Three Most Important Topics (MIT) Today

Transcription

1 Three Most Important Topics (MIT) Today Electrons in periodic potential Energy gap nearly free electron Bloch Theorem Energy gap tight binding Chapter 1 1

2 Electrons in Periodic Potential We now know the crystal structure, Bravais lattice, and reciprocal lattice. The reciprocal lattice had a context derived from diffraction a wave effect We now extend this to understand how electrons will behave in periodic potential of our the crystal 2 h 2 () r V() r () r E () r ; V() r V( r T) 2m ψ + ψ = ψ = + Because this is a wave problem in periodic potential, a good way of understanding this is through the k- space rather than the real space. Chapter 1 2

3 Energy Gap Near Free Electron Model V~0 At Bragg planes, for a one-dimensional lattice G G 2 Δ k = G k ( ) = ( ) 2 2 G nπ G nπ k = = ; k' = = 2 a 2 a At k=π/a 2 2 ( + ) = ( + ) cos ( x/ a) 2 2 ( ) = ψ( ) sin ( πx/ a) Chapter 1 3 e ikx ψ ( + ) = exp( iπx/ a) + exp( iπx/ a) = 2cos( πx/ a) ψ ( ) = exp( iπx/ a) exp( iπx/ a) = 2isin( πx/ a) n ψ π Bragg Plane e ik x 0 π/a 2π/a n The different charge distribution causes variation of the electrostatic potential energy in the field of the positive ion cores k

4 Energy Gap Near Free Electron Model Kittel Fig 7-3 Chapter 1 4

5 Energy Gap Near Free Electron Model Kittel Fig 7-2 Chapter 1 5

6 Bloch Theorem The solution of Schrodinger equation for a periodic potential: 2 h 2 () r V() r () r E () r ; V() r V( r T) 2m ψ + ψ = ψ = + We know n(r+t) =n(r) Ψ nk (r+t) 2 = Ψ nk (r) 2 ikt ψ ( r+ T) = e ψ ( r)or ψ nk nk ikr () r = e u () r with u () r = u ( r+ T) nk nk nk nk Modulated plane wave Probability of finding electron is not constant anymore. n and k are two quantum numbers in S.E. k is the wave vector index and n is called band index. Periodic function coming from the influence of the lattice potential. The general proof of the Bloch Theorem can be done by expanding V(r) and Ψ (r) into a Fourier series and then rewriting the Schrödinger equation. Chapter 1 6

7 Implication of Bloch Theorem The Bloch theorem is quite remarkable, because, as said before, it imposes very special conditions on any solution of the Schrödinger equation, no matter what the form of the periodic potential might be. The quantity k, while still being the index of multiple solutions and the wave vector of the plane wave that is part of the wave function (and which may be seen as the "backbone" of the Bloch functions), has lost its simple meaning: It can no longer be taken as a direct representation of the momentum p of the wave via p = Һ k. The proof could be seen by imposing the momentum operator on the Bloch function and understand that the Bloch function is no longer the momentum eigenstate (Hamiltonian and momentum operators are not commutable). The momentum of the electron moving in a periodic potential is no longer constant ; for the standing waves resulting from reflections at the Brillouin zones boundary it is actually zero (because the velocity is zero), while k is not. Chapter 1 7

8 Implication of Bloch Theorem Instead of associating k with the momentum of the electron, we may identify the quantity k, which is obviously still a constant, with the socalled crystal momentum P, something like the combined momentum of crystal and electron. The crystal momentum P, while not a "true" momentum, still has many properties of momentums, in particular it is conserved during all kinds of processes and obey the relationship of Һ (dk/dt)=-ef under the influence of external field. The index n represents the fact that there are multiple solutions to S.E for given k. ikr ψ () r = e u () r put it back to S.E. k k H u () r = E u () r with B.C. u () r = u ( r+ T) k k k k k k similar to the particle in a box problem, you are going to find multiple solutions of u k (r) which we label with the band index n. Chapter 1 8

