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Electron-Electron Interactions DRAGICA VASILESKA PROFESSOR ARIZONA STATE UNIVERSITY Classification of Scattering Mechanisms S c a tte rin g M e c h a n is m s D e fe c t S c a tte rin g C rys ta l D e fe c ts N e u tra l Im p u rity C a rrie r-C a rrie r S c a tte rin g A llo y Io n ize d L a ttic e S c a tte rin g In te rva lle y In tra va lle y A c o u s tic D e fo rm a tio n p o te n tia l O p tic a l P ie zo e le c tric N o n p o la r A c o u s tic P o la r O p tic a l Treatment of the Electron-Electron Interactions Electron-electron interactions can be treated either in: K-space, in which case one can separate between Collective plasma oscillations Binary electron-electron collisions Real space Molecular dynamics Bulk systems (Ewald sums) Devices (Coulomb force correction, P3M, FMM) K-space treatment of the Electron-Electron Interactions Electron Gas As already noted, the electron gas displays both collective and individual particle aspects. The primary manifestations of the collective behavior are: Organized oscillations of the system as a whole – plasma oscillations Screening of the field of any individual electron within a Debye length Collective excitations In the collective excitations each electron suffers a small periodic perturbation of its velocity and position due to the combined potential of all other electrons in the system. The cumulative potential may be quite large since the longrange nature of the Coulomb potential permits a very large number of electrons to contribute to the potential at a given point The collective behavior of the electron gas is decisive for phenomena that involve distances that are larger than the Debye length For smaller distances, the electron gas is best considered as a collection of particles that interact weakly by means of screened Coulomb force. Collective behavior, Cont’d For the collective description to be valid, it is necessary that the mean collision time, which tends to disrupt the collective motion, be large compared to the period of the collective oscillation. Thus: coll 2 p m * 2 N e2 D 1/ 2 Examples for GaAs: ND=1017 cm-3, p =2×1013, coll >>2/p 1/coll <<3×1012 1/s ND=1018 cm-3, p =6.32×1013, coll >>2/p 1/coll <<1013 1/s ND=1019 cm-3, p =2×1014, coll >>2/p 1/coll <<3×1013 1/s Collective Carrier Scattering Explained Consider the situation that corresponds to the mode q=0, when all electrons in the system have been displaced by the same amount u, as depicted in the figure below: u d - - - - - - - - - - - - - - E + + + + + + + + + + + + + + Collective Carrier Scattering Explained Because of the positive (negative) surface charge density at the bottom (top) slab, an electric field is produced inside the slab. The electric field can be calculated using a simple parallel capacitor model for which: C A Q d V appl neuA E Ed neu The equation of motion of a unit volume of the electron gas of concentration n is: 2 nm * d u dt 2 2 neE 2 d u dt 2 p u 0, p 2 2 n e u ne 2 m * 1/ 2 Collective Carrier Scattering Explained Comments: Plasma oscillation is a collective longitudinal excitation of the conduction electron gas. A PLASMON is a quantum of plasma oscillations. PLASMONS obey Bose-Einstein statistics. An electron couples with the electrostatic field fluctuations due to plasma oscillations, in a similar manner as the charge of the electron couples to the electrostatic field fluctuation due to longitudinal POP. Collective Carrier Scattering Explained The process is identical to the Frӧhlich interaction if plasmon damping is neglected. Then: 2 ab q max m * e p N 0 ln ab 2 ( k ) 4 k q min 1 em q max ( N 1) ln 0 q em min Note on qmax: Large qmax refers to short-wavelength oscillations, but one Debye length is needed to screen the interaction. Therefore, when qmax exceeds 1/LD, the scattering should be treated as binary collision. qc=min(qmax,1/LD) Collective Carrier Scattering Explained Importance of plasmon scattering Plasma oscillations and plasmon scattering are important for high carrier densities When the electron density exceeds 1018 cm-3 the plasma oscillations couple to the LO phonons and one must consider scattering from the coupled modes www.engr.uvic.ca/.../Lecture%207%20-%20Inelastic%20Scattering.ppt