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Computational Electronics Computational Electronics A.1 Derivation of the Boltzmann Transport Equation Kinetic theory: We need to derive an equation for the single particle distribution function f(v,r,t) (classical) which gives the probability of finding a particle with velocity between v and v+dv and in the region r to r+dr • We assume that v and r are given simultaneously which neglects quantum mechanical nature of particles. • f(v,r,t) allows us to calculate ensemble averages over velocity and space (particle density, current density, energy density, etc.): At dr dvAv, r, t f v, r, t Computational Electronics • Consider a hypervolume in phase space ds jv,r, t j(r,v,t) is the flux density V j(r,v,t)ds is flux through hypersurface ds S • Consider the particle balance through the hyper-volume V n drdvnv, r, t jv, r, t ds drdv t V t Coll S V Time rate of change of # particles in V Leakage through S Time rate of change due to collisions drdvGr, v, t R r, v, t V Time rate of change due to G-R mechanisms Computational Electronics • The flux density is written in terms of the time derivatives of the ‘position’ variables in 6D: j x, y , z,v x ,v y ,v z v x n v, r, t aˆ x v y naˆy v z naˆz Fy Fx ˆ Fz ˆ F ˆ nbv nbv nbv with v m m m m x y z • Applying the divergence theorem in 6D jv, r, t ds drdv jv, r, t S V where the divergence of j is n n n Fx n Fy n Fz n j vx vy vz x y z m v x m v y m v z Computational Electronics which is written more compactly as: F j v r n v n m • Particle balance is therefore: n F n n 0 drdv v r n v n m t Coll t G R V t Normalizing, we get the classical form of the Boltzmann transport equation: f r, v, t F f f v r f v f t m t Coll t First two terms on the rhs are the streaming terms Computational Electronics GR • For Bloch electrons in a semiconductor, we could have considered a 6D space x,y,z,kx,ky,kz where k is the wavevector and 1 v k E k • The semi-classical BTE for transport of Bloch electrons is therefore f r, k, t 1 F f f k E k r f k f t t Coll t G R Computational Electronics A.2 Collisional Integral Assume instantaneous, single collisions which are independent of the driving force and take particles from k to k (out scattering) or from k to k (in scattering). Out scattering kz k ky k kx Computational Electronics In scattering (A) Out Scattering nr, k, t nr, k, t kkt where kk is the transition rate per particle from k to k Distribution function is: f r, k, t nr, k, t N Take limit as t0 f r, k, t f r, k, t kk 1 f r, k, t t OUT where the last term in the brackets accounts for the Pauli exclusions principle (degeneracy of the final state after scattering). Computational Electronics (B) In Scattering By an analogous argument, the rate of change of the distribution function due to in scattering is: f r, k, t f r, k, t kk 1 f r, k, t t IN Total rate of change of f (r,k,t) around k is a sum over all possible initial and final states k: f r, k, t f r, k, t 1 f r, k, t kk t Coll k In scattering f r, k, t 1 f r, k, t kk Out scattering Computational Electronics (C) Boltzmann Equation with Collision Integral The sum over final states k may be converted to an integral due to the small volume of k-space associated with each state: V 3 dk 8 k The BTE becomes: fk 1 F k E r fk k fk t V dkfk 1 fk kk fk 1 fk kk 3 8 Computational Electronics A.3 Scattering Theory What contributes to kk ? How do we calculate kk’? Scattering Mechanisms Defect Scattering Crystal Defects Impurity Neutral Carrier-Carrier Scattering Alloy Lattice Scattering Intervalley Intravalley Acoustic Ionized Deformation potential Computational Electronics Optical Piezoelectric Nonpolar Acoustic Polar Optical B.1 Time Evolution of Quantum States When the