Direct and Indirect Semiconductor Allowed values of energy can be plotted vs. the propagation constant, k. Since the periodicity of most lattices is different in various direction, the E-k diagram must be plotted for the various crystal directions (complex). -Direct bandgap: a minimum in the conduction band and a maximum in the valence band for the same k value -Indirect bandgap: a minimum in the conduction band and a maximum in the valence band at a different k value Ei E f hv k k k k where k i f photon f photon 0 Ei E f Ephonon k k k k k k where k 0 i f photon phonon f phonon phonon GaAs : the minimum conduction band energy and maximum valence band energy occur at the same k-value. direct band gap semiconductor semiconductor lasers and other optical devices Si, Ge, GaP, AlAs : indirect band gap semiconductor A transition must necessarily include and interaction with the crystal so that crystal momentum is conserved.
Extension to Three Dimensions Electrons traveling in different directions encounter different potential patterns and therefore different k-space boundaries. The (100) plane of a face-centered cubic crystal showing the [100] and [111] directions.
The k-space Diagrams of GaAs and Si Plot the [100] [111] direction
Intrinsic Material A perfect semiconductor crystal with no impurities or lattice defects No charge carrier at 0 K EHP generation at higher temperature For intrinsic material n p n i At a given temperature there is a certain concentration of electron-hole pairs n i If a steady state carrier concentration is maintained, there must be recombination of EHPs at the same rate at which they are generated. r i g i
Extrinsic Material Doping is the most common technique for varying the conductivity of semiconductors By doping, a crystal can be altered so that it has a predominance of either electrons (N-type) or holes (P-type) Extrinsic: the equilibrium carrier concentrations n 0 and p 0 are different from the intrinsic carrier concentration n i [P-type] [N-type]
For the case of N-type doping The energy level of the 5 th electron is inside the energy gap but very close to the bottom of the conduction band. At RT, it gets enough energy to jump into the conduction band, becoming a free electron. It leaves behind positively charged donor atoms, which is immobile. For the case of P-type doping The acceptor atom introduces a localized energy level into the energy gap which is close to the top of the valence band. As a consequence, an electron from the valence band jumps onto this level, leaving behind a mobile hole and creating a negatively charged immobile acceptor atom. Effects of (a) N-type and (b) P-type doping in energy-band model presentation (C.B., conduction band; V.B., valence band.)
donor level column V impurities column III impurities acceptor level
Donors and Acceptors in the Band Model Energy levels of donors and acceptors - Shallow levels: acceptor and donor levels with small ionization energies, such as As, P, Sb, and B - Deep levels: impurity levels with large ionization energies, such as Au, Cu, Pt..
Ionization Energy of Donor and Acceptor (Binding Energy) Estimated by modifying the theory of the ionization energy of a hydrogen atom. Energy required for electron in solid to make a transition from the donor level to the conduction band and become (quasi) free. Hydrogen atom in vacuum e, m 0 r r 3 n=1 n= n=3 0 r Si r 3 n=3 r 1 e, m * n= n=1 r 1 = 11.7 for Si r Donor atom in Si atom n =, E = 0 n =, E d = n = 3, E 3 n = 3, E d3 n =, E n = 1, E 1 n =, E d n = 1, E d1 hydrogen atom Donor atom E =E E ion n, n n=1,, 3, Eion E E1 Ec Ed ~6meV d n=1, E * d1 =E d 0 mn 13.6( ) ( ) ev 4 m0q 0 r m0 E ion E E1 13. 6eV 3 0
Electrons and Holes Comparison between Bonding and Energy Band Model Completely empty E g : equal to the energy required to break a bond E g No broken bond No electron in conduction band and no empty state in valence band - E + At 0 K - E + Completely filled No current flow - - broken bond E + electron in conduction band E g E + empty states in valence band Electron hole pair (EHP) generation Broken bond Electron (in conduction band and) Hole (empty state in valence band) With excitation with thermal or optical energy Carriers: - Electrons in conduction band - Holes (empty state) in valence band
