
. When nearing the edges of the lattice, there are problems with the boundary condition. Therefore, we can represent the ion lattice as a ring following the Born–von Karman boundary conditions. If is the length of the lattice so that , then the number of ions in the lattice is so big, that when considering one ion, its surrounding is almost linear, and the wavefunction of the electron is unchanged. So now, instead of two boundary conditions we get one circular boundary condition: \psi (0)=\psi (L). If is the number of ions in the lattice, then we have the relation: . Replacing in the boundary condition and applying Bloch's theorem will result in a quantization for : \psi (0) = e^ u(0) = e^ u(L) = \psi (L) u(0) = e^ u(L)=e^ u(N a) \to e^ = 1 \Rightarrow kL = 2\pi n \to k = n \qquad \left( n=0, \pm 1, \dots, \pm \frac \right).


Kronig–Penney model

The Kronig–Penney model (named after
Ralph Kronig Ralph Kronig (10 March 1904 – 16 November 1995) was a German physicist. He is noted for the discovery of particle spin and for his theory of X-ray absorption spectroscopy. His theories include the Kronig–Penney model, the Coster–Kronig tr ...
and
William Penney William George Penney, Baron Penney, (24 June 19093 March 1991) was an English mathematician and professor of mathematical physics at the Imperial College London and later the rector of Imperial College London. He had a leading role in the d ...
) is a simple, idealized quantum-mechanical system that consists of an infinite periodic array of
rectangular potential barrier In quantum mechanics, the rectangular (or, at times, square) potential barrier is a standard one-dimensional problem that demonstrates the phenomena of wave-mechanical tunneling (also called "quantum tunneling") and wave-mechanical reflection. ...
s. The potential function is approximated by a rectangular potential: Using
Bloch's theorem In condensed matter physics, Bloch's theorem states that solutions to the Schrödinger equation in a periodic potential can be expressed as plane waves modulated by periodic functions. The theorem is named after the Swiss physicist Felix Bloch, ...
, we only need to find a solution for a single period, make sure it is continuous and smooth, and to make sure the function is also continuous and smooth. Considering a single period of the potential:
We have two regions here. We will solve for each independently: Let ''E'' be an energy value above the well (E>0) * For 0 < x < (a-b): \begin \frac \psi_ &= E \psi \\ \Rightarrow \psi &= A e^ + A' e^ & \left( \alpha^2 = \right) \end *For -b : \begin \frac \psi_ &= (E+V_0)\psi \\ \Rightarrow \psi &= B e^ + B' e^ & \left( \beta^2 = \right). \end To find ''u''(''x'') in each region, we need to manipulate the electron's wavefunction: \begin \psi(0 And in the same manner: u(-b To complete the solution we need to make sure the probability function is continuous and smooth, i.e.: \psi(0^)=\psi(0^) \qquad \psi'(0^)=\psi'(0^). And that and are periodic: u(-b)=u(a-b) \qquad u'(-b)=u'(a-b). These conditions yield the following matrix: \begin 1 & 1 & -1 & -1 \\ \alpha & -\alpha & -\beta & \beta \\ e^ & e^ & -e^ & -e^ \\ (\alpha-k)e^ & -(\alpha+k)e^ & -(\beta-k)e^ & (\beta+k)e^ \end \begin A \\ A' \\ B \\ B' \end = \begin 0 \\ 0 \\ 0 \\ 0 \end. For us to have a non-trivial solution, the determinant of the matrix must be 0. This leads us to the following expression: \cos(k a) = \cos(\beta b) \cos
alpha(a-b) Alpha (uppercase , lowercase ) is the first Letter (alphabet), letter of the Greek alphabet. In the system of Greek numerals, it has a value of one. Alpha is derived from the Phoenician alphabet, Phoenician letter ''Aleph#Origin, aleph'' , ...
\sin(\beta b) \sin
alpha(a-b) Alpha (uppercase , lowercase ) is the first Letter (alphabet), letter of the Greek alphabet. In the system of Greek numerals, it has a value of one. Alpha is derived from the Phoenician alphabet, Phoenician letter ''Aleph#Origin, aleph'' , ...
To further simplify the expression, we perform the following approximations: b \to 0; \quad V_0 \to \infty; \quad V_0 b = \mathrm \Rightarrow \beta^2 b = \mathrm; \quad \alpha^2 b \to 0 \Rightarrow \beta b \to 0; \quad \sin(\beta b) \to \beta b; \quad \cos(\beta b) \to 1. The expression will now be: \cos(k a) = \cos(\alpha a)+P \frac, \qquad P= \frac. For energy values inside the well (''E'' < 0), we get: \cos(k a) = \cos(\beta b) \cosh
alpha(a-b) Alpha (uppercase , lowercase ) is the first Letter (alphabet), letter of the Greek alphabet. In the system of Greek numerals, it has a value of one. Alpha is derived from the Phoenician alphabet, Phoenician letter ''Aleph#Origin, aleph'' , ...
\sin(\beta b) \sinh
alpha(a-b) Alpha (uppercase , lowercase ) is the first Letter (alphabet), letter of the Greek alphabet. In the system of Greek numerals, it has a value of one. Alpha is derived from the Phoenician alphabet, Phoenician letter ''Aleph#Origin, aleph'' , ...
with \alpha^2 = and \beta^2 = \frac. Following the same approximations as above ( b \to 0; \, V_0 \to \infty; \, V_0 b = \mathrm), we arrive at \cos(k a) = \cosh(\alpha a) + P \frac with the same formula for ''P'' as in the previous case \left(P = \frac\right).


