Electron states in a periodic potential. Assume the electrons do not interact with each other. Solve the single electron Schrodinger equation: KJ =

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1 Electo states i a peiodic potetial Assume the electos do ot iteact with each othe Solve the sigle electo Schodige equatio: 2 F h 2 + I U ( ) Ψ( ) EΨ( ). 2m HG KJ = whee U(+R)=U(), R is ay Bavais lattice vecto Bloch theoem: The eigestates ca be witte as a plae wave times a cell-peiodic fuctio u (), u (+R)= u (). ik ( ) = e u ( ). ( 1) Ψ Theefoe, ude lattice taslatio, we have v ik R Ψ ( + R) = e Ψ ( ). ( 2) Covesely, a state satisfyig the taslatioal popety (2) ca always be witte i fom (1). Why? Simply defie ik u ( ) = e Ψ ( ).

2 Poof of the Bloch theo.: Fist we show that the H commutes with lattice taslatio opeatos T R, whee T R Ψ() Ψ(+R). T[ H( ) Ψ( )] = H( + R) Ψ( + R) R = H( ) TΨ( ), fo Ψ( ). Theefoe [T R, H]=0 fo ay R, ad we ca fid the simultaeous eigestates of H ad T R Futhemoe, it is easy to see that T R T R = T R T R = T R+R, which meas that C(R)C(R )=C(R+R ) C(R)=e ik R Note: e ik R is the eigevalue of T R, which depeds o the wave vecto k. Theefoe, we ca label the Bloch states with two idices (,k) bad idex ad wave vecto idex R HΨ = ε Ψ, TΨ = C( R) Ψ, C( R) = 1. HΨ T R Ψ R ik R TΨ ( ) Ψ ( + R) = e Ψ ( ). Q.E.D. R = ε Ψ, ik R = e Ψ.

3 Allowed values of k ae detemied by the B.C. Peiodic B.C. Ψ ( + N a ) = Ψ ( ), i =1,2, ( N1N 2N = N ) i i in k a i i e = 1, i = 1, 2, Nik ai = 2πmi, mi Z, i = 1, 2, m k N b m N b m N b 1 2 = This is simila to the discussio i chap 2 about the allowed values of k fo a fee electo i a empty cubic box. Volume elemet Notice that fom the 1 st lie above we also have That is, thee ae N k-poits i a pimitive cell of the R- lattice, whee N is the total umbe of pimitive cells i a cystal. b b b 1 2 k = = 1 2 N1 N2 N N b b b 1 ( 2π ) =, a1 ( a2 a ) = v = N v = ( 2π ), V 1 ( ) V N as i the fee - electo case. b1 ( b2 b ) k = N

4 Domai of k-vectos Recall that the eigevalues of T R ae e ik R, ad we label the eigestates by (,k). Howeve, (,k+g) labels the same eigestate sice k =k+g gives the same eigevalues e ik R. Theefoe, we oly eed to defie k withi oe pimitive cell of the R-lattice (othes ae edudat) Usually choose the Wige-Seitz cell as a domai (WS cell i R-lattice is called the 1 st Billoui zoe) The 1 st BZ of a bcc lattice

5 Example: The Koig-Pey model i oe-dimesio Let b 0, but keeps bv 0 a costat ( Diac comb) The eegy spectum is ε k ε 2k ε 1k Sometimes it is coveiet to exted the domai of k with the followig equiemet Ψ ε = ε, ( ) = Ψ ( ). + G, k + G, k

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7 Numbe of electo states i a eegy bad Thee ae N k-poits i a eegy bad, b1 ( b2 b ) = N k Each k-poit ca be occupied by two electos (spi up ad dow). each eegy bad has 2N seats fo electos. If a solid has oe valece electo pe pimitive cell, the the eegy bad is half-filled (coducto) If two electos pe pimitive cell, the eithe (a) o eegy ovelap o (b) eegy ovelap isulato coducto Fo example, all alkali metals ae coductos, while alkai eath elemets be coducto/isulato.

8 Femi suface Fo fee electos, the FS is a sphee. Fo electos i a lattice, the FS is detemied by ε whee ε F is the eegy of the top-laye electos. Fo case (a), thee is o Femi suface Fo case (b), thee ae two baches of Femi sufaces Desity of states ( k ) = ε F, Fo fee electos i D, g(ε) E Fo electos i a lattice, it s moe complicated g z g( ε) = g ( ε), whee d k ( ε) = 2 ( ( k )) ( π ) δ ε 2 ε A typical DOS, with seveal cusps (va Hove sigulaities)

9 Fou types of va Hove sigulaity The spectum must cotai at least oe citical poit of each of the type S 1 ad S 2. (Zima, p.50) 2D case: at least oe saddle poit fo the suface E(k)