Electron Band Gap

1. Electron Band Gap Wesley Matson

2. Content of Presentation • Project Goals • Background Physics • Details of code • Results from program • Discuss importance of Energy Gap

3. Project Goals • Write a program using python to integrate the Schrödinger equation • Create a 1-D array of potential wells to simulate a crystal lattice • Show there is a forbidden energy gap that increases in size as the number of potential wells increase

4. 2dsin(0)=nl d=spacing of parallel atomic planes 0=angle of incidence l=wavelength of incident beam Requires l<=2d Bragg reflection of electron waves in crystals cause energy gap. At locations of Bragg reflection solution to the Schrödinger equation do not exist. Braggs Law of Diffraction

5. Defined as the primitive zone in reciprocal lattice Made up of all the wave vectors that satisfy the diffraction condition Wave vectors outside of Brillouin zone do not lead to solution of the Schrödinger Equation Brillouin Zone

6. Origin of Energy Gap • In one dimension, solution to the Schrödinger equation at the boundaries of Brillouin are standing waves. • Standing waves made up of equal parts of left and right moving waves • Waves traveling in different directions have different energy, corresponding to energy gap

7. Define regions were the potential is zero, the top of the potential wells Include parameter z so one run of the program can loop over multiple well configurations def V(x,z): if (x<0.5): return 0 if (x>1 and x<1.5): return 0 if (x>2 and x<2.5): return 0 if (x>3 and x<3.5): return 0 if (x>4 and x<4.5): return 0 if (x>5 and x<5.5): return 0 if (x>6 and x<6.5): return 0 if (x>7 and x<7.5): return 0 if (x>8 and x<8.5): return 0 if (x>9 and x<9.5): return 0 if x>z: return 0 else: return H Creating the Potential Function

8. Signswitch Function def signswitch(Enow): philow=integrate(Enow,N,dx) phihigh=integrate(Enow+0.1,N,dx) if ((philow*phihigh)<0): return energyfinder(Enow,Enow+0.1) else: return signswitch(Enow+0.1) • Compares the sign of the end of the wave function at two different energies • If the sign is different there is a solution between the two points • If the sign is different rerun the function again with next energy difference

9. Imports two energies from the signswitch function Finds the midpoint and determines the sign of the last point of the wave function Replaces it with the energy value that matches its sign Loops until a desired accuracy has been reached def energyfinder(upper, lower): upperpsi=integrate(upper,N,dx) lowerpsi=integrate(lower,N,dx) while(fabs(upper-lower)>EPS): E=(upper+lower)/2.0 midpsi=integrate(E,N,dx) if(midpsi*lowerpsi>0.0): lowerpsi=midpsi lower=E else: upperpsi=midpsi upper=E # normalize sum=0.0 for i in range(0,N): sum+=phi[i]*phi[i]*dx allowedenergy=gcurve(gdisplay=pic,color=color.green) for x in arange(0,20,0.01): allowedenergy.plot(pos=(x,E/-H)) print E return E Energyfinder Function

10. Emax=0 Elow=H phi[0]=1.0 p[0]=0.0 while (Elow<Emax): Enew=signswitch(Elow) Elow=Enew+0.00000001 • Set energy range and even or odd solution • First call Signswitch with low energy • Signswitch uses integrate function to find first energy range • Energyfinder bisects to find solution • Recall Signswith with energy solution plus small step • Loop until max energy to find all solutions in range

11. Energy Band Gap • Energy gap is clearly seen between top and bottom allowed energy levels • 1 large energy gap along with 3 smaller • Generated from 5- 15 wells, width ½, spacing of 1

12. Energy Level Spacing • As number of wells increase, energy level spacing decreases by app. 1/n • As number of wells increase, number of energy levels increase by n

13. Results • Expected energy gap to increase with the number of wells app. 1/n • Expected number of solutions to be constant • Energy gap was constant with increase in wells • Energy Level spacing deceased by 1/n

14. Importance of Energy Gap • Conductivity is extremely sensitive to how energy bands are filled and magnitude of energy gap • Metals and conductors have a energy bands filled 10-90% • Insulators have filled energy gaps • Conductivity has a range of 10^32 orders of magnitude

15. Semiconductors • Feature conduction band that is empty at absolute zero that fills up as the temperature increases • Conductivity varies with temperature • Conductivity depends of Eg/Kt • As temperature increases electrons are thermally excited into the conduction band