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Even if the potential is not a well, similar gaps occur

This text explains the formation of energy bands and band gaps in periodic potentials, specifically focusing on the nearly-free-electron model. It discusses the different types of wavefunctions and energy bands, as well as the appearance of band gaps at each Brillouin Zone edge.

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Even if the potential is not a well, similar gaps occur

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  1. Even if the potential is not a well, similar gaps occur 0 < x < a BAND 2 Forbidden band gap E(k) (a.u.) -b < x < 0 Ka/ BAND 1 The left graph shows the energy bands. For values of K where the left side of the equation has a magnitude < 1, then k is real and energy bands are allowed. ka/

  2. –/a /a To better understand why only the edge of the BZ is effected, compare to the free-electron model Free electron dispersion Let’s slowly turn on the periodic potential

  3. U U1 x -b 0 a a+b 2a+b 2(a+b) Electron Wavefunctions in a Periodic Potential(Another way to understand the energy gap) Consider the following cases: Wavefunctions are plane waves and energy bands are parabolic: –/a /a Electrons wavelengths much larger than atomic spacing a, so wavefunctions and energy bands are nearly the same as above

  4. Electron Wavefunctions in a Periodic PotentialU=barrier potential Consider the following cases: Wavefunctions are plane waves and energy bands are parabolic: –/a /a Electrons wavelengths much larger than a, so wavefunctions and energy bands are nearly the same as above Electrons wavelengths approach a, so waves begin to be strongly back-scattered by the potential: Electrons waves are strongly back-scattered (Bragg scattering) so standing waves are formed:

  5. These two approximate solutions to the S. E. at have very different potential energies. has its peaks at x = a, 2a, 3a, … at the positions of the atoms, where U is at its minimum (low energy wavefunction). The other solution, has its peaks at x = a/2, 3a/2, 5a/2,… at positions in between atoms, where U is at its maximum (high energy wavefunction). Either: Or: a Due to the ±, there are two such standing waves possible: The nearly-free-electron model (Standing Waves) Nodes at ions Nodes midway between ions

  6. a Strictly speaking we should have looked at the probabilities before coming to this conclusion: The nearly-free-electron model Symmetric and Antisymmetric Solutions Different energies for electron standing waves

  7. -2π/a –π/a π/a 2π/a BAND GAPS APPEAR AT EACH BRILLOUIN ZONE EDGE Summary: The nearly-free-electron model The periodic potential U(x) splits the free-electron E(k) into “energy bands” separated by gaps at each BZ boundary. The energy difference between + and - causes this split. Gaps in the energy bands form. (Remember to look in 3D though because there might not be a gap along another direction. It mainly matters if there is a gap around EF. E- Eg E+

  8. -2π/a –π/a π/a 2π/a BAND GAPS APPEAR AT EACH BRILLOUIN ZONE EDGE Approximating the Band Gap cos2(x)-sin2(x)= cos(2x) E- Eg E+ For square potential: U(x) =Uo for specific values of x (changes integration limits)

  9. How to Determine the Energy Bands • If most energy bands look a lot like free electron bands with some modifications at the Brillouin Zone edge, then we mainly need to know how to draw the free electron bands! • But now along all interesting directions within the Brillouin Zone.

  10. Empty Lattice Bands for bcc Lattice For the bcc lattice, let’s plot the empty lattice bands along the [100] direction in reciprocal space. What makes the band diagrams for a bcc look different than an fcc? E k –/a  /a More generically:

  11. General reciprocal lattice translation vector: Let’s use a simple cubic lattice, for which the reciprocal lattice is also simple cubic: And thus the general reciprocal lattice translation vector is: (but only certain values allowed) Empty Lattice Bands for bcc Lattice For the bcc lattice, let’s plot the empty lattice bands along the [100] direction in reciprocal space.

  12. The maximum value(s) of x, y, and z depend on the reciprocal lattice type and the direction within the 1st BZ. For example:  H ky N H To H=[100] 0 < x < 1, y=0=z kx 0 < x < ½, 0 < y < ½, z=0 To N=[110] Energy Bands in BCC (Pay attention: Will have similar HW problem) We write the reciprocal lattice vectors that lie in the 1st BZ as: Remember that the reciprocal lattice for a bcc direct lattice is fcc! Here is a top view, from the + kz direction:

  13. Group: Plot the Empty Lattice Bands for bcc Lattice Thus the empty lattice energy bands are given by: Along [100], we can enumerate the lowest few bands for the y = z = 0 case, using only G vectors that have nonzero structure factors (h + k + l = even, otherwise S=0): What if we plotted along [110]? {G} = {000} What other values of h,k,l? {G} = {110} Is it just 110 or the family? {G} = {200}

  14. Empty Lattice Bands for bcc Lattice: Results Thus the lowest energy empty lattice energy bands along the [100] direction for the bcc lattice are: A good approximation for BCC monovalent metals.

