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Full Band Energy Gap Part Filled Band E Solid State Theory Physics 545 Band Theory III Each atomic orbital leads to a band of allowed states in the solid Band of allowed states Gap: no allowed states Band of allowed states Gap: no allowed states Band of allowed states Independent Bloch states F1 0 Solution of the tight binding model -2 α= 10 is periodic in k. Apparently have -4 γ = 1 an infinite number of k -states for -6 E(k) each allowed energy state. -8 In fact the different k-states all -10 -12 equivalent. -14 -16 ikRk.R Bloch states ψ (r + R) ≡ e ψ (r) -18 -4 -2−π/a 0π/a 2 4 Let k = k′́ + G where k′ is in the first Brillouin zone k [111] direction andGid G is a reci procal ll latti ce vector. ψ(r + R) ≡ eik′.ReiG.Rψ(r) But G.R = 2πn, n-integer. Definition of the reciprocal lattice. So eiG.R = 1 and ψ(r + R) ≡ eik′.Rψ(r) eik.R ≡ eik′.R k′ is exactly equivalent to k. The only independent values of k are those in the first Brillouin zone. Reduced Brillouin zone scheme The onlyyp independent values of k are those in the first Brillouin zone. Discard for |k| > π/a Results of tight binding calculation 2π/a -2π/a Displace into 1st BZB. Z. Results of nearly free electron calculation Reduced Brillouin zone scheme Extended, reduced and periodic Br illoui n zone sch emes PidiZPeriodic Zone R Rdeduced dZ Zone E xtend ddZed Zone All allowed states correspond to k-vectors in the first Brillouin Zone. Can draw E(k) in 3 different ways The number of states in a band Independent k-states in the first Brillouin zone, i.e. ⏐kx⏐ < π/a etc. 2πn FiniteFinite ccrystal:rystal: oonlynly ddiscreteiscrete k-states aallowedllowed k = ± x , n = 0,1,2,.... etc. x L x Monatomic simple cubic crystal, lattice constant a, and volume V. One allowed k state per volume (2π)3/V in k-space. Volume of first BZ is (2π/a)3 Total number of allowed k-states in a band is therefore 3 ⎛ 2π ⎞ (2π) V ⎜ ⎟ = = N ⎝ a ⎠ V a 3 Precisely N allowed k-states i.e. 2N electron states (Pauli) per band This result is true for any lattice: each primitive unit cell contributes exactly one k-state to each band. Metals and insulators In full band containing 2N electrons all states within the first B. Z. are occupied. The sum of all the k-vectors in the band = 0. A partially filled band can carry current, a filled band cannot Insulators have an even integer number E of electrons per primitive unit cell. With an even number of electrons per unit cell can still have metallic behaviour due to ban overlap. EF Overlap in energy need not occur in the same k direction 0 π k a MtldMetal due t o overlapping bands E E E EF 0 π k 0 π k a a 0 π k a Empty Band Partially Energy Gap Filled Band Part Filled Band EF Full Band Part Filled Band Energy Gap Full Band INSULATOR METAL METAL or SEMICONDUCTOR or SEMI-METAL Bands in 3D Germanium In 3D the band structure is much more complicated than Figure removed to in 1D because crystals do not reduce file size have spherical symmetry. The form of E(k) is dependent upon the direction as well as the magnitude of k. • Chemical bonds and electron bands. a) Number of electrons in any band is finite because the density of states is finite. E 3/ 2 1/ 2 top 8π 2m(E) E 8π 2 3/ 2 1/ 2 ρ (E) = N = m(E) E dE 3D 3 h3 ∫ h Ebottom b) Bands are formed from molecular orbitals. Filling of Energy Bands ⇒ An important property of a full band is that it is UNABLE to carry a net current since for each state in the band we can identify a corresponding state with equal and OPPOSITE momentum that is filled by an electron. To drive a net current through the crystal it is necessary to induce an IMBALANCE in the filling of momentum states ⇒ For an energy band that is filled completely however this requires that we excite electrons ACROSS the forbidden gap. E • Situation in which the lowest energy band is filled completely with electrons ENERGY GAP • the only way in whhhich a net current can f low is to excite electrons across the energy gap • if the energy gap is large however excitati on cannot be achieved and so no net current is allowed to flow k −π/a π/a ⇒By the same arguments if the energy band is PARTIALLY filled then it should be very EASY to generate a net current flow in the crystal ⇒ In this situation the forbidden gap lies FAR away from the highest filled electron states and so it is easyyg to use an electric field to generate an imbalance in the filling of momentum states ⇒ A small applied voltage will therefore generate a LARGE current as we discussed previously for free electrons E E ENERGY GAP ENERGY GAP k k −π/a π/a −π/a π/a NO APPLIED ELECTRIC FIELD SMALL ELECTRIC FIELD APPLIED ⇒Electronic band theory presents us a natural scheme for CLASSIFYING different types of materials ⇒ METALS should be materials whose uppermost energy band is only PARTIALLY filled with electrons. ⇒ This