Modeling of Electron Transport in Thin Films with Quantum and Scattering Effects

Transcription

1 Modelng of Electron Transport n Thn Flms wth Quantum and Scatterng Effects Anuradha Bulusu Advsor: Prof. D. G. Walker Interdscplnary Program n Materal Scence Vanderblt Unversty Nashvlle, TN

2 Motvaton L Devce λ de Brogle Gate Insulator e - tunnelng quantum and scatterng effects Source Channel e - scatterng e - confnement Dran cannot be gnored. Insulator L devce < ι r assumpton of mnmal devaton from local equlbrum no-longer vald BTE moments cannot be appled to extreme non-equlbrum. Increased need for mcroelectronc devce smulaton models that ncorporate both quantum and scatterng effects wthout beng computatonally ntensve.

3 Nanoscale Devce Models Contnuum Lattce Boltzmann Transport Equaton Drft-Dffuson, Conducton Eq. Non- Equlbrum NEGF Current Work Quantum Monte Carlo Monte Carlo Hydrodynamc Drft dffuson Atomstc DFT/ Molecular Dynamcs Quantum NEGF: Non-Equlbrum Green s functon Non- Equlbrum Electrons Contnuum

4 Advantages of Non-Equlbrum Green s Functon Method Incorporaton of quantum nterference effects such as tunnelng and dffracton, not possble through the Boltzmann equaton. Mathematcally accurate approach to nclude rgorous scatterng (electron-phonon scatterng, surface scatterng etc). Elmnates perodc boundary condtons as outgong waves are planar. Multscale formulaton: Can be used to solve atomstc systems to mesoscopc systems.

5 Non-Equlbrum Green s Functon Formalsm Orgnal Schrödnger wave equaton Modfed wave equaton wth self-energy terms Green s functon ( H + U ) ψ (r) ε ψ (r) α = ( + U + Σ + Σ + Σ ) ψ ( r) = ε ψ ( ) H 1 2 s α α α r G( E) E 0 + I H Σ1 Σ2 Σ = s α α 1 Devce Current I + q = Trace [ ] [ ] n Γ A f Trace Γ G

6 Nonequlbrum Green s Functon Method Choose Hamltonan H for devce Compute selfenergy functon for contacts coupled to the devce Σ 1, Σ 2 Intal guess for Hartree Potental U Compute selfenergy functon for scatterng effects Σ s Combne Σ 1, Σ 2, Σ s, U, H, µ 1, µ 2 n the retarded Green's functon to obtan charge densty ρ Compute potental U self-consstently du = ƒ(ρ) Converged? Obtan termnal currents as the dfference n the nflow and outflow electron densty ρ

7 Example of a Very-Small Scale Devce Problem Z X Source µ 1, Σ 1 Devce Ε, Σ s Dran µ 2, Σ 2 2-D nanoresstor connected to source and dran contacts. µ 1, µ 2 - source and dran chemcal potentals. Y µ 1 µ 2 = qv Energy states along z-axs are quantzed. Electron-phonon scatterng assumed to be ndependent of electron Σ ( E) Denergy. G( E) s = o e.g. D o = 0.1eV 2 E ph = 20meV

8 I-V Characterstcs of Slcon Thn Flms wth Incoherent Near-Elastc Scatterng 2.50E-07 n = 5x10 18 cm -3 Ballstc 2.00E-07 Eph = 0.2meV Dran Current (A) 1.50E E-07 Eph = 2meV Eph = 10meV 5.00E-08 Eph = 20meV 0.00E Dran Voltage (V)

9 Resstvty (Ohm-m) Effectve Channel Resstvty wth Incoherent Scatterng Phonon Energy (mev) Contnuous densty of states from source and dran contacts spll over nto the channel causng energy level broadenng. Dscrete energy states n narrow channel reduce the number of energy states avalable for the ncomng electrons ntroducng resstance at the contacts. Increase n phonon energy from 0 to 20m ev for varous resstors ncreases correspondng channel resstvty by a factor of 4.

10 Thermoelectrc Propertes of Slcon Thn Flms 2.0E E-07 n = 5x10 18 cm -3 Eph = 2meV Dran Current (A) 1.0E E E E-08 Eph =10meV Eph = 20meV E-07 Dran Voltage (V) Predcted Seebeck coeffcent value of 250µ V/K matches well wth experments. (Geballe, T. H. et.al, Physcal Revew, 98, 4, 1955).

11 Seebeck Coeffcent for SGe Superlattces 2.50E-06 n = 8x10 19 cm -3 Dran Current (A) 2.00E E E E-07 Ballstc Eph = 2meV 0.00E E-07 Dran Voltage (V) Calculated Seebeck value of 100 µ V/K matched well wth expermentally measured values of 312µ V/K (B.Yang et. al. Appled Physcs Letters, 80, 10, 2002).

12 Conclusons and Future Work The NEGF formalsm successfully couples quantum effects wth electron-phonon scatterng. A 33% to 70% drop n channel current was notced due to near-elastc electron-phonon scatterng. Predcted Seebeck coeffcent values are n good agreement wth experment for Slcon and SGe superlattces. Extenson of the present work to nclude energydependent electron-phonon scatterng.

13 Acknowledgements Prof. Supryo Datta, Dept. of Electrcal and Computng Engneerng, Purdue Unversty, West Lafayette, IN. Vanderblt Dscovery Grant. Vanderblt Insttute of Nanoscale Scence and Engneerng (VINSE) Fellowshp.

14 Seebeck Coeffcent of Slcon Thn Flms 1200 (submtted to Journal of Appled Physcs) 1050 Seebeck Coeffcent ( µv/k) Predcted Measured E+17 1.E+18 1.E+19 1.E+20 Dopng level (cm -3 )

15 Non-Equlbrum Green s Functon Formalsm Modfed wave equaton wth self-energy terms Relaton between lfetme of the egen state and self energy ( ) H + U + Σ + Σ + Σ ψ ( r) = ε ψ ( r) γ = = τ s α 2 Im Σ α α Energy level broadenng Green s functon Spectral functon Devce Current Γ = ( Σ Σ + ) 1,2, s 1,2, s 1,2, s G( E) E 0 + I H = s + ( G( E) G ( E) ) Σ Σ Σ 1 2 A( E) γ = D( E) = = 2π + I + q = Trace ( E ε ') ( γ / 2) [ ] [ ] n Γ A f Trace Γ G Σ s ( E) = D G( E) o