Hamiltonian is time dependent, the state or the wavefunction of the system will be also time dependent. In other words, an electron will have a probability to transfer from one state (molecular orbital) to another. The transition probability can be obtained from the time-dependent Schrödinger Equation (t ) i H(t ) t One the initial wavefunction, (0), is known, the wavefunction at a given later time can be determined. If H is time independent, we can easily find that (t ) an e iEnt / n n Computational Electronics Suppose that a system has initial (t=0) Hamiltonian, H0 (time independent), and is at an initial eigenstate, k. Under an external influence, described by H’ (time dependent), the system will change state. For example, a molecule moves close to an electrode surface to feel an increasing interaction with the electrode. The combined Hamiltonian is Hˆ (t ) Hˆ 0 (0) Hˆ ' (t ) the combined Hamiltonian should be a linear combination of the initial eigenstates, The wavefunction of the system corresponding to (t ) Cnk (t ) n n From mathematical point of view, this is always possible since the initial eigenstates, n, form a complete set of basis. The physical picture is that the system under the influence of the external perturbation will end up in a different state with a probability given by |Cnk|2. The indices nk mean a transition from kth eigenstate to nth eigenstate. How fast the transition is or the transition rate is given by wnk Computational Electronics d |C nk (t ) |2 dt B.2 Time-Dependent Perturbation Theory Now we determine the transition rate according to the above definition. We assume that the initial state of the system is (0) k the external perturbation, H’, is switched on at t=0. The time dependent Schrödinger Eq. is (t ) i ( H 0 H ' ) (t ) t For simplicity, we can rewrite this equation as (t ) Cnk '(t )e iEnt / n n Note that Cnk’(t) is different from Cnk(t), but |Cnk’(t)|2=| Cnk(t)|2 and we can omit the prime. dCnk iEnt / iEnt / i e n Cnk e H ' n. dt n n Computational Electronics Multiplying by k’ and integrate, we obtain dCk 'k i ( Ek ' E n ) t / i e k ' | H ' | n Cnk dt n After considering that n are normalized orthogonal functions. Note that the initial condition becomes Cnk (0) nk In general, solving above equation set is not easy, but we can obtain approximate solution using perturbation theory when H’ is small comparing to H0. Let us denote the solution in the absence of H’ as Cnk(0), we have dC i k 'k dt (0) 0 So Ck’k(0) is independent of time and the initial condition is Ckk ' (0) kk ' Computational Electronics Cnk (0) nk We replace Cnk on the right hand side with Cnk(0) and obtain the first order correction dC i k 'k dt (1) e i ( E k ' Ek ) t / k'| H '| k k ' | H ' | k in the above equation is often denoted as H’k’k and it measured the coupling strength between the k’ and k states. Solving we have t Ck ' k 1 H k' 'k e i ( Ek ' Ek ) t / dt, i 0 One important case is that H’ is fixed once switched on. In this case, 1 ' e i ( Ek ' Ek ) t / 1 C k 'k ( t ) H k 'k i i( Ek ' Ek ) / Computational Electronics B.3 Fermi-Golden Rule Thus, we can obtain 2 1 2t t ' 2 sin (( E k ' E k )t / ) ' 2 | C k 'k ( t ) | 2 | H k 'k | | H | ( Ek ' Ek ) k 'k [( Ek ' Ek ) / ]2 2 2 So the transition rate is wk 'k 2 2 | H k' 'k |2 ( E k ' E k ) We can conclude that (1) the transition rate is independent of time, (2) the transition can occur only if the final state has the same energy as the initial state. The later one reflects energy conservation. In the case when the energy levels are continuous band, the number of states near Ek’ for an interval of dEk’ is In the case when the energy levels are continuous band, the number of states near Ek’ for an interval of dEk’ is rEk’)dEk’ , where