ME597PHYS57000Fall Semester 2009Lecture 02Electron States in Solids B - PDF document

1 ME597/PHYS57000Fall Semester 2009Lecture
ME597/PHYS57000Fall Semester 2009Lecture 02Electron States in Solids Bloch FunctionsKronigPenney ModelDensity of States What happens when the electronion interaction is more realistically approximated?the 1D KronigPenney Model . . . baUo L

2 =N(a+b) Model A: Note that U(x)=U(x+a+
=N(a+b) Model A: Note that U(x)=U(x+a+b) U(x) The periodic delta function Kronig Penney Model (1930) U(x)=UNote that U(x)=U(x+a) Uo L=NaModel B: x Model C:Etc… Mathematicians have shown that solutions to Hills equation must obey F

3 loquets theorm (aka as Blochs theo
loquets theorm (aka as Blochs theorem) which states that solutions must be of the form: ()()ikxkkxuxe where is some function (that δeπenδs on k) which must have the same periodicity as the lattice. ux ux are known as Bloch

4 functionsAny Unifying Principles?Whenev
functionsAny Unifying Principles?Whenever U(x) is periodic, the resulting Schrödinger equation belongs to a class of δifferential equations known as Hills equation Blochstheoremsstatesthatmatterwhattheformtheperiodic potential,t

5 hesolutionsmusthavecertainpropertiesthat
hesolutionsmusthavecertainpropertiesthatsatisfyverygeneralconditions Consider(x)“correctionfactor”generatesolutionsforperiodicpotentialsstartingfromthetrivialikxsolutionsforconstantpotentialThe(x)arerelatedatomicwavefunctionsindextha

6 tdistinguishesthevarioussolutionsWhat
tdistinguishesthevarioussolutionsWhats it Mean? JS Blakemore, πg 231Plot of (x) Under these circumstances, what is the relationshiπ between E anδ k?? a ux ikx Due to the translation symmetry of U(x),an electron has same probabil

7 ty to be at equivalent points in the lat
ty to be at equivalent points in the lattice.This means () should be somehow related to ().Accordingly, ()()1,2,3xaxmaxm+==This means ()() where J is some function to be determined.It follows that () should be related to () as ()()

8 Accordingly, for N unit cells in a 1D cr
Accordingly, for N unit cells in a 1D crystal, thenxaJxxmaJxxmaxNa+==+) would satisfy ()()What are the implications for J?JxxNa=+ What determines “Bloch form” for 22 Implication is that J must be one of N roots of

9 unity, i.e. 1 ; 1mNwhere cos2sin2==≤â
unity, i.e. 1 ; 1mNwhere cos2sin2==≤≤ m=1m=2 112N 222N m 2mmN Unit circleFor translation through m lattice constants: 2 These translational requirements for can be m

10 et if we write ()()provided () has same
et if we write ()()provided () has same periodicity as ().By specifying m, we specify how many complete cycles the exponential completexuxeuxUx s in a distance L, since for a given m, oscillates through m complete cycles as x goes from 0 to Na

11 .2mWriting K=,we then have()(). Na ixx
.2mWriting K=,we then have()(). Na ixxuxe The Bloch form A few consequences (remember this!) The effect of a translation: ()() with Since ()(),(0)(0)iKxiKaxmaxKxuxeu+==== If we assume the solutions are of the form ()()AND we

12 require that ()()()(i.e., wavefunction s
require that ()()()(i.e., wavefunction solutions repeat after L=Na),then it follows that (i) ()(), and(ii)iKxiKxLxuxexLuxLexuxuxL+=+==+ When does 1?only when 2; m=integeriKLiKLKLmLaN== Another way of saying the same thing Either argment

13 implies that if we choose to write as
implies that if we choose to write as or ;1,2,....then we can write() ()and Bloch's theorem will be explicitly satisfied.iKxNaLxuxe If K has DISCRETE values, then so will E(K)!The bottom line is that K now has DISCRETE values 8Na K= 2Na 4N

14 a 6Na 0 2a . . . . 2KL . . . .1
a 6Na 0 2a . . . . 2KL . . . .1 2 3 4 . . . . . N a Uo (x) Apply these general ideas to KronigPenney delta function model 12 12 ' Require that ()(') and require thatxa dd dxdx −= =ï

15 §  ï
§   (x) sin()cos()cos() and limb is width of barrier (see earlier slidewhen P0, α=K (free electron case)when P, E= n(bound electron case)KaaPPbU=+→∞ Often called the “energy functional

16 8; x= a x= a+ -2.0-1.00.01.02.03.0
8; x= a x= a+ -2.0-1.00.01.02.03.0 02468 Ka cos(Ka) no real solutionsno real solutionsno real solutions real solutionsPlotting it out Delta-function Kronig PennyP=1.65; first two bands 0.05.010.015.020.025.0 0.000.250.500.751.00 K (in un

17 its of a/π) E(eV) E is no longer singl
its of a/π) E(eV) E is no longer single valued with KEnergy gaps appear no solutions for real KE vs. K no longer free electronlike imaginary solutions for k Source: Blakemore, πg 213Compare to freeelectron case Slope velocity Check out

18 the Kronig Penney aππlet athttπ://fer
the Kronig Penney aππlet athttπ://fermilaasueδu/schmiδt/aππlets/kπ/πlugkπhtml Ψ]&#x/MCI; 10;&#x 000;&#x/MCI; 10;&#x 000;Im[m[Ψ Ψ 21016202016ixikxkakeeefTa==⇒=⇒= ~4a The real part of is in red, and the imagi

19 nary part is in blue. The magnitude is p
nary part is in blue. The magnitude is plotted in orange. The potential is shown in black The “Bloch Functions” plot shows the periodic Bloch functions. The color code is identical to the top plot. To emπhasize “atomic” like

20 nature of Bloch functions, choose energy
nature of Bloch functions, choose energy near bottom of square well.Applet notes Finally, where are the electrons? position: (x)(x)charge: (x)*(x)energy: define density of states n(E) E=g(E)f(E) Number of particles per unit volume with e

21 nergy between E and Number of energy st
nergy between E and Number of energy states per unit volume in the energy interval FermiDirac Probability that an electron is actually in the energy state E ()()()limnEEnEdnEdE+∆− From n(E), define the Density of States (DOS): Number o

22 f electron states per unit volume per un
f electron states per unit volume per unit energy at energy E In 3D:In 2D, V → Area In 1D, V Length Given E(k), what is n(E)? E E+ /L =No. of states111lim()kEndnLELdE∆=×=×∆=∆=≡= Units: mIn 1D: Uo a Add e

23 lectrons 111in 1D ()dEdEdKππ == Del
lectrons 111in 1D ()dEdEdKππ == Delta-function Kronig PennyP=1.65; first two bands 0.05.010.015.020.025.0 0.000.250.500.751.00 K (in units of a/π) E(eV) DOS Delta-function Kronig Penny P=1.65; first two bands 0.05.010.015.020.025.0 0.0

24 20.040.060.0 DOS (in units of 10-1 Ener
20.040.060.0 DOS (in units of 10-1 Energy (eV) filled (f=1)empty (f=0) An example of a 3D band structure and DOS (PbSe) Y. Zhang, et al, Phys. Rev. B, 024304 (2009).Band 1Band 2 Peaks in DOS Source: Owens and Poole, pg. 502For idealized iso