Kronig's Penny Model.pptx

Presentation on kronig's penny model

1. Chameli Devi Group of Institutions Title - Kronig’s Penny Model Submitted by : Piyush Gupta Submitted to : Abhay Tambe

2. Introduction The Kronig – penney model is a simplified model for an electron in a one- dimensional periodic potential. The possible states that the electron can occupy are determined by the Schrodinger equation, In the case of Kronig -Penney model, the potential V(x) is a periodic square wave. The virtue of this model is that it is possible to analytically determine the energy eigen values and eigen functions . It is also possible to fine analytic expressions for the dispersion relation (E vs. k) and the electron density of states.

3. Solution of the Schrodinger equation for the kronig -penny potential Since the Kronig -Penney potential exhibits translational symmetry, the energy eigenfunctions of the Schrödinger equation will simultaneously be eigenfunctions of the translation operator. As we often do in solid state physics, we proceed by seeking the eigenfunctions of the translation operator. The translation operator  T  shifts the solutions by one period,  Tψ ( x ) =  ψ ( x + a ). Notice that any function of the form, is an eigenfunction of the translation operator with eigenvalue  e ika . The eigenfunctions of the translation operator can be readily constructed from any two independent solutions of the one-dimensional Schrödinger equation. A convenient choice is,

4. The solutions in region 1 (0 <  x  <  b ) are, while the solutions in region 2 ( b  <  x  <  a ) are , Here, For energies where  k 1  or  k 2  are imaginary, the solutions are still real since cos( i θ ) =  cosh (θ) and sin( i θ ) =  i sinh (θ).

5. Any other solution can be written as a linear combination of ψ 1 ( x ) and ψ 2 ( x ). In particular, ψ 1 ( x  +  a ) and ψ 2 ( x  +  a ) can be written in terms of ψ 1 ( x ) and ψ 2 ( x ). These solutions are related to each other by the matrix representation of the translation operator.  The elements of the translation matrix can be determined by evaluating the equation above and its derivative at  x  = 0. The eigenfunctions and eigenvalues λ of this 2 × 2 matrix are easily determined to be, where and

6. If periodic boundary conditions are used for a potential with  N  unit cells, then applying the translation operator  N  times brings the function back to its original position, The eigenvalues of the translation operator are therefore the solutions to the equation λ N  = 1. These solutions are, where  j  is an integer between - N /2 and  N /2,  L  =  Na  is the length of the crystal, and  k j  = 2π j/L  are the allowed  k  values in the first Brillouin zone. The dispersion relation can be determined by first calculating α for a specific energy, solving for the eigenvalues λ and then solving the equation above for the wavenumber  k , Whether the eigenvalues are real or imaginary depends on the magnitude of α. If α² > 4, the eigenvalues will be real and the solutions fall in a forbidden energy gap. If α² < 4, the eigenvalues will be a complex conjugate pair λ +  =  e ika  and λ -  =  e - ika .