r is the density of states. The transition rate from k state to the states near Ek’ is then w r ( Ek ' ) wk 'k dEk ' This is Fermi Golden rule, Computational Electronics 2 ' 2 | H | r ( Ek ) k ' k 2 (23.18) Assumptions made: (1) Long time between scattering (no multiple scattering events) (2) Neglect contribution of other c’s (Collision broadening ignored) kk Pkk 2 kk 2 Vs Ek Ek t Computational Electronics B.4 Total Scattering Rate Calculation • For the case when we have general matrix element (with qdependence), the procedure for calculating the scattering rate out of state k is the following (k ) kk ' k' where 2 1 2 3 1 2 3 2 k ' dk ' d (cos )dkk ' 1 2 k ' dk ' d (cos ) M (q) ( Ek ' Ek 2 2 1 0 k' k q 1 2 k' 2 0 1 dk ' d (cos ) M (q) ( Ek ' Ek Computational Electronics 1 2 0 ) 0 ) • The integration over k’ can be converted into integration over qwavevector and the integration over cos() together with the function that denotes conservation of energy will put limits on the q-values: qmin and qmax for absorption and emission. The final expression that needs to be evaluated is: ab,em (k ) qmax m 2 k 3 q M (q) 2 dq qmin • where ab min q em min q k k 1 k k 1 Ek Ek Computational Electronics , ab max , q ab max q k k 1 k k 1 Ek Ek q-vector qmax(ab) qmax(em) qmin(ab) qmin(em) -1 Computational Electronics +1 cos() 3.5 x 10 5 3.5 3 2.5 2.5 2 2 1.5 1.5 1 1 0.5 0.5 0 0.5 1 1.5 2 2.5 3 polar angle 5 Absorption Process 3 Emission Process 0 x 10 3.5 0 0 0.5 1 1.5 2 2.5 3 3.5 polar angle Histogram of the polar angle for polar optical phonon emission and absorption. In accordance to the graph shown on the previous figure for emission forward scattering is more preferred for emission than for absorption. Computational Electronics Special Case: Constant Matrix Element For the special case of constant matrix element, the expression for the scattering rate out of state k reduces to: ( k ) mM 02 k 3 1 Ek The top sign refers to absorption and the bottom sign refers To emission. For Elastic scattering we can further simplify to get: ( k ) Computational Electronics mM 02 k 3 C.1 Elastic Scattering Mechanisms (A) Ionized Impurities scattering (Ionized donors/acceptors, substitutional impurities, charged surface states, etc.) • The potential due to a single ionized impurity with net charge Ze is: 2 Ze 0 Vi r 4r mks units • In the one electron picture, the actual potential seen by electrons is screened by the other electrons in the system. Computational Electronics What is Screening? lD - Debye screening length - r Example: + 3D: 1 screening cloud r r exp r l D 1 - Ways of treating screening: • Thomas-Fermi Method static potentials + slowly varying in space • Mean-Field Approximation (Random Phase Approximation) time-dependent and not slowly varying in space Computational Electronics • For the scattering rate due to impurities, we need for Fermi’s rule the matrix element between initial and final Bloch states n, k Vi r n, k V 1 drun* ,k eikrVi r un,k eikr Since the u’s have periodicity of lattice, expand in reciprical space V 1 dre ikrVi r eikre iGrUnnkk G G 1 V dre Vi r e e ikr G ikr iGr * iGr dr un,k r un,k r e • For impurity scattering, the matrix element has a 1/q type dependence which usually means G0 terms are small 1 V dre Vi r eikr drun* ,k run,k r Vi q Ikk ikr Computational Electronics nn • The usual argument is that since the u’s are normalized within a unit cell (i.e. equal to 1), the Bloch overlap integral I, is approximately 1 for n=n [interband(valley)]. Therefore, for impurity scattering, the matrix element for scattering is approximately k Vi r k 2 Vi q 2 2 4 Z e 2 2 ; V volume 2 2 V q l sc where the scattered wavevector is: q k k • This is the scattering rate for a single impurity. If we