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Kronig-Penney model

The Kronig-Penney model is a simplified theoretical model used in solid state physics to study the behavior of electrons in a periodic potential. It provides insights into the electronic band structure of a one-dimensional crystal lattice.

It is also possible to find analytic expressions for the dispersion relation (E vs. k) and the electron density of states.

In the Kronig-Penney model,

  • the crystal lattice is approximated as a periodic array of infinitely high potential barriers separated by regions of zero potential.
  • The model assumes that the potential within each barrier is uniform and that the barriers are infinitely narrow.
  • To analyze the behavior of electrons in this model, the Schrödinger equation is solved for the electron wave function within each region of the lattice.
  • The boundary conditions are applied to ensure the continuity of the wave function and its derivative across the potential barriers.

We assume that the electron inside an array of equally spaced potential wells or potential barriers, all which have the same width and depth/heigh.

In the case of the Kroning-Penney model, the potential V(x) is a periodic square wave.

The Schrödinger equation determines the permissible states in which an electron can exist within the Kronig-Penney model.

$$\frac{-\hbar^2}{2m}\frac{\partial^2\psi}{\partial x^2}+v(x)\psi=E\psi$$

Solution of the Schrödinger equation for the Kronig-Penney potential

We assume,$$E<V_0$$

For  $$0\leq x\leq a$$

$$ψ=A sin⁡αx+B cos⁡αx$$ 

  $$\frac{d\psi}{dx}=A\alpha\;\sin\alpha x+B\alpha\;\cos\alpha x$$

where, $$\alpha=\frac{\surd(2mE)}ℏ$$

For $$-b≤x≤0$$

$$\frac{-\hbar^2}{2m}\frac{\partial^2\psi}{\partial x^2}+(E-v_0)\psi=0$$

Solution is periodic, barrier of limited thickness- solution doesn’t continue decay to zero. There is tunneling between wells.

$$\psi=C\;\sin h(\gamma x)+D\;\cos h(\gamma x)$$

$$\frac{d\psi}{dx}=C\gamma\;\sin h(\gamma x)+D\gamma\;\cos h(\gamma x)$$

$$\gamma=\frac{\surd(2m(V_0-E))}{\hbar}$$

ψ must be contuinues at x=0, and B=D

$$Also,\;\;\frac{d\psi}{dx}\;\;must\;be\;continuous\;at\;x=0,\;so\;A\backslash alpha=\backslash gamma\;C$$

Since the Kronig-Penney potential exhibits translational symmetry, the energy eigenfunctions of the Schrödinger equation will simultaneously be eigenfunctions of the translation operator.

we proceed by seeking the eigenfunctions of the translation operator. The translation operator T shifts the solutions by one period,

$$T\psi(x)\;=\;\psi(x\;+\;a)$$

Notice that any function of the form

$$\psi_k(x)=e^{ikx}U_k(x)$$

Is an eigen is an eigen function of translational operator with eigenvalue $$e^{ikx}$$

$$\psi_k(x)=e^{ik(x+a)}U_k(x+a)=e^{ikx}U_k(x)$$

In solid state physics, functions like this are said to have Bloch form. The convention for describing electron waves in a periodic medium is to express them in terms of the eigenfunctions of the translation operator.

These eigenfunctions have a well defined frequency and form a complete set that can be used to describe any wave.

An eigenfunction is specified by the k that appears in the expression for the eigenvalue. It turns out that the physical interpretation of k is nearly the same as the wave number of harmonic waves (usually also called k).

The eigenfunctions of the translation operator can be readily constructed from any two linearly independent solutions of the one-dimensional Schrödinger equation. A convenient choice is,

$$\frac{d\psi(x)}{dx}=e^{ik(x+a)}\frac{d\psi(x+a)}{dx}$$

There for,

$$Asin\alpha x+Bcos\alpha x=e^{ik(x+a)}(Asinh\gamma x+Bcos\alpha a)$$

Now take X= a and a=-b

Then,

$$A\;\sin\alpha a+B\;\cos\alpha a=e^{ik(a+b)}(A\;\sinh(-\gamma b)+B\;\cosh(-\gamma b))$$

$$A\lbrack\;\sin(\alpha a)+e^{ik(a+b)}\;\sinh(\gamma b)\rbrack+B\lbrack \;\cos(\alpha a)-e^{ik(a+b)}\;\cosh(\gamma b)\rbrack=0—–(1)$$

