1. The Monte Carlo method for the solution of charge transport in semiconductors
洪于晟

2. Abstract
With the Monte Carlo method, we apply this method to solutions of transport problems in conductors.
The presentation will put attentions on the evaluations of the integrated scattering probabilities.
For their direct use. A collection of results obtained with Monte Carlo simulations will be presented.

3. Information Flow Flowchart of a typical Monte Carlo program.
Basic formula
Definition of the physical system
Initial conditions of motion
What’s assumption in application?
Flight duration
Self-scattering
What’s the random number?
Simulation results
Within sophisticated theorem

4. Information Flow Flowchart of a typical Monte Carlo program.
Basic formula
Definition of the physical system
Initial conditions of motion
What’s assumption in application?
Flight duration
Self-scattering
What’s the random number?
Simulation results
Within sophisticated theorem

5. Flowchart of a typical Monte Carlo program

6. Basic formula
During each collision (each new flight), the relation of external force of carrier in semiconductor is
From the differential cross section of this mechanism, a new k state after scattering is randomly chosen as initial state of the new free flight.

7. Definition of the physical system
The parameters that characterize the material, the least known, usually taken as adjustable parameters.
We also define the parameters that control the simulation, such as the duration of each subhistory, the desired precision of the results, and so on.

8. Initial conditions of motion
For the information of stationary, homogeneous, transport process, the time of simulation must be long enough.
In order to avoid the undesirable effects of an inappropriate initial choice and to obtain a better convergence, the simulation is divided into many subhistories.

9. Information Flow Flowchart of a typical Monte Carlo program.
Basic formula
Definition of the physical system
Initial conditions of motion
What’s assumption in application?
Flight duration
Self-scattering
What’s the random number?
Simulation results
Within sophisticated theorem

10. Flight duration
P[k(t)]dt is the probability that a carrier in the state k suffers a collision during the time dt.
Assume the probability that has no suffered another collision is Y(t)

11. Flight duration t ↑, collision events↑, Y(t) ↓. Thus, Y=
The probability P(t) that the carrier will suffer its next collision during dt is
P(t)dt =

12. But, what’s the problem ?
The problem is: the integral for the collision of this situation is too complex.

13. self-scattering
Rees (1968, 1969) has devised a very simple method to overcome this difficulty.
If Γ ≡ 1/ τ0 is the maximum value of P(k) in the region of k space.

14. self-scattering A new fictitious “self-scattering” is introduced.
If carrier undergoes such a self-scattering, it state k’ after the collision is taken to be equal to its state k before collision.
It means ‘”as if no scattering at all had occurred”
Generally, it is sufficient that Γ can be not less than the maximum value of P(k).

15. self-scattering = With the constant P(k) = Γ = 1/ τ0
The probability P(t) is approximate as diffusion
P(t)

16. Information Flow Flowchart of a typical Monte Carlo program.
Basic formula
Definition of the physical system
Initial conditions of motion
What’s assumption in application?
Flight duration
Self-scattering
What’s the random number?
Simulation results
Within sophisticated theorem

17. What’s the random number
0 ≦ P(t) τ0 = exp(-t/ τ0 ) ≦1
Set the random number r = P(t) τ0 , 0≦r≦1
Thus, the time t between each collision is
t = - τ0 ln(r)

18. Information Flow Flowchart of a typical Monte Carlo program.
Basic formula
Definition of the physical system
Initial conditions of motion
What’s assumption in application?
Flight duration
Self-scattering
What’s the random number?
Simulation results
Within sophisticated theorem

19. Sampling particle

20. Within sophisticated theorem

21. Reference
Rev. Mod. Phys. 55, (1983) Carlo Jacoboni and Lino Reggiani Gruppo Nazionale di Struttura della Materia, Istituto di Fisica dell'Università di Modena, Via Campi 213/A, Modena Italy

22. Thanks for your attention Wish you have a good summer vacation