Quantum and Non-local Transport Models in Crosslight Device Simulators. Copyright 2008 Crosslight Software Inc.

Quantum and Non-local Transport Models in Crosslight Device Simulators Copyright 2008 Crosslight Software Inc. 1

Introduction Quantization effects Content Self-consistent charge-potential profile. Space charge distribution and recombination. Non-local quantum transport models Intraband tunneling. Interband tunneling (tunneling junction). Mini-band tunneling (superlattice). Two-Fermi-level quantum well trapping model. Mean free path-controlled quantum escape. 2

Introduction:Device evolution Optoelectronic devices: Laser diode: from double heterostructure laser to quantum well laser and VCSELs. LED: from bulk to MQW and quantum dots. From red (GaAs) to blue (GaN) emissions. Solar cell: from single crystalline cell to tandem thin film and multiple junction cells. Silicon IC: CMOS gate length from several microns down to deep submicrons. Semiconductor from pure Si to strained Si on SiGe and with various dielectric/drain contact materials. Modeling techniques: Inclusion of quantum mechanics (QM) into classical driftdiffusion equation solvers. Modification of local drift-diffusion transport to include non-local quantum transport effects. 3

Integrated Quantum-Drift Drift-Diffusion Diffusion Model 4 Poisson s Equation Electron/Hole Drift-Diffusion Model (Energy transport) Potential profile Space charge Injection current Tunneling current correction Wave Mechanics (Quantization/ Tunneling/ Multi-band k.p Theory)

Introduction Quantization effects Content Self-consistent charge-potential profile. Space charge distribution and recombination. Non-local quantum transport models Intraband tunneling. Interband tunneling (tunneling junction). Mini-band tunneling (superlattice). Two-Fermi-level quantum well trapping model. Mean free path-controlled quantum escape. 5

Quantization and quantum confinement in a quantum well 6 Schematics of quantized state of a quantum well with (a) 2D concentration distributed entirely within the well and (b) concentration modulated by the envelop wave function.

Quantum/classical regions in a MOSFET 7 Electron distribution Vg=2 Electron distribution and Id-Vd showing effect of quantum confinement. Lg=0.17 um Vg=1 Id-Vd

Example: self-consistent solution in InGaN/GaN MQW LED 8 Without polarization charge With polarization charge Remark: well known situation in MQW LED and LD using InGaN/GaN QW

Introduction Quantization effects Content Self-consistent charge-potential profile. Space charge distribution and recombination. Non-local quantum transport models Intraband tunneling. Interband tunneling (tunneling junction). Mini-band tunneling (superlattice). Two-Fermi-level quantum well trapping model. Mean free path-controlled quantum escape. 9

10 Quantized levels non-polar polarized Remark: notice the difference in elec/hole wave function peak positions.

11 Carrier profile and radiative recombination non-polar polarized Remark: notice that the good overlap of electrons and holes results in larger radiative recombination and emission

IQE vs. current for typical MQW 12 Remark: the well-known IQE droop effect can easily be simulated using polarization charge on MQW and heterojunction interfaces.

Introduction Quantization effects Content Self-consistent charge-potential profile. Space charge distribution and recombination. Non-local quantum transport models Intraband tunneling. Interband tunneling (tunneling junction). Mini-band tunneling (superlattice). Two-Fermi-level quantum well trapping model. Mean free path-controlled quantum escape. 13

Quantum tunneling correction 14 J TE bar J DD JTE Thermionic emission J DD J DD J TE bar J DD Enhanced by tunneling factor from quantum model. Numerically, it means current from one mesh point goes to a remote mesh point. Drift-diffusion

15 Quantum-DD model works! Comparison of APSYS simulation of gate leakage current of an NMOS with thin gate oxide with experimental data (ref: IBM J. Res. Develop., vol. 43, No. 3, pp. 327-337, May 1999.).

Demonstration: RTD simulation 16

Demonstration: InGaN LED superlattice blocker 17 Without SL With SL blocker Structure based on Ref: Lee et al. Appl. Phys. Lett. 88, p.111101-1 (2006), which experimentally found improved emission power for LED with superlattice blocker.

Simulated emission power and IQE 18 2D Simulation by APSYS indeed shows improved emission power and IQE, similar to experimental observation.

Simulated quantum tunneling effect 19 (a) (b) Remark: quantum tunneling enhancement factor is actually larger for most of bias ranges. So the SL LED performed better NOT because of quantum interference effect as originally designed [see (a)]. Instead, the SL countered the internal fields and helped increase the overall barrier potential (b).

Introduction Quantization effects Content Self-consistent charge-potential profile. Space charge distribution and recombination. Non-local quantum transport models Intraband tunneling. Interband tunneling (tunneling junction). Mini-band tunneling (superlattice). Two-Fermi-level quantum well trapping model. Mean free path-controlled quantum escape. 20

21 About tunneling junction model Application: Solar cell, VCSEL, bipolar cascade laser, LED. Critical for design of many devices. Numerical issues: Equivalent carrier local generation has convergence issues. Improved convergence using equivalent mobility which is difficult to estimate. New approach: physically based TJ current across junction implemented within drift-diffusion solver.

Tunneling junction lets e<-- -->h non-locally 22 Numerical challenge: current flow across p-n junction through many mesh points. Example structure Ref: APL, 71, p3752, (1997)

Simulated I-V I V in both forward and reverse dir 23 Current (0.2A/div) experimental Voltage Remark: careful adjustment of contact resistance is necessary to get a good fit of experimental data. Negative resistance only appears within rather small range of contact resistance.