Movement of Hole originally missing bond newly missing bond - E + - E + movement of empty state (movement of missing bond) Water analogy electrons in valence band E partially filled water in tube electrons in conduction band no water flow completely filled water in tube - + no electron flow water flow - E e + electron flow bubble flow filled water with some bubble E - + Holes (empty states) in valence band: moves like positively charged free particle empty state flow Electrons in conduction band: moves like free electron
Effective Mass - The electrons and holes in a crystal interact with a periodic coulomb field in the crystal. - They surf over the periodic potential of the crystal, and therefore m n and m p are not the same as the free electron mass, m 0. q Acceleration mn q Acceleration m p electrons holes The electron wave function is the solution of the three-dimensional Schrö dinger wave equation For free electron, m 0 V ( r ) E The solution becomes the plane wave as; 0 m E 0, V ( r ) 0, or constant Aexp[ j( k r t)], where k m 0 E
For electrons in the crystal, m0 [ ( ) ] 0, V r E the periodic crystal potential m* E 0 by neglecting V (r ) V ( r ) 0 and introducing new mass m* called effective mass The solution looks like plane wave with m*! (m n for electron and m p for hole) m* E Aexp[ j( k r t)], where k By adopting effective mass concept, the carriers in solids can be treated as almost free carriers. The calculation of effective mass must take into account the shape of the energy bands in three-dimensional k-space. E Assuming the E-k relationship has spherical symmetry, an electric field,, would accelerate an electron wave packet with k p E d E 1 dk m* m* m* p k Effective mass d E / dk Acceleration, a q m n q d E dk k
For particles moving in a crystal, there are an internal force in addition to an externally applied force. dv Ftotal Fext Fint m0a m0, m0 rest mass of particle dt F F m a F for free particle, int 0, ext total 0 In a crystal, F F total ext F E int m* a m*: new directly related to the external force, assuming F int 0 E e, m 0 e, m * solid correspond to V( r) 0 F ext in vacuum F m a q glass marble in semiconductor F F F m* a q, neglecting F total 0 total ext int int F ext : gravitational force fast drop vacuum slow drop water F int : viscosity of the liquid
For parabolic energy band, the electron mass is inversely related to the curvature of the (E, k) relationship m* d E / dk The curvature of the band determines the electron effective mass, m*. Electrons in valence band The mass calculated by m* will have a negative value d E / dk Valence band e - with (-) charge & (-) mass moves in an electric field in the same direction as h + with (+) charge & (+) mass Selecting holes as the valence-band carriers (the minimum kinetic-energy position of holes at the peak)
For a band centered at k=0, the E-k relationship near the minimum is usually parabolic: m E = k E * C m* d E / dk d E / dk negative negative effective mass Wider small d E / dk heavy mass E-k relationship for parabolic band with isotropic effective mass in 3-D: m x * = m y * = m z * = m* E ( kx k y kz ) m* k k 1 x y kx m* E m* E m* E Equation for sphere in k space Sharper large d E / dk light mass (a) E k diagram and (b) spherical constant-energy surface for GaAs The radius of the sphere stands for energy and the surface of the sphere is same energy, which is called constant energy surface.
The real E-k diagram of Si is more complicated (indirect semiconductor). The bottom of E C and top of E V appear for different values of k. (b) ellipsoidal constant-energy surfaces in the conduction band. There are 6 equivalent minima along [100] direction (a) E k diagram of Si large d E / dk small light hole d E / dk heavy hole Read subsection 1.5.;How to measure the effective mass? E-k relationship for parabolic band with anisotropic effective mass in 3-D: m x * m y * m z *. In Si, m x * = m y * m z *. k k E ( ) m m m x y kz * * * x y z k k x y kx 1 * * * mxe mye mze Equation for ellipsoid in k space The constant energy surface is not sphere, but ellipsoid.