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M.C. Chang

Dept of Phys. M.C. Chang. Energy bands (<<) Nearly-free electron limit Bragg reflection and energy gap Bloch theorem The central equation Empty-lattice approximation. (>>) Tight-biding limit.  For history on band theory, see 半導體的故事 , by 李雅明 , chap 4. Density distribution.

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M.C. Chang

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  1. Dept of Phys M.C. Chang • Energy bands • (<<) Nearly-free electron limit • Bragg reflection and energy gap • Bloch theorem • The central equation • Empty-lattice approximation • (>>) Tight-biding limit  For history on band theory, see半導體的故事, by李雅明, chap 4

  2. Density distribution Nearly-free electron (NFE) model • NFE Model is good for Na, K, Al… etc, in which the lattice potential is only a small perturbation to the electron sea. Free electron plane wave Consider 1-dim case, when we turn on a lattice potential with period a, the electron wave will be Bragg reflected when k=±π/a, which forms two different types of standing wave. (Peierls, 1930)

  3. Energy dispersion These 2 standing waves have different electrostatic energies. The difference is the energy gap. For U(x)=U cos(2πx/a) Electron’s group velocity is zero near the boundary of the 1st Brillouin zone Q: where are the energy gaps when U(x)=U1 cos(2πx/a)+U2 cos(4πx/a)?

  4. Energy bands (Kittel, p.170) • The Kronig-Penny model (1930) • Let b0, but keeps bV0 a constant (called a Dirac comb), then Calculated electron energy dispersion

  5. Consider periodic BC, • Ψ(x+Na)=Ψ(x)=CNΨ(x) • C=exp(i2πs/N), s=0,1,2… N-1 General theory for an electron in a weak periodic potential • |Ψ(x)|2is the same in each unit cell. Bloch theorem (1928) • The electron states in a periodic potential is of the form • where uk(r)= uk(r+R) is a cell-periodic function The cell-periodic part uk(x) depends on the form of the periodic potential. A simple proof for 1-dim: Ψ(x+a)=CΨ(x) • Write Ψ(x)as uk(x) exp(i2πsx/Na) • then uk(x) = uk(x+a)

  6. Allowed values of k are determined by the B.C. • Periodic B.C. • (3-dim case) • This is similar to the discussion in chap 4 about the allowed values of k for a phonon. • Therefore, there are N k-points in the 1st BZ, where N is the total number of primitive cells in a crystal.

  7. insulator conductor Difference between conductor/insulator (Wilson, 1931) • There are N k-points in an energy band, each k-point can be occupied by two electrons (spin up and down). • each energy band has 2N “seats” for electrons. • If a solid has one valence electron per primitive cell, then the energy band is half-filled (conductor) • If two electrons per primitive cell, then there are 2 possibilities • (a) no energy overlap or (b) energy overlap • For example, all alkali metals are conductors, while alkali earth elements can be conductor/insulator. (Boron?)

  8. How do we determine uk(x) from the potential U(x)? (1-dim case) Keypoint: go to k-space to simplify the calculation Fourier transform 1. the lattice potential G=2n/a (U0=0) 2. the wave function k=2n/L Schrodinger equation Schrod. eq. in k-space aka. the central eq. • Solve one k at a time, but notice each C(k) is related to other C(k-G) (G) • For a given k, there are many eigen-energies n, with corresponding eigen-vectors Cn.

  9. k -g 0 g • If k is close to 0, then the most significant component of1k(x) is exp[ikx](little superposition from other plane waves) • If k is close to g/2, then the most significant components of1k(x)and2k(x) are exp[i(k-g)x] and exp[ikx], others can be neglected. Focus on one k in the first BZ (and a specific n) The Bloch state nk(x) is a superposition of… exp[i(k-g)x], exp[ikx], exp[i(k+g)x] … The energy ofnk(x) is n(k) (g=2/a) exp[i(k+g)x] exp[ikx] exp[i(k-g)x]

  10. cutoff for kg/2! For example, U(x) = 2U cos2x/a = U exp(2ix/a)+U exp(-2ix/a) There are only 2 components Ug=U-g=U (g=2/a) Matrix form of the central eq. • The eigenvalues εn(k) determines the energy band • The eigenvectors {Cn(k-G), G} determines the actual form of the Bloch states check What are the eigen-energies and eigen-states when U=0?

  11. Energy levels near zone boundaryk=g/2 Cut-off form of the central eq. Energy eigenvalues

  12. Kittel, p.225 • Sometimes it is convenient to extend the domain of k with the following requirement

  13. Kittel, p.227 NFE model in more than 1 dim (a rough discussion) Band structure εn(kx,ky) in 2-dim Origin of energy gaps: Bragg reflection k Laue condition G/2 Bragg reflection whenever k hits the BZ boundary

  14. M  X Folding in 2-dim, the “empty lattice” approximation • How to fold a parabolic “surface” back to the first BZ? • Take 2D square lattice as an example k-space 2/a

  15. In reality, there are energy gaps at BZ boundaries because of the Bragg reflection The folded parabola along ΓM is different M  X Folded parabola along ΓX (reduced zone scheme) • Usually we only plot the major directions, for 2D square lattice, they are ΓX, XM, MΓ 2/a

  16. Folding in 3-dim • The energy bands for an “empty” fcc lattice Actual Band structure for copper (fcc crystal, 3d104s1)

  17. Tight binding model (details in chap 9) Covalent solid, d-electrons in transition metals Alkali metal, noble metal

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