explains why these materials are GOOD conductors of electricity ⇒ We expect that insulators on the other hand should be materials whose energy bands are either COMPLETELY full or empty so that an energy gap BLOCKS current flow in these materials E FORBIDDEN GAP FORBIDDEN GAP FILLED STATES FILLED STATES METAL INSULATOR Band structure of metals monovalllent metals multivalent metals, semimetals ⇒What types of elements p roduce p artial or comp lete fillin g of ener gy bands? ⇒ The GROUP I elements should be good METALS since these elements have only ONE valence eltlectron, whereas coordiditination number is 6-12. ⇒ If we have a crystal composed of N atoms there will therefore be N valence electrons which will HALF-FILL a singggyle energy band ⇒ The GROUP IV elements should be INSULATORS since these elements have FOUR valence electrons and so in an N-atom crystal there will 4N valence electrons that FILL two energy bands completely E FILLED STATES FILLED STATES FILLED STATES FILLING OF ENERGY LEVELS BY THE VALENCE ELECTRONS OF GROUP I & IV GROUP I GROUP IV ELEMENTS Semiconductors • In certain materials known as SEMICONDUCTORS however the energy gap that separates the highest filled band in the ground state from the lowest empty band is SMALL * Such materials are INSULATORS at zero temperature since their ground state is one in which the energy bands are either completely full or empty * Since the forbidden gap is small however electrons can be EXCITED across it at higher temperatures to PARTIALLY fill the next band ⇒ The material will no longgper be an insulator at this temperature but will CONDUCT electricity E E FORBIDDEN GAP FILLED STATES FILLED STATES FILLED STATES FILLED STATES INSULATOR INSULATOR SEMICONDUCTOR SEMICONDUCTOR T = 0 T > 0 T = 0 T > 0 Some general COMMENTS on semiconductors •The energy band that holds the valence electrons in the ground state is known as the VALENCE BAND. It is usually formed by Bonding Orbitals. •The lowest empty band is known as the CONDUCTION BAND. It is usually formed by antibonding orbitals. ⇒ The energy gap that separates these bands is usually denoted as Eg * Room tempera ture semicond uct ors are generally materi al s in which Eg is a FEW eV (≤ 3 eV) ⇒ This should be compared to a thermal energy of approximately 40 meV that is available to electrons at room temperature (300 K) E SEMICONDUCTOR Eg (eV) 0 K 300 K CONDUCTION BAND Si 1.17 1.11 Ge 0.74 0.66 Eg InSb 0.23 0.17 InAs 0.43 0.36 InP 1.42 1.47 VALENCE BAND GaP 2.32 2.25 GaAs 1521.52 1431.43 GaSb 0.81 0.68 AlSb 1.65 1.60 SEMICONDUCTOR T = 0 Concept of a hole • At higher temperatures electrons in semiconductors may be excited into the conduction band where they are able carry an electrical current * Each electron leaves behind an EMPTY state in the valence band and to account for current flow in semiconductors we must ALSO consider the role of these HOLE states * If the valence band is COMPLETELY filled, then the total crystal momentum of this ban d is equal to ZERO since for any occupidied k-state we can ident ify an corresponding filled state with OPPOSITE momentum E • The total crystal momentum in a filled energy band is exactly equal to zero • to illustrate this consider the total momentum due to occupation of states 1 & 2 ENERGY GAP • state 1 corresponds to an electron with positive 2 1 momentum while state 2 corresponds to one with equal and opposite momentum • the net crystal momentum of electrons occupying states 1 & 2 is zero and this pairing can be repeated k for all other states in the band −π/a π/a • When the valence band is completely filled with electrons we can write k = 0 ∑ i (17) * if we excite AN electron from the state with wavenumber ke in the valence band into the conduction band equation (17) for the valence band may now be REWRITTEN as (18) ∑ ki = − ke ki ≠ ke * The empty state in the valence band may therefore be viewed as a HOLE which has OPPOSITE momentum to the electron that was excited out of that state kh = − ke (19) • Since the hole corresponds to a missing electron its energy may be written as (20) Eh (kh ) = − Ee (−ke ) = − Ee (ke ) * Equation 10.4 shows that electrons and holes have OPPOSITE energy scales since moving DOWNWARD in the valence band implies INCREASING hole energy • Moving downwards in the valence band corresponds E to iiincreasing hlhole energy • in the picture shown here hole 1 therefore has more energy than hole 2 ENERGY GAP • this is not too difficult to understand if we think of HOLE 2 the total energy of the electrons left in the band HOLE 1 • since hole 1 corresponds to a missing electron from k a lower energy electron state than hole 2 the total −π/a π/a energy of electrons in the band is higher for hole 1 than for hole 2 • While the energy scales are oppositely directed for electrons
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