assume that there are Ni impurities in the whole crystal, and that scattering is completely uncorrelated between impurities: Vi kk Ni Z 2e 4 ni Z 2e 4 2 2 2 2 2 V q l sc V q 2 l2 sc where ni is the impurity density (per unit volume). Computational Electronics • The total scattering rate from k to k is given from Fermi’s golden rule as: ki k 2ni Z 2e 4 Ek Ek 2 2 2 V q l sc If is the angle between k and k, then: q k k k 2 k 2 2kk cos 2k 2 1 cos • Comments on the behavior of this scattering mechanism: - Increases linearly with impurity concentration - Decreases with increasing energy (k2), favors lower T - Favors small angle scattering - Ionized Impurity-Dominates at low temperature, or room temperature in impure samples (highly doped regions) • Integration over all k gives the total scattering rate k : 2 ni Z e m * 4k i k 2 3 3 2 2 2 8 sc k qD 4k qD 2 4 Computational Electronics ; qD 1 / l (A1) Neutral Impurities scattering • This scattering mechanism is due to unionized donors, neutral defects; short range, point-like potential. • May be modeled as bound hydrogenic potential. • Usually not strong unless very high concentrations (>1x1019/cm3). (B) Alloy Disorder Scattering • This is short-range type of interaction as well. • It is calculated in the virtual crystal approximation or coherent potential approximation. • Limits mobility of ternary and quaternay compounds, particularly at low temperature. • The total scattering rate out of state k for this scattering mechanism is of the form: 3/ 2 kalloy Computational Electronics 2 E 2m * 2 2 E1/ 2 (C) Surface Roughness Scattering • This is a short range interaction due to fluctuations of heterojunction or oxide-semiconductor interface. • Limits mobility in MOS devices at high effective surface fields. High-resolution transmission electron micrograph of the interface between Si and SiO2 (Goodnick et al., Phys. Rev. B 32, pp. 8171, 1985) 2.71 Å 3.84 Å Modeling surface-roughness scattering potential: H' (r,z) Voz (r) Vo z Vo(z)(r) random function that describes the deviation from an atomically flat interface Computational Electronics • Extensive experimental studies have led to two commonly used forms for the autocovariance function. • The power spectrum of the autocovariance function is found to be either Gaussian or exponentially correlated. Comparison of the fourth-order AR spectrum with the fits arising from the Exponential and Gaussian models (Goodnick et al., Phys. Rev. B 32, pp. 8171, 1985) SPECTRUM OF HRTEM ROUGHNESS =0.24 nm AR model Commonly assumed power spectrums for the autocovariance function : Gaussian model x=0.74 nm Exponential model x=0.94 nm • Gaussian: • Exponential: q2 2 SG (q) exp 4 2 2 SE (q) 2 2 1 q2 2 2 32 Wave vector (Å-1) • Note that is the r.m.s of the roughness and is the roughness correlation length. Computational Electronics • The total scattering rate out of state k for surface-roughness scattering is of the form: ksr m* e 1 k Ndepl 0.5Ns E 3 2 sc 1 k 2 2 1 k 2 2 2 2 4 where E is a complete elliptic integral, Ndepl is the depletion charge density and Ns is the sheet electron density. • It is interesting to note that this scattering mechanism leads to what is known as the universal mobility behavior, used in mobility models described earlier. m Increasing substrate doping Eeff Computational Electronics e 0.5Ns Ndepl sc The Role of Interface Roughness: D. Vasileska and D. K. Ferry, "Scaled silicon MOSFET's: Part I - Universal mobility behavior," IEEE Trans. Electron Devices 44, 577-83 (1997). Phonon 2 Mobility [cm /V-s] Coulomb -1 (aN + bN 400 s ) depl N 300 200 -1/3 s experimental data uniform step-like (low-high) retrograde (Gaussian) 100 12 10 13 10 -2 Inversion charge density N [cm ] s Interface-roughness