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Published byClarence Walton Modified over 9 years ago
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The Dirac Comb: A First Glance at Periodic Potential Jed Brody
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2 Outline Review of basic quantum mechanics Solution to the Dirac comb problem Some semiconductor devices I made
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3 The Schrödinger Equation Given V(x) and boundary conditions, find (x) and E that satisfy We generally find an infinite number of solutions: 1 (x) and E 1 2 (x) and E 2 3 (x) and E 3. All energies except E 1, E 2, E 3, etc. are forbidden!!!
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4 Example: Infinite Square Well Plot allowed energies in units of :
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5 Continuity of and d /dx is continuous everywhere d /dx is continuous everywhere except where V is infinite If V(x)=- (x-a), Outline of derivation: Integrate Schrödinger equation from a- to a+ Let 0
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6 Commuting Operators If operators A and B commute ([A,B]=AB-BA=0), then we can find eigenfunctions of A that are also eigenfunctions of B. Proof: Let [A,B] operate on a1, an eigenfunction of A with eigenvalue a 1 (A a1 =a 1 a1 ):
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7 The Dirac Comb Consider an infinite, one-dimensional chain of regularly spaced atoms Suppose each positively charged atomic core acts on electrons as a delta-function well (not realistic) Observe: V(x+a)=V(x)
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8 Bloch’s Theorem If V(x+a)=V(x), then the solutions to the Schrödinger equation satisfy (x+a)=e iKa (x) for some constant K. Suppose we know (x) only between 0 and a: (x)=sin( x/a),0<x<a Replace x with (x+a): (x+a)=sin[ (x+a)/a],0<x+a<a -a<x<0 Bloch’s theorem gives (x)=e -iKa (x+a), (x)=e -iKa sin[ (x+a)/a],-a<x<0
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9 First Step of Proof: D and H commute Define the displacement operator: Df(x)=f(x+a) H is the Hamiltonian, (-ħ 2 /2m)d 2 /dx 2 +V(x), for a periodic potential: V(x+a)=V(x) If D commutes with (-ħ 2 /2m)d 2 /dx 2 and with V(x), then D commutes with H D commutes with (-ħ 2 /2m)d 2 /dx 2 because taking the second derivative of a function and then shifting it to the left is the same as shifting it to the left and then taking its second derivative [D,V(x)]f(x)=[DV(x)-V(x)D]f(x) =DV(x)f(x)-V(x)f(x+a) =V(x+a)f(x+a)-V(x)f(x+a) =0 because V(x+a)=V(x)
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10 Completion of Proof D and H commute, so we can choose eigenfunctions of H that are also eigenfunctions of D Let D act on one of these eigenfunctions: D (x)= (x), where is the eigenvalue of D Thus since D (x)= (x+a), (x+a)= (x) Note that (x+2a)= 2 (x), (x+3a)= 3 (x), etc. To keep (x) from blowing up or vanishing upon repeated applications of D, | |=1 (x+a)=e iKa (x)
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11 Restrictions on K The electrical properties of a solid deep in its bulk are independent of edge effects. To avoid edge effects, use periodic boundary conditions Wrap the x axis around on itself, forming a loop of length Na; N is the very large number of potential wells After travelling a distance Na, you return to your starting point: (x+Na)= (x) Bloch’s theorem gives (x+Na)=e iNKa (x), so e iNKa =1 NKa=2 n K=2 n/(Na), n=0,±1,±2,...
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12 Solving the Dirac Comb Problem The potential is a series of delta functions: In between delta functions (e.g.,0<x<a), V(x)=0:
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13 Replace x with (x+a): Apply Bloch’s theorem, (x)=e -iKa (x+a): (x) must be continuous at x=0: Conditions on (x) at x=0
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14 Equation To Find Allowed Energies Eliminating A and B from previous equations gives On LHS, we know K=2 n/(Na), where n=0,±1,±2,… On RHS, define z ka=a(2mE) 1/2 /ħ and m a/ħ 2 Example: N=8 n0±1 ±2 ±3 ±4 ±5 ±6 2 n/N 0± /4 ± /2 ±3 /4 ± ±5 /4±3 /2 cos(2 n/N)12 1/2 /20-2 1/2 /2-1-2 1/2 /20
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15 N=8 Solution =5
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16 N=50 Solution
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