9 Reduced Zone Scheme Assume we know a Bloch function with k lies outside the 1 st Brillouin Zone, we can always find a reciprocal lattice vector G such that k = k + G lies within the 1 st B.Z. Periodic in T ψ () r = e u () r =e ( e u ()) r ik ' r ikr igr nk ' nk ' nk ' ikr e unk ' ( r) = ψ nk ' ( r) E = E nk ' n' k This is a significant result that allows any wave vector k outside of the 1 st B.Z. can be represented by a wave vector k, differed by G, in the 1 st B.Z but with different band index n, i.e. E nk E nk Chapter 1 9

10 Reduced Zone Scheme Extend Zone Scheme E band n 0-2π/a -π/a π/a 2π/a Reduced Zone Scheme E band n 2 k band n 1 band n 0-2π/a -π/a π/a 2π/a 1 st B.Z. Chapter 1 10 k

11 Orbit Model: Connection to Chemistry Atom can be approximated with the core and the valence electrons (similar to the pseudo-potential approximation: The core is the nucleus and the filled states of electrons with the same major quantum number n, such as 1S 2, 2S 2, 2P 6, etc... Valence electrons are: outermost electrons in orbital concepts easiest to break off from the nucleus participate in the chemical reactions can break the Coulombic potential (ionization) and contribute to conduction dominant in chemical and semiconductor discussions (why the periodic table is very important) Chapter 1 11

12 Covalent Bonds: H 2 ψ A ψ B antibonding E A E A bonding ψ A + ψ B ψ A - ψ B Kittel, p.73 Wavefunction overlap controls the formation of the bonding and antibonding states. Strongest bond with anti-parallel spin and Pauli Exclusion Principle s modification of charge distribution through spin-dependent Coulomb energy Chapter 1 12

13 Tight Binding Approach for Energy Bands Two atoms are brought together, two separated energy levels are formed for each level of the isolated atom. For N atoms, N levels (bands) are formed for each level of the isolated atom. (Pauli exclusion) The bandwidth of each band is determined by the wavefunction overlap. Larger overlap (shorter nearest neighbor distance), larger bandwidth. Chapter 1 13

14 Band Structure: Semiconductor Band Structure The unique electron momentum and energy relation (E-k dispersion) of particular materials including the periodic potential disturbance from the ion cores (QM problem). Indirect bandgap direct bandgap Chapter 1 14

15 1 st BZ for Diamond/Zincblende Lattice Γ: Zone center (k=0) X : Zone boundary along <100> direction L : Zone boundary along <111> direction Pierret Fig Chapter 1 15

16 E E-k Relationship Effective Mass Zone Boundary 2 nd band m 2 * G/2 m 1 * Free electron 1 st band k m 1 * >m 2 * Near the band maximum (minimum), the E-k relation can be approximated by the parabolic dispersion: 2 2 h k Ek ( ) = E0 + * 2m d E = * 2 2 m h dk dω 1 d E vg = = dk h dk Curvature of the parabola Slope of the parabola In semiconductors, m* is usually smaller than m 0 (free electron mass) due to the larger curvature at the band extreme. Effective mass is even a simpler concept than the band structure. Together with the concept of the bandgap, classical mechanisms can be applied to describe the electron motion in the solid without dealing with QM!! Chapter 1 16

17 Constant Energy Surface Ek ( ) k k = E + h + h + h * * * 2m1 2m2 2m3 k Chapter 1 17

18 Constant Energy Surface 6 ellipsoids Chapter 1 18

19 Electron and Hole The hole is an alternative description of a band with one missing electron. The motion for a hole is that of a particle of positive charge e. E F E F E k k k v e j e v F h Chapter 1 j h 19

20 Density of States (Particle in the 3D Box) E C E V E C =0 x=0 x=l E TOP h 2 h ψ() r = ( + + ) ψ() r = Eψ() r m 2m x y z Assume ψ( r) = ψ ( x) ψ ( y) ψ ( z) ; E = E + E + E x y z x y z 2 2 h d ψ x ( x) = E ( ) 2 xψ x x 2m dx B.C. ψ ( x) = ψ ( x+ L) Circular B.C. x x (See Kittel for periodic B.C.) ik xx π π π ψ x = e where kx = 0, ±, ±, ± K L L L ik y ikxx y ψ () r = Ae e e ik z z k = 2 me/ h = k + k + k x y z Chapter 1 20