Computational Electronics C.2 Inelastic Scattering Mechanisms C.2.1 Some general considerations • The Electron Lattice Hamiltonian is of the following form: H He Hl Hep He Electron Hamiltonia n; Hl lattice Hamiltonia n Hep Electron Phonon coupling where Hen,k En,k n,k n,k eikrun,k Bloch states • For the lattice Hamiltonian we have: H l l E l l El x q x,q l nq1nq 2nq 3... n x q 1 2 Second quantized representation, where nq is the number of phonons with wave-vector q, mode x. Computational Electronics • Phonons: The Fourier expansion in reciprocal space of the coupled vibrational motion of the lattice decouples into normal modes which look like an independent set of Harmonic oscillators with frequency xq x labels the mode index, acoustic (longitudinal, 2 transverse modes) or optical (1 longitudinal, 2 transverse) q labels the wavevector corresponding to traveling wave solutions for individual components, • The velocity and the occupancy of a given mode are given by: v xq nqx xq e q 1 xq / kBTl ; Bose Einstein distribution 1 Computational Electronics (1) For acoustic modes, lim q 0 vxq xq q u x , acoustic velocity. (2) For optical modes, velocity approaches zero as q goes to zero. Room temperature dispersion curves for the acoustic and the optical branches. Note that phonon energies range between 0 and 60-70 meV. Computational Electronics • The Electron-Phonon Interaction is categorized as to mode (acoustic or optical), polarization (transverse or longitudinal), and mechanism (deformation potential, polar, piezoelectric). During scattering processes between electrons and phonon, both wavevector and energy are conserved to lowest order in the perturbation theory. This is shown diagramatically in the figures below. Absorption: k k q Ek Ek q k, Ek q , q Computational Electronics Emission: k k q Ek Ek q k, Ek q , q • For emission, E k q must hold, otherwise it is prohibited by conservation of energy. Therefore, there is an emission threshold in energy • Emission: nq nq 1 Absorption: nq nq 1 C.2.2 Deformation Potential Scattering Replace Hep with the shift of the band edge energy produced by a homogeneous strain equal to the local strain at position r resulting from a lattice mode of wavevector q (A) Acoustic deformation potential scattering • Expand E(k) in terms of the strain. For spherical constant energy surface E k E 0 k E1 e2 Computational Electronics where: u r dilation of volume of unit cell E1 Deformatio n potential const. E1 Deformatio n potential and u is the displacement operator of the lattice 1/ 2 eq,x aq,xe iqr aq*,xe iqr u r x q 2NMq eq,x polarization vector x • Taking the divergence gives factor of e·q of the form: eq,x q q for longitudin al modes eq,x q 0 for transverse modes Therefore, only longitudinal modes contribute. Computational Electronics • For ellipsoidal valleys (i.e. Si, Ge), shear strains may contribute to the scattering potential E k E 0 k Ed Eu ezz u ezz zˆ ; ezz is component of the strain tensor z Scattering Matrix Element: Assuming q ul q, then: Vac 2 E12q nq 1 1 2Vrul upper absorption lower emission • At sufficient high temperature, (equipartition approximation): kBTl nq nq 1 q Computational Electronics • Substituting and assuming linear dispersion relation, Fermi’s rule becomes kack 2 2 2 E 2 1 kBTl Vac Ek Ek q Ek Ek q 2 Vrul • The total scattering rate due to acoustic modes is found by integrating over all possible final states k’ ac k 2E12kBTl V 2 4 dk k Ek Ek q 2 3 Vrul 8 0 where the integral over the polar and azimuthal angles just gives 4. • For acoustic modes, the phonon energies are relatively small since q 0 as q 0 Computational Electronics • Integrating gives (assuming a parabolic band model) kac m *kE12 kBT l 2 ; c r u l l 3 cl where cl is the