Band theory Handout - University of California, Berkeley...

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University of California, Berkeley EE230 - Solid State Electronics Prof. J. Bokor Band theory Handout.fm Band structure (read Kittel, ch. 7) Now we include the crystal potential . This will lead to a more complicated dispersion for the electron states. First consideration is that the potential is periodic in the space lattice: We then expect the electron density is also periodic. Note, it is not necessarily that is periodic in the space lattice. Based on the above, we can write: where is primitive lattice vector, and λ is an arbitrary phase. for 1D, periodic BC’s, , so So we can write: This is called a Bloch function. with , L Na. General 3D crystal case Since periodic in lattice, make a Fourier analysis: G: reciprocal lattice vectors Vr () Ψ r Ψ r λΨ ra i + = a i λ N 1 = n 12 N ,, , = Ψ x U k x e i 2 π nx Na = k 2 π n L --------- = U k xa + U k x =
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- 43 - University of California, Berkeley EE230 - Solid State Electronics Prof. J. Bokor The sum over G only goes over reciprocal lattice vectors, as proven previously. Now make a Fourier analysis of : here k goes over all values allowed by the BC’s; is not periodic in the lattice. Writing down the Schrodinger equation: plugging in the Fourier series for : multiply by and integrate over V: [use and ] then change dummy variable back to k. Define kinetic energy This is called the central equation. It provides the starting point for all band structure theory. It actually describes a set of simultaneous equations for a set of coefficient for wavevector k , C(k) . Notice that C(k) is only coupled to the set of vectors C(k-G). Ψ r () Ψ r Ck e ik r k = Ψ r Ψ r h _ 2 k 2 2 m ---------- U G e iG r EC k G + e k 0 = e ik ' r e ' r e V d V V δ kk ' = e ' r e iG k + r V d V δ ' G , = k '
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- 44 - University of California, Berkeley EE230 - Solid State Electronics Prof. J. Bokor The wavefunction corresponding to a particular value of k is similarly composed of a super- position of all components with wavevectors k+G , where we sum over all G vectors: We can thus write: The electron wavefunction representing a particular k vector is written as a product of the function u k , which is periodic in the lattice, times a plane-wave factor. This result is known as the Bloch theorem. Reduced zone scheme The k label is not unique. It actually labels a superposition of wavevectors separated by all possible G vectors. We can choose any one of these as the label for this wavefunction. We make the convention to choose k in 1st BZ. To see how this works, consider free-electron (empty lattice): V(r) = 0 For outside 1st BZ, find so that is in 1st BZ. Drop the prime and define the band index, n. then Ψ k r () Ck G e ik G r G = e iG r G e ik r = k G k ' kG =
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- 45 - University of California, Berkeley EE230 - Solid State Electronics Prof. J. Bokor Label Ga /2 π 2mE(000)/ 2mE( k x ,0,0)/ 1 000 0 2,3 4-7 010, 0 0, 001, 00 8-11 110,101, 1 0, 10 12-15 10, 01, 0, 0 E n k x k y k z ,, () h _ 2 2 m ------- k x G nx + 2 k y G ny + 2 k z G nz + 2 ++ [] = 0 1 2 3 4-7 8 - 1 2 5 E π a -- k x , 00 (100) direction Display entire band- structure in 1st BZ h _ 2 h _ 2 k x 2 100 100 , 2 π a ------ ⎝⎠ ⎛⎞ 2 G
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Band theory Handout - University of California, Berkeley...

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