1. Semiconductor Device Modeling
Sts. Cyril and Methodius University - Skopje FACULTY OF ELECTRICAL ENGINEERING AND INFORMATION TECHNOLOGIIES
Semiconductor Device Modeling
Lecture 6. Hydrodynamic Model
1. Extensions of the Drift-Diffusion Model
2. Stratton’s Approach
3. Balance Equation Model
4. Discretization of Hydrodynamic Equations
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2. Electronic Structure, Lattice Dynamics Electromagnetic Fields
Device Simulation
Before we explain the hydrodynamic model…
Electronic Structure, Lattice Dynamics
Transport Equations
Electromagnetic Fields
Device Simulation
J, r
E, B

3. Hierarchy of Semiconductor Simulation Models
Schrödinger Equation
Boltzmann Transport Equation
Monte-Carlo Particle-Based App.
Hydrodynamic and Energy Balance
Transport Approaches
Drift-Diffusion
Approaches
Compact Approaches

4. Diffusive vs. Ballistic Transport
‘Classical’
Transport regime depends on length scale:
l - Phase coherence length
lin - Inelastic mean free path
le - Elastic mean free path
‘Quantum’

5. Characteristic Length-Scales

6. Nanoscale Transistors

7. Review of Drift-Diffusion Approach
Valid when diffusive transport dominates
Mobility is a good parameter
T=300K
Silicon
Temperature dependence of the Electron and Hole Mobility Si Ge
GaAs
Electrons
∝T −2.4
∝T −1.7
∝T −1.0
Holes
∝T −2.2
∝T −2.3
∝T −2.1

8. Extension of the validity of Drift-Diffusion Approach
Introduction of the field-dependent mobility
High-field region
Long-channel
device S
Low-field region D
Short-channel
device S D

9. Extension of the validity of Drift-Diffusion Approach [1]
Velocity Saturation
Intervalley
scattering

10. Extension of the validity of Drift-Diffusion Approach [2]
Velocity Saturation (cont’d)

11. Extension of the validity of Drift-Diffusion Approach [3]
Velocity Saturation Implications (cont’d)

12. Extension of the validity of Drift-Diffusion Approach [4]
Velocity Overshoot in Bulk Si

13. Hydrodynamic Simulations of Devices
In summary, we need to use at least HYDRODYNAMIC MODEL to describe velocity overshoot in nanoscale devices.

14. Hydrodynamic Simulations of Devices [2]
Importance of Velocity Overshoot
90 nm Channel Length Device

15. Isothermal Hydrodynamic Simulations of Devices [3]
Current Characteristics
Hydrodynamic
Drift-diffusion
Isothermal

16. Drift-Diffusion Simulations of Devices
Velocity Along the Channel – Drift-Diffusion Model

17. Hydrodynamic Simulations of Devices [4]
Velocity Along the Channel – Hydrodynamic Model

18. Hydrodynamic Simulations of Devices [4]
Current Characteristics - Drift-Diffusion (DD-isothermal) vs Hydrodynamic+ Lattice Temperature (HF-LT)
HD-isothermal
DD-isothermal
HD-LT
DD-LT

19. Drift-Diffusion Simulations of Devices
Velocity – DD (Isothermal)
Velocity – DD + Lattice Temperature

20. Drift-Diffusion vs Hydrodynamic Simulations of Devices
Velocity – Drift Diffusion + Lattice Temperature
Velocity – Hydrodynamic+ Lattice Temperature

21. Monte Carlo Results for Velocity Overshoot

22. Hydrodynamic Simulator
theory
Basic hydrodynamic (balance) equations - derivation
Ensemble relaxation rates and their calculation

23. Basic Hydrodynamic Equations

24. 1.Carrier Density
Balance equation for the carrier density is obtained by assuming that f(p)=1:
Using v=vd+c, one gets the final form of the balance equation for the carrier density:

25. 2. Momentum
Momentum Balance equation is obtained by assuming that f(p)=p:
This leads to the following momentum balance equation:

26. 2. Momentum (cont’d)

27. 3. Energy
The Energy Balance equation is obtained by assuming that f(p)=E(p):
The Energy Balance equation is then of the form:

28. Closure To have a closed set of equations, one either:
(a) ignores the heat flux altogether
(b) uses a simple recipe for the calculation of the heat flux:
Substituting T with density of the carrier energy, the momentum and energy balance equations become:

29. Complete Balance Equations
More convenient set of balance equations in terms of n, vd and w:

30. Momentum Balance Equation
The DD model is obtained by simplifying the momentum balance equation, which in 1D is of the form:
In steady-state, one gets for the momentum balance equation that:
where:

31. DD Model
The DD model is then obtained by making the following assumptions:
The distribution function is close to the equilibrium, and
The energy gradient field is zero.
In the extended DD model m(E) and D(E) are assumed to depend upon the local field values only.

32. Ensemble Relaxation Rates

33. Ensemble Relaxation Rates (cont’d)

34. Momentum Relaxation Rates

35. Energy Relaxation Rates

36. Incorporation of Non-parabolicity
The nonparabolicity of the conduction band can be included by using the energy-dependent effective masses. If one uses a hyperbolic band model, then:
The energy dependent effective mass is then given by:

37. Discretization of Hydrodynamic Equations [1]
For simplicity, the equations will be discretized on an equally spaced meshes. In 1D and in finite-difference operator notation, the RHS’s of the equations that need to be solved are:

38. Discretization of Hydrodynamic Equations [2]
(a) Forward-time centered-space scheme (FTCS):
(b) Upwind scheme:
(c) Lax-Wendroff scheme:

39. Discretization of Hydrodynamic Equations [3]
(d) DuFort-Frankel scheme:
(e) Leapfrog scheme:

40. More on the Velocity Overshoot

41. More on the Velocity Overshoot (cont’d)
d >> 20 nm for electrons in Si
Why do we observe velocity overshoot?
The energy relaxation time in Si is larger than the momentum relaxation time
At first, the electric field simply displaces the distribution function with little change on its shape

42. FD-SOI Device Generations Examined
(simulation results obtained with Silvaco ATLAS- commercial device simulator)
feature
14 nm
25 nm
90 nm
Tox
1 nm
1.2 nm
1.5 nm
VDD 1V
1.2 V
1.4 V
Source/drain doping = 1020 cm-3
Channel doping = 1018 cm-3

43. What happens in a 14 nm FD SOI Device?

44. 14 nm FD-SOI Device

45. 25 nm FD-SOI Device

46. 25 nm FD-SOI Device
VG=1.2 V
Hydrodynamic
Drift-diffusion

47. 90 nm FD-SOI Device

48. 90 nm FD-SOI Device
hydrodynamic
VG=1.4 V
Drift-diffusion

49. Results from Hydrodynamic Simulations

50. Problems with Hydrodynamic Model -
Dependence upon energy relaxation time
VG=1 V
LG = 14 nm
0.3 ps
0.2 ps
=0.1 ps

51. 180 nm FD SOI Device Results with Velocity Saturation
Tsi = 72 nm
NA = 1018 cm-3
Results with Velocity Saturation
Results with Velocity Overshoot