Electron-Electron Interactions (Binary Collisions) This scattering mechanism is closely related to charged impurity scattering and the interaction between the electrons can be approximated by a screened Coulomb interaction between point-like particles, namely: H e e ( r12 ) e 2 4 r12 e r12 / L D Then, one can obtain the scattering rate in the Born approximation as one usually does in BrooksHerring approach. Binary Collissions To write the collision term, one needs to define a pair transition rate S(k1,k2,k1’,k2’), which represents the probability per unit time that electrons in states k1 and k2 collide and scatter to states k1’ and k2’, as shown diagramatically in the figure below: k2’ k1’ r1 k1 r2 k2 Binary Collisions, Cont’d The pair transition rate is defined as: S ( k 1 , k 2 , k 1' , k 2 ' ) 2 M 12 2 E k1 ' E k 2 ' E k1 E k 2 M 12 k 1' , k 2 ' H ee ( r12 ) k 1 , k 2 Since the interaction potential depends only upon the distance between the particles, it is easier to calculate M12 in a center-of-mass coordinate system, to get: M 12 e V 2 1 2 2 q 1 / LD , q 2 k 12 k 12 ' 2 Binary Collisions Scattering Rate To evaluate the scattering rate due to binary carrier-carrier scattering, one weights the pair transition rate that a target carrier is present and by the probability that the final states k1’ and k2’ are empty: 1 ( k1 ) S (k k2' 1 , k 2 , k 1' , k 2 ' ) f ( k 2 )[1 f ( k 1 ' )][ 1 f ( k 2 ' )] k2 Note that a separate sum over k1’ is not needed because of the momentum conservation -function. For nondegenerate semiconductors, we have: 1 ( k1 ) S (k k2' k2 1 , k 2 , k 1' , k 2 ' ) f (k 2 ) Binary Collisions Scattering Rate In summary: 1 ( k1 ) k2 4 f (k 2 ) 2 m * ne L D 3 2 4 k k2 k1 k 2 2 2 1 / LD Incorporation of the electron-electron interactions in EMC codes For two-particle interactions, the electron-electron (hole-hole, electron-hole) scattering rate may be treated as a screened Coulomb interaction (impurity scattering in a relative coordinate system). The total scattering rate depends on the instantaneous distribution function, and is of the form: ee k 0 m ne 4 2 V 3 f k k k k0 2 k k 2 0 2 Screening constant There are three methods commonly used for the treatment of the electronelectron interaction: A. Method due to Lugli and Ferry B. Rejection algorithm C. Real-space molecular dynamics (A) Method Due to Lugli and Ferry • This method starts form the assumption that the sum over the distribution function is simply an ensemble average of a given quantity. • In other words, the scattering rate is defined to be of the form: ee k 0 4 2 nm n e L D 3 2 4 N k ki k ki • The advantages of this method are: i 1 2 1 / LD 2 1. The scattering rate does not require any assumption on the form of the distribution function 2. The method is not limited to steady-state situations, but it is also applicable for transient phenomena, such as femtosecond laser excitations • The main limitation of the method is the computational cost, since it involves 3D sums over all carriers and the rate depends on k rather on its magnitude. (B) Rejection Algorithm • Within this algorithm, a self-scattering mechanism, internal to the interparticle scattering is introduced by the following substitution: k k0 k k0 2 1/ 2 LD 1 2 LD • When carrier-carrier collision is selected, a counterpart electron is chosen at random from the ensemble. • Internal rejection is performed by comparing the random number with: k k0 k k0 2 1 / LD 2 • If the collision is accepted, then the final state is calculated using: cos r 1 where: 2r 1 g (1 r 2 2 )L D , where r angle ( g, g ' ) g k k 0 ; g ' k 'k 0 ' The azimuthal angle is then taken at random between 0 and 2. • The final states of the two particles are then calculated using: 1 ' k0 k0 k k0 ' 2 1 2 g ' g g ' g Real-Space Treatment of the Electron-Electron Interactions BULK SYSTEMS SEMICONDUCTOR DEVICE MODELING Bulk Systems (C) Real-space molecular dynamics • An alternative to the previously described methods is the real-space treatment proposed by Jacoboni. • According to this method, at the observation time instant ti=it, the total force on the electron equals the sum of the interparticle coulomb interaction between a particular electron and the other (N-1) electrons in the ensemble. • When implementing this