Band gaps in the Kronig–Penney model

In the previous paragraph, the only variables not determined by the parameters of the physical system are the energy ''E'' and the crystal momentum ''k''. By picking a value for ''E'', one can compute the right hand side, and then compute ''k'' by taking the \arccos of both sides. Thus, the expression gives rise to the
dispersion relation In the physical sciences and electrical engineering, dispersion relations describe the effect of dispersion on the properties of waves in a medium. A dispersion relation relates the wavelength or wavenumber of a wave to its frequency. Given the ...
. The right hand side of the last expression above can sometimes be greater than 1 or less than –1, in which case there is no value of ''k'' that can make the equation true. Since \alpha a \propto \sqrt, that means there are certain values of ''E'' for which there are no eigenfunctions of the Schrödinger equation. These values constitute the
band gap In solid-state physics and solid-state chemistry, a band gap, also called a bandgap or energy gap, is an energy range in a solid where no electronic states exist. In graphs of the electronic band structure of solids, the band gap refers to t ...
. Thus, the Kronig–Penney model is one of the simplest periodic potentials to exhibit a band gap.


Kronig–Penney model: alternative solution

An alternative treatment to a similar problem is given. Here we have a ''delta'' periodic potential: V(x) = A\cdot\sum_^\delta(x - n a). is some constant, and is the lattice constant (the spacing between each site). Since this potential is periodic, we could expand it as a Fourier series: V(x) = \sum_K \tilde(K)\cdot e^, where \tilde(K) = \frac\int_^dx\,V(x)\,e^ = \frac\int_^ dx \sum_^ A\cdot \delta(x-na)\,e^ = \frac. The wave-function, using Bloch's theorem, is equal to \psi_k(x) = e^ u_k(x) where u_k(x) is a function that is periodic in the lattice, which means that we can expand it as a Fourier series as well: u_k(x)=\sum_ \tilde_k(K)e^. Thus the wave function is: \psi_k(x)=\sum_\tilde_k(K)\,e^. Putting this into the Schrödinger equation, we get: \left frac-E_k\right\tilde_k(K)+\sum_\tilde(K-K')\,\tilde_k(K') = 0 or rather: \left frac-E_k\right\tilde_k(K)+\frac\sum_\tilde_k(K')=0 Now we recognize that: u_k(0)=\sum_\tilde_k(K') Plug this into the Schrödinger equation: \left frac-E_k\right\tilde_k(K)+\fracu_k(0)=0 Solving this for \tilde_k(K) we get: \tilde_k(K)=\frac=\frac\,u_k(0) We sum this last equation over all values of to arrive at: \sum_\tilde_k(K)=\sum_\frac\,u_k(0) Or: u_k(0)=\sum_\frac\,u_k(0) Conveniently, u_k(0) cancels out and we get: 1=\sum_\frac Or: \frac\frac=\sum_\frac To save ourselves some unnecessary notational effort we define a new variable: \alpha^2 := \frac and finally our expression is: \frac\frac=\sum_\frac Now, is a reciprocal lattice vector, which means that a sum over is actually a sum over integer multiples of \frac: \frac\frac=\sum_^\frac We can juggle this expression a little bit to make it more suggestive (use
partial fraction decomposition In algebra, the partial fraction decomposition or partial fraction expansion of a rational fraction (that is, a fraction such that the numerator and the denominator are both polynomials) is an operation that consists of expressing the fraction as ...
): \begin \frac\frac &= \sum_^\frac \\ &=-\frac\sum_^\left frac-\frac\right\\ &=-\frac\sum_^\left frac-\frac \right\\ &=-\frac\left sum_^\frac - \sum_^\frac \right\end If we use a nice identity of a sum of the cotangent function
Equation 18
which says: \cot(x)=\sum_^\frac-\frac and plug it into our expression we get to: \frac\frac = -\frac\left cot\left(\tfrac-\tfrac\right) - \cot\left(\tfrac+\tfrac\right)\right/math> We use the sum of and then, the product of (which is part of the formula for the sum of ) to arrive at: \cos(k a)=\cos(\alpha a)+\frac\sin(\alpha a) This equation shows the relation between the energy (through ) and the wave-vector, , and as you can see, since the left hand side of the equation can only range from to then there are some limits on the values that (and thus, the energy) can take, that is, at some ranges of values of the energy, there is no solution according to these equation, and thus, the system will not have those energies: energy gaps. These are the so-called band-gaps, which can be shown to exist in ''any'' shape of periodic potential (not just delta or square barriers). For a different and detailed calculation of the gap formula (i.e. for the gap between bands) and the level splitting of eigenvalues of the one-dimensional Schrödinger equation see Müller-Kirsten.Harald J. W. Müller-Kirsten, Introduction to Quantum Mechanics: Schrödinger Equation and Path Integral, 2nd ed., World Scientific (Singapore, 2012), 325–329, 458–477. Corresponding results for the cosine potential (Mathieu equation) are also given in detail in this reference.


