N ational Technical University of Ukraine “Kyiv Polytechnic Institute” Authors: Fedyay Artem, Volodymyr Moskaliuk, Olga Yaroshenko Q uantum transport simulation tool, supplied with GUI Presented by: Fedyay Artem 13, April 2011Kyiv, Ukraine ElNano XXXI Department of physical and biomedical electronics

2 Overview Objects of simulation Physical model Computational methods Simulation tool Examples of simulation

3 Objects to be simulated Layered structures with transverse electron transport: - resonant-tunneling diodes (RTD) with 1, 2, 3, … barriers; - Supperlattices Reference topology  (example):

4 Physical model. Intro ENVELOPE FUNCTION (EFFECTIVE MASS) METHOD in case of homogeneous s/c and flat bands (Bloch waves)  Envelope of what? of the electron wave function:  What if not flat-band?

5 Physical model. Type ENVELOPE FUNCTION (EFFECTIVE MASS) METHOD

6 Model’s restriction h/s with band wraps of type I (II) TYPE III InAs-GaSb Band structures  sketches TYPE I GaAs – AlGaAs GaSb – AlSb GaAs – GaP InGaAs – InAlAs InGaAs – InP TYPE II InP-Al 0.48 In 0.52 As InP-InSb BeTe–ZnSe GaInP-GaAsP Si-SiGe

7 Physical model. Type  What do we combine? we combine semiclassical and “ quantum-mechanical ” approaches for different regions Sometimes referred to as “COMBINED” (*) homogeneous, (**) almost equilibrium  high-doped (*) nanoscaled heterolayers, (**) non-equilibrium  intrinsic

8 Physical model. Electron gas

9 Physical model. Master equations. (1 band)

10 Physical model. Electrical current. Coherent component Coherent component of current flow is well described by Tsu-Esaki formulation:

11 Physical model. Electrical current. (!) Coherent component

12 Physical model.   Which equation  L and  R are eigenfunction of? We need |  L | 2 and |  R | 2 for calculation of CURRENT and CONCENTRAION

band model. What for?

band model. What for?  Current re-distribution between valleys  changing of a total current  Electrons re-distribution  changing potential ! [100]

15 Physical model.  Г,  X  It was derived from k.p-method that instead of eff.m.Schr.eq. it must be a following system: which “turns on” Г-X mixing at heterointerfaces (points z i ) by means of . It of course reduces to 2 independent eff.m.Shcr.eqs. for X and Г-valley

16 Physical model. Boundary conditions for Schr. eq. We have to formulate boundary conditions for Schrödinger equation. They are quite natural (Q uantum T ransmission B oundary M ethod ). Wave functions in the reservoirs are plane waves. Transmission coefficient needs to be found for current calculation

17 Physical model. Features Combined quazi-1D. Self-consistent (Hartee approach). Feasibility of 1 or 2-valley approach. Scattering due to POP and Г-X mixing is taking into acount.

18 Scientific content circumstantial evidence: direct use of works on modeling of nanostructures

19 Computational methods Numerical problems and solutions : ?Computation of concentration n(z) needs integration of stiff function  using adaptive Simpson algorithm; ?Schrodinger equation have to be represented as finite-difference scheme, assuring conservation, and needs prompt solution  using of conservative FD schemes and integro- interpolating Tikhonov-Samarskiy method; ?Algorithm of self-consistence with good convergence should be used to find V H  using linearizing Gummel’s method ?Efficient method for FD scheme with 5-diagonal matrix solution (appeared in 2-band model, corresponding to Schrödinger equation)  direct methods, using sparse matrix concept in Matlab (allowing significant memory economy)

20 Let’s try to simulate Al 0.33 Ga 0.64 As/GaAs RTD

21 Application with GUI

22 Emitter

23 Quantum region

24 Base

25 Materials data-base (1-valley case) mГ(x),mГ(x), (!) Each layer supplied with the following parameters: x – molar rate in Al x Ga 1-x As  E c (x)=U00*x m Г (x)=m00+km*x,  E c (x) – band off-set  (x) is dielectric permittivity  (x)= e00+ke*x

26 Settings

27 Graphs

28 Calculation: in progress (few sec for nsc case)

29 Calculation complete

30 Concentration

31 Potential (self-consistent)

32 Concentration (self-consistent)

33 Transmission probability (self-consistent)

34 Local density of states g (E z, z) (self-consistent)

35 Local density of states g (E z, z) ( in new window with legend)

36 Distribution function N (E z, z) ( tone gradation)

37 I-V characteristic (non self-consistent case)

38 Resonant tunneling diode (2 valley approach) m X  E Х-Г  (!) Each layer supplied with additional parameters: CB in Г and X-points

39 LDOS in Г and X-valleys X-valley: barriers  wells Г-valley

40 Transmission coefficient 2 valleys Г – valley only both Г and X valleys  (*) Fano resonances (**) additional channel of current

41 Another example: supperlattice AlAs/GaAs 100 layers LDOS  CB profile 

42 Try it for educational purposes! Simulation tool corresponding to 1-band model w/o scattering will be available soon at: (!) Open source Matlab + theory + help Today you can order it by 2-band model contains unpublished results and will not be submitted heretofore THANK YOU FOR YOUR ATTENTION