21 Density of States (Particle in the 3D Box) k x g(e) k z k ~E 1/2 E 2π 4π 3π From above: kx, ky, kz = 0, ±, ±, ± K L L L 2π 3 one allowed state per volume of ( ) in k space L k y (4 π k / 3) V 3π N total state N( k) = 2 = k k = ( ) (2 π / L) 3π V / h k h 3π N 2/3 V 2mε 3/2 E = = ( ) N( ε ) = ( ) 2 2 2m 2m V 3π h The number of states per volume per unit energy or dn m 2mE Density of States (DOS) ge ( ) = = 2 3 dedv π h DOS is zero at the band edge in QM sense. g(e)de is the available states from E to E + de, not the carrier concentration yet! Chapter 1 21

22 Density of States in Specific Material In the real material considering the periodic potential, the derivation of the DOS is much more complicate. But at the band extreme, it can be approximated by the similar parabolic relation with correction on the electron effective mass. conduction band valence band * * mn 2 mn( E Ec) gc( E) = E E 2 3 π h * * mp 2 mp( Ev E) gv( E) = E E 2 3 π h c v Chapter 1 22

23 Fermi Distribution (Occupancy of States) How S states is filled by N electrons? The carrier distribution will be described by the Fermi functions (both electrons and holes are Fermions. In contrast, photons and phonons are Bosons). This describes the number of carriers in a specific energy level. Given the Fermi energy E F of the system (E F is a constant in equilibrium), the Fermi distribution function with respect to the energy of a single-particle state E is: f ( E) = ( E E )/ k 1 + e 1 F B T Derived from statistic mechanics and thermodynamics Chapter 1 23

24 f ( E) = ( E E )/ k 1 + The Fermi Function e 1 F B T if E >> E F, f(e) exp( (E F -E)/k B T),which is the Boltzmann distribution for electrons if E << E F, f(e) 1 - exp(- (E F -E)/k B T) or holes for exp( -(E F -E)/k B T), if E = E F, f(e) = 0.5 Below E F : occupied f(e) 0.5 Above E F : empty T=0 f(e) T>0 0.5 E F E F Chapter 1 24

25 Material Characterization by Bandgap Insulator (in room temperature) : E gap > 4eV (SiO 2 : 9.1eV, Si 3 N 4 : ~5eV) Conductor: E gap < 3 kt ~ 0.1eV, or even negative bandgap Semiconductor: anything in between, depending on what temperature we are talking about. Regular semiconductor in room temperature: Si: E gap = 1.12 ev; Ge: E gap = 0.66eV and GaAs: E gap = 1.42 ev: this is the system that can be dealt with most easily. For wide-bandgap materials (E gap ~ 2-3eV), conduction is usually governed more by the structural imperfection For narrow-bandgap materials (E gap ~ 0.25eV), conduction is usually governed by DOS of available states. Chapter 1 25

26 Material Characterization by Bandgap Metal Semi-metal Semiconductor Insulator Chapter 1 26

27 Free Carrier Concentration in Materials Kittel, Fig. 8-1 Chapter 1 27

28 Intrinsic Semiconductors Intrinsic case (electrical properties of pure semiconductor) All conduction caused by thermal breakup of electron-hole pairs n=p=n i, cm -3 for Si, for GaAs, and for Ge at 300K There are about atoms/cm 3, and valence electrons Because the energy for generating electron-hole pairs is from the thermal energy, n i increases exponentially with temperature. Chapter 1 28

29 N-type Semiconductors Charge species: n and N D+ and n N D+ in bulk N-type semiconductors for charge neutrality. Chapter 1 29

30 P-type Semiconductors Charge species: p and N A and p N A in bulk N-type semiconductors for charge neutrality. Chapter 1 30

31 Doping in Semiconductors Use of doping (impurity) to create conduction carriers more easily Ionization energy of hydrogen The Bohr model: E B = q 4 m*/2(4πε s ε 0 h) 2 E h m*/m 0 ε s2 0.1eV This is because the carrier looks at the ionized impurity through the sea of other Si atoms and valence band, and the bond strength needs to be corrected with the dielectric constant (polarization effect) and the effective mass (periodic potential) There is solid solubility limit depending on temperature. Above the limit, the dopants will form segregates. N-type E B (ev) P-type E B (ev) Donors Acceptors P B As Al Sb Ga In 0.16 Chapter 1 31