By considering (a+b)=c and taking the derivative,

$$A\alpha\lbrack\;\cos\;(\alpha c)+e^{ik(a+b)}\;\cos h\;(\gamma b)\rbrack+B\lbrack\;-\sin\;(\alpha c)-\;e^{ik(a+b)}\;\sin h\;(\gamma b)\rbrack=0—–(2)$$

Equation 1 & 2 have a non trivial solition (i.e, a spolution other than A=B=0 only)

If

$$\frac{\gamma^2-\alpha^2}{2\gamma\alpha}\;\sin (\alpha c)\;\sinh (\gamma b)+\cos (\alpha c) \cosh (\gamma b)= \cos (k(b+c)—–(3)$$

At this point it is convenient to consider the special case in which 0b and infinity

While the product  remains constant.

Let

$$P=(\frac{(mV_0bc)}{H^2})$$

Then eqn 3 became

$$\frac P{\alpha c}\;\;\sin(\alpha c)+\cos(\alpha c)=\cos(kc)\;\;——(4)$$

This can be solved graphically

The solution for $$P=\frac{3\mathrm\pi}2$$

(corresponding to the higher barrier) shown below, it plotted by using the 4th equation RHS and LHS

 

We can have a solution for this when K and

$$\frac P{\alpha c}\;\;\sin(\alpha c)+\cos(\alpha c)\leq 1$$

And as cos (kc) always we have a valid solution for range of   for a particular energy region

There for we need to find the  for which there is no valid, as

$$E=\frac{h^2\alpha^2}{2m}$$

This means there are disallowed regions of energy, .e, energy gap

Energy band gap Energies and wave functions for electrons in crystal(periodic potential V(v))

(a) isolated potential well

  • Only discrete energies E allowed
  • Wavefunctions are standing waves

 

(b) Free Electrons

$$\psi_k(x)=Ce^{ikx}$$

$$\psi_k(x.t)=Ce^{i(kx-ωt}$$

  • Plane wave solutions
  • Any energy allowed, $$E=\frac{h^2k^2}{2m}$$
  • Parabolic E versus k

C) periodic potential

  • The ability of electrons to tunnel between barrier wall spreads out the discrete energy levels seen for isolated wells in to bands
  • We fine a number of bands of energies are allowed
  • We have restricted K values $$(-π/a≤k≤π/a)$$
  • The wave functions are bloch wave $$\psi_k(x)=e^{ikx} U_k(x)
  • $$psi_k(x,t)=e^{i(kx-ωt}U_k(x)$$
  • Which is a modulating travelling wave
  • Wave functions act like free electrons(almost)
  •  

General result from Kronig_penny model

  • If potential barriers between well is strong, energy bands are narrowed and spaced apart
    • Correspond to crystals in which electrons are tightly bond to ion core and wavefunction do mot overlap much with adjacent core and also true for lowest energy bands
  • If potnetial barriers between wells is weak, energy bands are wide and spaced close together
    • This is a typical situation for metal with =weakly bonded electrons. Here there ”nearly free electron model” works
  • If potnetial barriers between wells is weak, energy bands are wide and spaced close together
    • This is a typical situation for metal with =weakly bonded electrons. Here there ”nearly free electron model” works

By solving the Schrödinger equation, it is found that the allowed energy states of the electrons form energy bands. Within these bands, the electrons have allowed energy values, while outside the bands, they are forbidden from having certain energies. The width and location of these energy bands depend on the properties of the crystal lattice, such as the spacing between the potential barriers and the height of the barriers.

The Kronig-Penney model helps explain phenomena such as energy bandgaps, where there are ranges of energies for which electrons cannot exist in a crystal lattice. It provides a simplified framework for understanding the electronic properties of real crystalline materials and is often used as a starting point for more advanced theoretical models in solid state physics.