Introduction Quantization effects Content Self-consistent charge-potential profile. Space charge distribution and recombination. Non-local quantum transport models Intraband tunneling. Interband tunneling (tunneling junction). Mini-band tunneling (superlattice). Two-Fermi-level quantum well trapping model. Mean free path-controlled quantum escape. 24

25 Mini-band tunneling in Crosslight Application: When supperlattice used as part of cathode/anode. May be used to alter potential profile (e.g., to reduce overall polarization field). Implementation: Included as part of non-local tunneling. Requires a single representative period for mini-band computation --> So choose a period carefully. Theory based on Ref: Physics Reports 357 (2002) 1-111

Simple test structures 26 21 period GaN/AlGaN n-i-n structure Comparative n-i-n homojunction Remark: tunneling region defined from n to n regions. Al fraction is 16 percent for the barriers and well/barrier width is 2.5nm. Polarization charge ignored for simplicity.

First two mini-bands 27 Remark: mini-band calculated according to a reference period in the middle at bias of one volt. Energy referenced to the potential of the above reference period.

Band diagrams at one volt bias 28 Remark: mini-band structure is recalculated at all small bias steps to ensure self-consistency.

Calculated I-V I V curves 29 Remark: homojunction with no barriers has the highest current as expected. Without miniband transport, superlattice has low current based on thermionic emission only. Resonance between injection current and miniband states caused enhanced carrier transport in superlattice.

Introduction Quantization effects Content Self-consistent charge-potential profile. Space charge distribution and recombination. Non-local quantum transport models Intraband tunneling. Interband tunneling (tunneling junction). Mini-band tunneling (superlattice). Two-Fermi-level quantum well trapping model. Mean free path-controlled quantum escape. 30

Two-Fermi Fermi-level quantum well trapping model 31 When the well is really narrow and deep, carriers may not reach local equilibrium. Treat it as a carrier trap, with trapping rates determined by phonon scattering theory. Alam, Hybersten & al., IEEE Tran. Elec. Dev., Vol.. 47, No. 10, Oct. 2000, p. 1917

Mean free path-controlled quantum escape 32 Given the mean free path of carrier, we expect direct flying over the quantum well/dot with probability of exp(-d/lambda) where D=well width & lambda=mean free path.

Reference Structure P-GaN 0.1 µm P-Al 0.15 Ga 0.85 N 0.01 µm In 0.18 Ga 0.82 N(3nm)/GaN(10nm) QW 5 N-GaN 0.5 µm N-GaN 2.5 µm Size: 300 µm 300 µm The polarization charge set on the interface of QWs is 80% of the theoretical value calculated based on the Ref. Appl. Phys. Letts, 80, 1204(2002). 33

34 Comparing the models

I-V V Curve 160 140 Current( ma ) 120 100 80 60 MFP model Trapping model DD model 40 20 0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0 6.5 Voltage( V ) Drift-diffusion model shows unrealistically high turn-on voltage 35

EL Spectrum 2.00E+022 Total Sp Rate 1/(s.ev.m) 1.50E+022 1.00E+022 5.00E+021 MFP modle Trapping model DD model 0.00E+000 0.35 0.40 0.45 0.50 0.55 0.60 Wavelength (µm ) 36

IQE Curve 1.0 0.9 Internal Quantum Efficiency 0.8 0.7 0.6 0.5 0.4 0.3 0.2 MFP model Trapping model DD model 0.1 0.0 0 20 40 60 80 100 120 140 160 180 Current( ma ) 37

38 The impact of lifetime parameter in Trapping model

Comparison Current( ma ) 180 160 140 120 100 80 60 40 tau=1e-8 tau=1e-10 tau=1e-12 tau=1e-14 Internal Quantum Efficiency 1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 tau=1e-8 tau=1e-10 tau=1e-12 tau=1e-14 20 0.1 0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 Voltage( V ) 0.0 0 20 40 60 80 100 120 140 160 180 200 Current( ma ) q_trap_tauis is the the time constant for for the the quantum well 2 Fermi levelor trapping quantum dot model. trapping model In Crosslight, the default value is 1 ps. 39

40 The impact of the mean free path parameter in Trapping model

Comparison 180 1.0 160 140 0.8 Current(mA) 120 100 80 mfp=0.01 mfp=0.011 mfp=0.012 mfp=0.013 IQE 0.6 0.4 mfp=0.01 mfp=0.011 mfp=0.012 mfp=0.013 60 40 0.2 20 0 0 1 2 3 4 Voltage(V) 0.0 0 20 40 60 80 100 120 140 160 180 Current(mA) mfp is the electron or hole mean free path. In the trapping model, it controls the transport of carriers that fly over and are not trapped. In Crosslight, the default value is 0.01 (um). 41

42 The impact of the mean free path parameter in MFP model

Comparison 180 1.0 Current( ma ) 160 140 120 100 80 60 40 mfp=0.008 mfp=0.009 mfp=0.010 mfp=0.011 mfp=0.012 Internal Quantum Efficiency 0.8 0.6 0.4 0.2 mfp=0.008 mfp=0.009 mfp=0.010 mfp=0.011 mfp=0.012 20 0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 Voltage( V ) 0.0 0 20 40 60 80 100 120 140 160 180 Current( ma ) In MFP model, mfp is used for the exponential factor to control the non local current flow intensity. In Crosslight, the default value is 0.01 (um). 43

44 Summary Conventional drift-diffusion solver has been modified to include various quantum and non-local models. Such models turn out to be critical when simulating devices involving quantum and non-local physics. Potential difficulty in such approach: all such advanced treatment require some validity judgment by users.