Finite lattice

In some cases, the Schrödinger equation can be solved analytically on a one-dimensional lattice of finite length using the theory of periodic differential equations. The length of the lattice is assumed to be L = N a, where a is the potential period and the number of periods N is a positive integer. The two ends of the lattice are at \tau and L + \tau, where \tau determines the point of termination. The wavefunction vanishes outside the interval tau,L+\tau/math>. The eigenstates of the finite system can be found in terms of the Bloch states of an infinite system with the same periodic potential. If there is a band gap between two consecutive energy bands of the infinite system, there is a sharp distinction between two types of states in the finite lattice. For each energy band of the infinite system, there are N - 1 bulk states whose energies depend on the length N but not on the termination \tau. These states are
standing waves In physics, a standing wave, also known as a stationary wave, is a wave that oscillates in time but whose peak amplitude profile does not move in space. The peak amplitude of the wave oscillations at any point in space is constant with respect t ...
constructed as a superposition of two Bloch states with momenta k and -k, where k is chosen so that the wavefunction vanishes at the boundaries. The energies of these states match the energy bands of the infinite system. For each band gap, there is one additional state. The energies of these states depend on the point of termination \tau but not on the length N. The energy of such a state can lie either at the band edge or within the band gap. If the energy is within the band gap, the state is a
surface state Surface states are electronic states found at the Surface (topology), surface of materials. They are formed due to the sharp transition from solid material that ends with a surface and are found only at the atom layers closest to the surface. The t ...
localized at one end of the lattice, but if the energy is at the band edge, the state is delocalized across the lattice.


See also

*
Empty lattice approximation The empty lattice approximation is a theoretical electronic band structure model in which the potential is ''periodic'' and ''weak'' (close to constant). One may also consider an empty irregular lattice, in which the potential is not even periodic ...
*
Nearly free electron model In solid-state physics, the nearly free electron model (or NFE model and quasi-free electron model) is a quantum mechanical model of physical properties of electrons that can move almost freely through the crystal lattice of a solid. The model ...
*
Crystal structure In crystallography, crystal structure is a description of ordered arrangement of atoms, ions, or molecules in a crystalline material. Ordered structures occur from intrinsic nature of constituent particles to form symmetric patterns that repeat ...


References


External links

*
The Kronig–Penney Model
by Michael Croucher, an interactive calculation of 1d periodic potential band structure using
Mathematica Wolfram (previously known as Mathematica and Wolfram Mathematica) is a software system with built-in libraries for several areas of technical computing that allows machine learning, statistics, symbolic computation, data manipulation, network ...
, from
The Wolfram Demonstrations Project The Wolfram Demonstrations Project is an open-source collection of interactive programmes called Demonstrations. It is hosted by Wolfram Research. At its launch, it contained 1300 demonstrations but has grown to over 10,000. The site won a Pa ...
. {{DEFAULTSORT:Particle In A One-Dimensional Lattice Condensed matter physics Electronics concepts Quantum models