Lecture 9Phys 446 Solid State Physics Lecture 9 Nov 9, 2007(Ch. 6.1-6.5)Last time: Finished with the band theory of metalsToday: SemiconductorsOrigin of the energy gapsWe focused on the energy valuesgot away from the zone edgesandnear the zone edges.Metals, Insulators, semimetals, semiconductorsinsulator(semi) metalmetalTight binding modelAssumptions:– atomic potential is strong, electrons are tightly bound to the ions– the problem for isolated atoms is solved: know wave functions φnand energies Enof atomic orbitals– weak overlapping of atomic orbitalsStart with 1D caseBloch function in the form: )(1),(12/1jnNjikXXxeNxkj−=∑=φψwhere Xj= ja – position of the jthatom, a – lattice constant;ψn(x- Xj) – atomic orbital centered around the jthatom – large near Xj, but decays rapidly away from it.Small overlap exists only between the neighboring atoms⎟⎠⎞⎜⎝⎛+=2sin4)(20kaEkEγOriginal energy level Enhas broadened into an energy band.The bottom of the band is E0- located at k = 0The band width = 4γ – proportional to the overlap integralFor small k, ka/2 << 1 (near the zone center)220)( kaEkEγ=−- quadratic dispersion, same as for free electron*2)(220mkEkE==−where - effective massγ12*22am==Generally: 1222*−⎟⎟⎠⎞⎜⎜⎝⎛=dkEdm =For k = π/2, you will find thatγ12*22am=−=The results obtained can be extended to 3D case.For simple cubic lattice, get ⎥⎥⎦⎤⎢⎢⎣⎡⎟⎠⎞⎜⎝⎛+⎟⎟⎠⎞⎜⎜⎝⎛+⎟⎠⎞⎜⎝⎛+=2sin2sin2sin4)(2220akakakEkEzyxγVelocity of the Bloch electron: kEv∂∂==11222*−⎟⎟⎠⎞⎜⎜⎝⎛=dkEdm =Generally, anisotropic: zyxi,jkkEmjiij,, 1*122=∂∂∂=⎟⎠⎞⎜⎝⎛=- inverse effective mass tensorEffective mass is determined by the curvature of dispersion)(1kkEv ∇==3D: Summary  Tight binding model – strong crystal potential, weak overlap. The band width increases and electrons become more mobile (smaller effective mass) as the overlap between atomic wave functions increases Concept of effective mass: in a periodic potential electron moves as in free space, but with different mass: Metals: partially filled bands; insulators – at 0 K the valence band is full, conductance band is empty. Semiconductors and semimetals.  Velocity of the Bloch electron:remains constant in perfectly periodic lattice Density of states. Simple case: zyxi,jkkEmjiij,, 1*122=∂∂∂=⎟⎠⎞⎜⎝⎛=)(1kkEv ∇==212322*221)( EmED⎟⎠⎞⎜⎝⎛==πSummaryVelocity of the Bloch electron:In the presence of an electric field the electron moves in k-space according to the relation: This is equivalent to the Newton’s second law if we assume that the electron momentum is equal to ħkDynamical effective mass: m* is inversely proportional to the curvature of the dispersion. In a general case the effective mass is a tensor: pc= ħk is called the crystal momentum or quasi-momentum. The actual momentum is given byCan show that p = m0v, where m0is the free electron mass, v is given by the above expression)(1kvkE∇==1222*−⎟⎟⎠⎞⎜⎜⎝⎛=dkEdm =zyxi,jkkEmjiij,, 1*122=∂∂∂=⎟⎠⎞⎜⎝⎛=kkψψ∇− =iPhysical origin of the effective massSince p = m0v - true momentum, one can write:The total force is the sum of the external and lattice forces.But So, we can write Lexttotdtdm FFFv+==0*00mmdtdmextFv=LextextmmFFF+=0*The difference between m* and m0lies in the presence of the lattice force FLConductors vs. SemiconductorsCrystal structure and bondingSemiconductors include a large number of substances of different chemical and physical properties. Main types of semiconductors:•Group IV semiconductors - Si, Ge.- Crystallize in the diamond structure (fcc lattice with a basis composed of two identical atoms) - covalent crystals, i.e., the atoms are held together by covalent bonds- the covalent electrons forming the bonds are hybrid sp3atomic orbitals•III-V (GaAs, InP, etc.), II-VI (ZnSe, ZnS) semiconductors and alloys- zinc blende structure (same as diamond but with two different atoms)or hexagonal wurtzite structure (GaN)- also covalent bonds, but polar - the distribution of the electrons along the bond is not symmetric•Some other compounds (I-VII, various oxides, halogenides, organic semiconductors...)Carrier density of metals and semiconductorsSemiconductors have higher resistance than metals (~ 10-2to 109Ohm⋅cm)Typical metals: 10-6Ohm⋅cmTemperature dependence of conductivity• Semiconductors have the property that the resistance increases with decreasing temperature – opposite to metal behavior• This is because the number of conduction electrons changes dramatically as a function of temperatureTemperature (K)electron concentration (cm-3)Band structureSemiconductor - a solid in which the highest occupied energy band, the valence band (VB), is completely full at T = 0°K However, the gap above this band is small, so that electrons may be excited thermally at room temperature from the valence band to the next-higher band –the conduction band (VB).Electrons are excited across the gap → the bottom of the conduction band is populated by electrons, and the top of the valence band - by holes. As a result, both bands are now only partially full → can carry a current if an electric field were appliedThe energy of the CB has the form:The energy of the VB may be written asBand structure parameters of some semiconductors(room temperature)The energy gap varies with temperature – due to a change in lattice constant. This affects the band structure, which is sensitive to the lattice constant.Band structure parameters of some semiconductors(room temperature)The energy gap varies with temperature – due to a change in lattice constant. This affects the band structure, which is sensitive to the lattice constant.Carrier concentration: intrinsic semiconductorsIn order to determine the number of carriers, recall the Fermi-Dirac distribution function:- probability that an energy level E is occupied by an electron at TFermi level in intrinsic semiconductors lies close to the middle of the band gap. The distribution functionDensity of states for electrons and holesThe distribution function and the conduction and valence bands of a semiconductor: First, calculate the concentration of electrons in the CB. The number of states in the energy range (E, E + dE) is equal to De(E)dE, where De(E) - the density of electron states. Each of these states has an