method, several things need to be taken into account: 1. The fact that N electrons are used to represent a carrier density n = N/V means that a simulation volume equals V = N/n. 2. Periodic boundary conditions are imposed on this volume, and because of that, care must be taken that the simulated volume and the number of particles are sufficiently large that artificial application from periodic replication of this volume do not appear in the calculation results. • Using Newtonian kinematics, the real-space trajectories of each particle are represented as: r (t t ) r (t ) v t and: 1 F(t ) 2 m * F(t ) v(t t ) v(t ) t m * t 2 Here, F(t) is the force arising from the applied field as well as that of the Coulomb interaction: F ( t ) q E r ( t ) i i • The contributions due to the periodic replication of the particles inside V in cells outside is represented with the 2 Ewald sum: e N 1 2 F(t ) ri 2 a i 4 i 1 ri 3V Simulation example of the role of the electron-electron interaction: • The effect of the e-e scattering allows equilibrium distribution function to approach Fermi-Dirac or Maxwell Boltzmann distribution. • Without e-e, there is a phonon ‘kink’ due to the finite energy of the phonon Semiconductor Device Modeling Ways of accounting for the short-range Coulomb interactions Long-range Coulomb interactions are accounted for via the solution of the Poisson equation which gives the so-called Hartree term If the mesh is infinitely small, the full Coulomb interaction is accounted for However this is not practical as infinite systems of algebraic equations need to be solved To avoid this difficulty, a mesh size that satisfies the Debye criterion is used and the proper correction to the force used to move the carriers during the freeflight is added Earlier Work – k-space treatment of the Coulomb interaction Good for 2D device simulations Requires calculation of the distribution function to recalculate the scattering rate at each time step and the screening which is time consuming Implemented in the Damocles device simulator K-space Approach Present trends – Real-space treatment Requires 3D device simulator, otherwise the method fails There are several variants of this method Corrected Coulomb approach developed by Vasileska and Gross Particle-particle-particle-mesh (p3m) method by Hockney and Eastwood Fast Multipole method Real Space Treatment Cont’d Corrected Coulomb approach and p3m method are almost equivalent in philosophy FMM is very different Treatment of the short-range Coulomb interactions using any of these three methods accounts for: Binary collisions + plasma (collective) excitations Screening of the Coulomb interactions Scattering from multiple impurities at the same time which is very important at high substrate doping densities 1. Corrected Coulomb approach A resistor is first simulated to calculate the difference between the mesh force and the true Coulomb force Cut-off radius is defined to account for the ions (inner cut-off radius) Outer cut-off radius is defined where the mesh force coincides with the Coulomb force Correction to the force is made if an electron falls between the inner and the outer radius The methodology has been tested on the example of resistor simulations and experimental data are extracted Corrected Coulomb Approach Explained Resistor Simulations MOSFET: Drift Velocity and Average Energy 2. p3m Approach Details of the p3m Approach Impurity located at the very source-end, due to the availability of Increasing number of electrons screening the impurity ion, has reduced impact on the overall drain current. 60% Impurity position varying along the center of the channel Current Reduction 50% V G = 1.0 V 40% V D = 0.2 V 30% 20% Source end Drain end 10% 0% 0 10 20 30 40 Distance Along the Channel [nm] 50 3. Fast Multipole Method Different strategy is employed here in a sense that Laplace equation (Poisson equation without the charges) is solved. This gives the ‘Hartree’ potential. The electron-electron and electron-ion interactions are treated using FMM The two contributions are added together Must treat image charges properly. Good news is that the surfaces are planar and the method of images is a good choice Idea The philosophy of FMM: Approximate Evaluation Ideology behind FMM Simulation