  15. The maximum value(s) of x, y, and z depend on the reciprocal lattice type and the direction within the 1st BZ. For example:  H ky N H To H=[100] 0 < x < 1, y=0=z kx 0 < x < ½, 0 < y < ½, z=0 To N=[110] Do Energy Bands in Other Directions for BCC One of your next homework problems: Do Gamma to N and Gamma to P, and H to N and N to P We write the reciprocal lattice vectors that lie in the 1st BZ as: Remember that the reciprocal lattice for a bcc direct lattice is fcc! Here is a top view, from the + kz direction:

  16. Two Common Approaches Do Non-Physicists Think Quantum Wells are Easy? A Physicist Thinks Quantum Wells are Easy (Kroniq-Penney Model or Nearly Free Electron Approx.) “Chemist’s View” -Start with atomic energy levels & build up the periodic solid by decreasing distance between atoms (tightbinding or linear combination of atomic orbitals U(x) x d

  17. bare atoms solid Approach 2: Tightbinding or Linear Combination of Atomic Orbitals (LCAO) • Assume the atomic orbitals ~ unchanged Atomic energy levels merge to form molecular levels & merge to form bands

  18. Compare to hydrogen • When atoms are covalently bonded electrons are shared by atoms • Example: the ground state of the hydrogen atoms forming a molecule • If atoms far apart, little overlap • If atoms are brought together the wavefunctions overlap and form the compound wavefunction, ψ1(r)+ψ2(r), increasing the probability for electrons to exist between atoms These two possible combinations represent 2 possible states of two atoms system with different energies

  19. LCAO: Electron in Hydrogen Atom(in Ground State) 5 4 Second hydrogen atom 3 2 Do you see a pattern? 1 0 Number of Nodes?

  20. c0 c1 c2 c3 c4 k=p/a k=p/2a k=0 a Group: For 0, 2 and 4 nodes, determine wavefunctions If there are N atoms in the chain there will be N energy levels and N electronic states (molecular orbits). The wavefunction for each electronic state is: Yk = Seiknacn= e0c0+eikac1+e2ikac2+e3ikac3+e4ikac4 • a is the lattice constant, • n identifies the individual atoms within the chain, • cnrepresents the atomic orbitals of each individual atom • k is a quantum # that identifies the wavefunction and tells us the phase of the orbitals. n The larger the absolute value of k, the more nodes one has

  21. c0 c1 c2 c3 c4 k=p/a k=p/2a k=0 a Infinite 1D Chain of H atoms k = p/a Yp/a = c0+(exp{ip})c1 +(exp{i2p})c2 +(exp{i3p})c3+(exp{i4p})c4+… Yp/a = c0 - c1 + c2 - c3 + c4 +… k = p/2a Yp/2a = c0+(exp{ip/2})c1 +(exp{ip})c2 +(exp{i3p/2})c3+(exp{i2p})c4+… Yp/2a = c0 + i c1 - c2 - i c3 + c4 +… k = 0 Y0 = c0+c1 +c2 +c3 +c4 +… k=0  orbital phase does not change when we translate by a k=p/a  orbital phase reverses when we translate by a

  22. Infinite 1D Chain of H atoms • What would happen if consider k> p/a? • If not obvious, try k=2 p/a. What is the wavefunction? Yk = S eiknacn

  23. Energy bands of a crystal • Tight binding model results in the same form of energy bands as in the nearly free electron model • Note the higher energy states are nearly free This method gives good core electron bands! The high bands are not as good because electrons act free!

  24. Effect of Orbital Overlapband width or dispersion=the difference in energy between the highest and lowest energy levels in the band • If we reduce the lattice parameter a (bring closer together) it has the following effects: • The spatial overlap of the orbitals increases • The band becomes more bonding (energy reduces) at k=0 • The band becomes more antibonding (energy up) k=p/a. • The increased antibonding is larger than the increased bonding. • The bandwidth increases. • The electron mobility increases. • Wide bands  Good orbital overlap  High carrier mobility

  25. Band Structure Linear H Chain E(k) EF 0 k p/a • The Fermi energy separates the filled states (E < EF at T = 0 K) from the empty states (E > EF at T = 0 K). • Here splits the band (each band holds 2 electrons per primitive cell) • A 1D chain of H atoms is predicted to be metallic because the Fermi level cuts a band (there is no gap so it takes only an infinitesimal energy to excite an electron into an empty state). • The band runs "uphill" (from 0 top/a) because the in phase (at k=0) combination of orbitals is bonding and the out of phase (at k=p/a) is antibonding.

  26. How Bands “Run” Yk = Seiknacn(tight binding) applies in general Itdoes not, however, say anything about whether the lowest energy should occur at k=0 or at the BZ edge. For a chain of H atoms (s orbitals) it is clear that E(k = 0) < E(k = n/a), meaning they prefer to be in phase. But consider a chain of p orbitals. Is being in phase preferred? Why?

  27. One electron in the p Orbital Runs Opposite to s

  28. Merging the individual potentials • We will refer to the difference of the potential U between a single atom and what forms from combining many • This difference will show up in overlap integrals

  29. Consequence of Bloch’s TheoremProbability * of finding the electron • Each electron in a crystalline solid “belongs” to each and every atom forming the solid • Very accurate for metals where electrons are free to move around the crystal!

  30. Just need Fourier components of UG(or UG along high symmetry parts of Fermi surface) The potential of a crystal with translational symmetry may be expanded in a Fourier series in the reciprocal lattice vectors G • Values of UG decrease rapidly with increasing G. • By pulling into Sch. Eq., it can be shown that the wavefunction may be expressed as a Fourier series summed over all values of the wavevector

  31. Where do energy gaps come from? Real materials have gaps in the available levels GaAs Bringing atoms close together. Even amorphous silicon has band gaps.

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