longitudinal elastic constant. Replacing k, using parabolic band approximation, finally leads to: kac 2m E12 kBT l 1 / 2 E 4 cl *3 / 2 • Assumptions made in these derivations: a) spherical parabolic bands b) equipartition (not valid at low temperatires) c) quasi-elastic process (non-dissipative) d) deformation potential Ansatz Computational Electronics (B) Optical deformation potential scattering (Due to symmetry of CB states, forbidden for -minimas) • Assume no dispersion: q 0 as q 0 Out of phase motion of basis atoms creates a strain called the optical strain. • This takes the form (D0 is optical deformation potential field) Vdo D0 ur ; D0 D0 eq zeroth order The matrix element for spherical bands is given by ac 2 Vkk D0 2 n k k q n 1 k k q 2 r V 0 0 which is independent of q . Computational Electronics 0 • The total scattering rate is obtained by integrating over all k’ for both absorption and emission kdo 1/ 2 1 m D0 n E 0 3 1 / 2 do 2r 0 n 1 E 0 E 0 *3 / 2 2 0 0 where the first term in brackets is the contribution due to absorption and the second term is that due to emission • For non-spherical valleys, replace m *3 / 2 mt m1l / 2 • The non-polar scattering rate is basically proportional to density of states kdo rE 0 Computational Electronics (C) Intervalley scattering • May occur between equivalent or nonequivalent sets of valleys - Intervalley scattering is important in explaining room temperature mobility in multi-valley semiconductors, and the NDR observed (Gunn effect) in III-V compounds - Crystal momentum conservation requires that qk where k is the vector joining the two valley minima Computational Electronics • Since k is large compared to k, assume q q and treat the scattering the same as non-polar optical scattering replacing D0 with Dij the intervalley deformation potential field, and the phonon coupling valleys i and j q ij • Conservation of energy also requires that the difference in initial and final valley energy be accounted for, giving n E Eij ij 1 / 2 kiv 1/ 2 2r ij n 1 E Eij ij E Eij 0 j 3/ 2 2 md j Dij 3 ij ij where the sum is over all the final valleys, j and Eij Emin j Emin i Computational Electronics C.2.3 Phonon Scattering in Polar Semiconductors • Zinc-blend crystals: one atom has Z>4, other has Z<4. • The small charge transfer leads to an effective dipole which, in turn, leads to lattice contribution to the dielectric function. • Deformation of the lattice by phonons perturbs the dipole moment between the atoms, which results in electric field that scatters carriers. • Polar scattering may be due to: optical phonons => polar optical phonon scattering (very strong scattering mechanism for compound semiconductors such as GaAs) acoustic phonons => piezoelectric scattering (important at low temperatures in very pure semiconductors) Computational Electronics (A) Polar Optical Phonon Scattering (POP) Scattering Potential: Microscopic model is difficult. A simpler approach is to consider the contribution of this dipole to the polarization of the crystal and its effect on the high- and low-frequency dielectric constants. • Consider a diatomic lattice in the long-wavelength limit (k0), for which identical atoms are displaced by a same amount. • For optical modes, the oppositely charged ions in each primitive cell undergo oppositely directed displacements, which gives rise to nonvanishing polarization density P. Transverse mode: Longitudinal mode: Computational Electronics k k • Associated with this polarization are macroscopic electric field E and electric displacement D, related by: D = E + P Here, we have taken into account the contribution to the dielectric function due to valence electrons • Assume D, E, P eik.r. Then, in the absence of free charge: ·D = ik ·D = 0 and E = ik E = 0 kD or D=0 k||E or E=0 • Longitudinal modes: P||k => D=0, (LO)=0 • Transverse modes: Pk => E=0, (TO)= Computational Electronics • The equations of motion of the two modes (in the k0 limit), for the relative displacement of the two atoms in the unit cell w=u1-u2 are: Transverse mode: u2 u2 u1 u1 u2 u1 2 d w 2 TOw 0 2 dt 2 TO 1 1 2C 2C M1 M 2 M Computational Electronics Longitudinal mode: u2 u2 k u1 u2 u1 u1 d 2w 1 * 2 TOw e E 2 M dt E ( t ) Ee * jt e E /M w 2 2 TO • The longitudinal displacement of the two atoms in the unit cell leads to a polarization dipole: *2 N * Ne / 2V M P ew 2 E 2 2V TO • The existence of a finite polarization dipole modifies the dielectric function: *2 Ne / 2V M D E P E 2 E ()E 2 TO 2 2 LO S TO 1 () 1 2 2 2 2 TO TO *2 1 Ne 1 2 Polarization S LO 2V M constant (0) Computational Electronics • The electric field associated with the perturbed dipole moment is obtained from the condition that, in the absence of macroscopic free charge: D ik D Eind P 0 D 0 and Eind • Consider only one Fourier component: P Eind ind e1 V (q ) e1 Vqe iqr i e1 qV (q ) e e V (q ) i q P V (q ) i P 2 q q e N * 1 iqr iqr V (r ) i e aqe aq e 2V 2M (N / 2)LO q q 1/ 2 e i 2VLO 2 1 1 1 iqr iqr 2 1 , LO aqe aq e q q (0 ) Computational Electronics Scattering Rate Calculation: • Matrix element squared for this interaction: Vk ,k ' 2 e 2 1 1 1 k'k q N 0 2 2 2 2VLO q • Transition rate per unit time from state k to state k’: k ,k ' 2 2 Vk ,k ' εk' εk ωL 0 2 e 1 1 1 k'k q N 0 k' k L0 2 2 2 VLO q • Total scattering rate per unit time out of state k: k k ,k ' k ,q k' q V 2 1 2 d d (cos ) q k ,q dq 3 ( 2) 0 1 0 Computational Electronics • Momentum and energy conservation delta-functions limit the values of q in the range [qmin,qmax]: absorption: qmin k k 1 LO E ( k ) qmax k k 1 LO E (k ) emission: qmin k k 1 LO E (k ) qmax k k 1 LO E (k ) E (k ) LO , emission threshold • Final expression for k m * e2 k 2 2kLO 1 E ( k ) 1 E ( k ) N0 sinh N0 1 sinh 1 LO LO Computational Electronics Discussion: 1. The 1/q2 dependence of k,k’ implies that polar optical phonon scattering is anisotropic, i.e. favors small angle scattering 2. It is inelastic scattering process 3. k is nearly constant at high energies 4. Important for GaAs at room-temperature and II-VI compounds (dominates over non-polar) Scattering rate Momentum relaxation rate Computational Electronics The larger momentum relaxation time is a consequence of the fact that POP scattering favors small angle scattering events that have smaller influence on the momentum relaxation. (B) Piezoelectric scattering • Since the polarization is proportional to the acoustic strain, we have P epz u • Following the same arguments as for the polar optical phonon scattering, one finds that the matrix element squared for this mechanism is: 2 Vkk ' 2 eepz Nq 12 12 k k'q 2rVq • The scattering rate, in the elastic and the equipartition approximation, is then of the form; 2 m * kBT eepz k2 k ln1 4 2 3 4 kr v s qD where qD is the screening wavevector. Computational Electronics Total Electron-Phonon Scattering Rate Versus Energy: Intrinsic Si GaAs In both cases the electron scattering rates were calculated by assuming non-parabolic energy bands. Computational Electronics 1. Acoustic Phonon Scattering 2. Intervalley Phonon Scattering 3. Ionized Impurity Scattering 4. Polar Optical Phonon Scattering Computational Electronics 5. Piezoelectric Scattering 5. Piezoelectric Scattering 6. Dislocation Scattering (e.g. GaN) where n’ is the effective screening concentration Ndis is the Line dislocation density 7. Alloy Desorder Scattering (Al xGa1-xAs) Where: d is the lattice disorder (0≤d≤1) Dalloy is the alloy disorder scattering potential Computational Electronics