Methodology Method of Images Resistor Simulations More on the Electron-Electron Interactions for Q2D Systems EXCHANGE-CORRELATION EFFECTS SCREENING OF THE COULOMB INTERACTION POTENTIAL Exchange-Correlation Correction to the Ground State Energy if the System Space Quantization • Poisson equation: 3 2 1 0 d 2V (z) e2 H [n(z) Na(z)] dz2 sc • Hohenberg-Kohn-Sham Equation: (Density Functional Formalism) VG>0 1’ 0’ EF z-axis [100] (depth) 4-band 2 2 * 2 Veff (z) y n (z) n y n (z) 2mz z [ ] 2-band V (z) V (z) V (z) V (z) eff H xc im [100]-orientation: Finite temperature generalization of the LDA (Das Sarma and Vinter) 2-band : mz=ml=0.916m0, mxy=mt=0.196m0 4-band: mz=mt, mxy= (ml mt)1/2 Exchange-Correlation Effects HF E EHF E corr EHF E kin exchange E corr Total Ground State Energy of the System Accounts for the error made with the Hartree-Fock Approximation Hartree-Fock Approximation for the Ground State Energy Accounts for the reduction of the Ground State Enery due to the inclusion of the Pauli Exclusion Principle Ways of Incorporating the Exchange-Correlation Effects: Density-Functional Formalism (Hohenberg, Kohn and Sham) Perturbation Method (Vinter) Subband Structure Importance of Exchange-Correlation Effects Vasileska et al., J. Vac. Sci. Technol. B 13, 1841 (1995) (Na=2.8x1015 cm-3, Ns=4x1012 cm-2, T=0 K) V 0.18 eff (z) Lower subband energies Increase in the subband separation Increase in the carrier concentration 0.14 at which the Fermi level crosses into the second subband second subband 0.1 Contracted wavefunctions Normalized wavefunction 0.06 first subband 0.02 0 20 40 60 Exchange-Correlation Correction: 80 Distance from the interface [Å] 100 Thick (thin) lines correspond to the case when the exchange-correlation corrections are included (omitted) in the simulations. Subband Structure Comparison with Experiments 50 depl =10 11 cm -2 Exp. data [Kneschaurek et al.] V (z)=V (z)+V (z)+V (z) 40 H eff V (z)=V 10 [m e V ] T = 4.2 K, N E n ergy E Infrared Optical Absorption Experiment: Kneschaurek et al., Phys. Rev. B 14, 1610 (1976) 60 eff 30 V (z)=V eff H H im SiO2 Al-Gate LED exc (z) (z)+V im far-ir (z) Si-Sample radiation 20 10 Vg 0 0 5x10 11 1x10 12 N 1.5x10 s 12 [cm 2x10 -2 ] 12 2.5x10 12 3x10 12 Transmission-Line Arrangement Subband Structure Comparison with Experiments Experimental data from: F. Schäffler and F. Koch (Solid State Communications 37, 365, 1981) Unprimed ladder [m eV ] Experimental data [Schäffler et al.] 40 30 T = 300 K N 20 depl = 6x10 10 [cm -2 Experimental data [Schäffler et al.] 40 T = 300 K 30 N ] 20 10 subb. appr. with V 5 subb. appr. with V 10 Primed ladder 50 1 '0 ' 10 [m e V ] 50 xc xc = 6x10 depl 10 [cm 10 subb. appr. with V (z) 5 subb. appr. with V (z) -2 10 ] xc xc (z) (z) Hartree approximation Das Sarma and Vinter Hartree approximation 0 0 0 5x10 11 10 12 1.5x10 N [cm s 12 2x10 -2 ] 12 2.5x10 12 3x10 12 0 5x10 11 10 12 1.5x10 12 N [cm s 2x10 -2 ] 12 2.5x10 12 3x10 12 Screening of the Coulomb Interaction What is Screening? - lD - Debye screening length r + Example: 3D: 1 r screening cloud r 1 exp r lD - 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 Diagramatic Description of RPA Polarization Diagrams Bare interaction = + + +... = Effective interaction (or ‘dressed’ or ‘renormalized’) Proper (‘irreducible’) polarization parts 1 - = + bare pair-bubble + +... Screening: Simulation results are for: Na=1015 cm-3, Ns=1012 cm-2 Re lative Polarization Function: 0) (q,0) /P (0) P (00 00(0,0) 1.2 3x10 6 2.5x10 6 Screening Wavevectors : 2 s qnm (q,0) e P (0) nm(q,0) 2k T=300 K 1 T=0K q s (q) 00 q s (q) T = 10 K 0.8 T = 40 K 2x10 11 6 q s (q) 0'0' T = 80 K 0.6 1.5x10 6 0.4 1x10 6 0.2 5x10 5 T = 300 K 0 0 2x10 6 4x10 6 6x10 Wavevector [cm 6 8x10 -1 ] 6 1x10 7 0 0 5x10 6 1x10 7 1.5x10 wavevector [cm -1 2D-Plasma Frequency: (q) pl 7 2x10 ] e 2N q s xy 2km* 7 Screening: Form-Factors: Na=1015 cm-3, Ns=1012 cm-2 0 0 Fij,nm(q) dz 10 10 -1 |F |F -2 |F |F 10 00,00 00,11 11,11 00,0'0' 11,0'0' 10 -1 10 -2 6 z z' | |F | 10 -3 |F | |F | 10 7 wavevector [cm 1.5x10 -1 ] 7 2x10 7 m n Off-diagonal form-factors |F 10 5x10 => | -3 0 j i Diagonal form-factors 0 |F 10 dz'y *j (z)y i(z)G˜(q,z,z')y *m(z')y n(z' ) | 01,00 | 01,11 | 01,01 01,0'0' | -4 0 5x10 6 1x10 7 wavevector [cm 1.5x10 -1 ] 7 2x10 7