# Si ellipsoids of constant energy?

1. Feb 19, 2006

### Baggio

Hi,

I understand that the conduction band in si is 6 fold degenerate but why do they form 6 ellipsoids of constant energy? i.e 2 for each axes. Are they just simply fermi surfaces?? How are they constructed? Thanks

2. Feb 19, 2006

### Gokul43201

Staff Emeritus
Because the band structure is a strong function of the crystal structure. For Si, the cubic symmetry dictates the reason for the 6-fold symmetry in the energy bands.

The conduction band for intrinsic Si, is not the Fermi surface. The Fermi surface lies between the valence band maximum and the conduction band minimum. It is, however, a constant energy surface, just like the other two.

Do you mean "How are they reconstructed (ie: theoretically or experimentally)?" I'm not sure I understand this question.

3. Feb 20, 2006

### Baggio

Right, another question.. in strained Si, intervalley phonon scattering is supressed how does this relate to an increase in mobility?.. Is this increase only in one direction or in all directions?.. If it's just in one direction why's this so?

Papers I have read are very vague in that they just state that there is an increase in mobility...

Thanks

4. Feb 20, 2006

### Gokul43201

Staff Emeritus
The mobility is simply a number proportional to the effective scattering time, given by :

$$\frac{1}{\tau_{eff}} = \frac{1}{\tau_{el-el}} + \frac{1}{\tau_{el-ph}} + \frac{1}{\tau_{imp}}$$

Processes which reduce the scattering time reduce the mobility. Phonon scattering (the cause for the second term above) is nothing but the process of collision with lattice sites. Suppressing phonon scattering leads to an increase in the phonon scattering time $\tau_{el-ph}$ and hence, an increase in mobility.

For a simplistic picture, Drude theory tells us that the drift velocity of an electron is simply

$$v_d = a\tau = (F/m)\tau = (eE/m)\tau$$
where $\tau$ is the time between collosions, or the scattering time.

From this, we calculate the mobility as

$$\mu \equiv \frac{v_d}{E} = \frac {e\tau}{m}$$

I've read that there is only an increase in the in-plane direction (the strained direction) while there is a decrease in the out-of-plane direction - in the case of Ge doped Si-MOSFETs. I don't know why this is true.

PS : I'm not very familiar with this area - I imagine the person closest to this field would be Dr. Transport.

Last edited: Feb 20, 2006
5. Feb 21, 2006

### Dr Transport

Thanks Gokul for the intro.

Everything said so far is correct to the zeroth order, a fine approximation, but an approximation. To get into transport in semiconductor systems one can or should I say needs to solve the Boltzmann eq directly. A long a tedious process at best. In Group IV semiconductors, III-V and II-VI materials you can do this quite nicely using $$\vec{k} \cdot \vec{p}$$ theory to calculate the band structure of the material. Using these wave functions one calculates the transitition matrix elements for the appropriate scattering mechanism. Each of these transition rates is then expanded in terms of combinations of symmetry adapted polynomials which correspond to a sum of angular momentum states ( $$l + l' = 0, 1, 2,\...$$).

If you calculate all the possible transitions, you'll be there for a very long time, years. If you use the symmetry of the lattice and its' associated group i.e. $$O_{h}, T_{d}$$ you can reduce the necessary number of computations because of symmetry. The transition rate is not only proportional to the difference between the incident and scattered $$\vec{k}$$ but the direction in the lattice. One also expands the Boltzmann eq in terms of the symmetry adapted polynomials and uses the expansion coefficients to solve for the conductivity and Hall mobilities.

Now why does the mobility change when strain is applied to the lattice. Very simply the band structure changes which is accompanied by a corresponding change in the wave functions which reduces the scattering rates for the different mechanisms.

For an overview of band structures, look at your well worn copy of Ashcroft and Mermin, Chapter 28: Homogeneous Semiconductors.

For a complete set of papers on transport in p-type Si look for papers by a Madarasz and Szmulowicz back in the mid '80's, mostly in Phys Rev B. Sz took it a little bit further and did the calculations for Ge. I worked some on III-V and mostly $$II-V-VI_{2}$$ or Chalcopyrites. Never did publish any of it, never thought I got too far but enough for a PhD, or could have been 2. Madarasz was my advisor, only PhD student he ever had that finished and the only student he ever thought had thick enough skin to dive into a problem like that and get anywhere.

6. Feb 22, 2006

### Baggio

Thanks, I think that's a little too much detail than what I was after! I think I understand now but I have a few more questions.

For strained silicon they say there's a reduction in intervalley scattering and that the f-phonon scatering is suppressed..what is f-phonon scattering? I think i't sht e90 degree scattering between valleys but what is the mechanism?

Also how does the valence band structure change in tensile strained silicon and how do the LH and HH effective masses change? For the conduction band the in plane electron mobility is enhanced but what about the hole mobility?

Thanks

Last edited: Feb 22, 2006
7. Feb 22, 2006

### Dr Transport

At the center of the zone, the HH and LH bands are degenerate, by straining the lattice you will separate them. This changes the effective masses and the wave functions. As for how the mobilities change, couldn't tell you off of the top of my head. The calculation for unstrained p-type silicon is difficult enough. I know that you would see an induced anisotropy in the conductivity and Hall mobilities because of the strain, i.e. $$x = y \neq z$$. Look at Chuang's book: Physics of Optoelectronic Devices, Chapter 4 deals with band structures and strains applied to the lattice. The problem that I have never looked into is exactly how does the associated material parameters like deformation potentials behave when a lattice is strained. Now if we assume that they do not change, we still have to look at an anisotropy in the scattering rates and preferred directions and how they affect the mobilities.

Interesting problems, just wish someone would pay me to live while I look for the answers.

8. Apr 13, 2007

### Andrew K

More papers of interest

For sake of completeness, I will mention what I have pulled up. I am not an expert, but I can address f-scattering and g-scattering (basically I can direct you to where these things are mentioned in literature relevant to the discussion).

"Piezoresistance effect in p-type Si" by Y. Ohmura, Phsical Review B (APS), Vol 42. Number 14 (1990)

This paper states that at lower strains, good models can be constructed to explain existing data (below 10^19 dyns/cm^2) without talking about intervalley transitions.

"Electron Mobility Model for Strained-Si Devices"
Siddhartha Dhar, et al, IEEE Transactions on Electron Devices, Vol 52, No. 4, April 2005.

This addresses the breaking of symmetry and the changes in intervalley scattering. Scattering mechanisms are broken down in this and other papers as acoustic scattering, g-type scattering (between equivalent valleys), f-type scattering (between nonequivalent valleys), and impurity scattering.

C Smith "Piezoresistance effect in germanium and silicon," Phys. Rev., vol 94 (1954)

This is probably the first paper to mention a population redistribution amongst nondegenerate valleys (degeneracy lifted by strain). Smith suggests that the ellipsoids (should not be, since at temperatures of interest, we are not at k=0, meaning we would not have ellipsoids-- nonetheless very instructive) distort due to strain. Examining two valleys, one will expand, and one will shrink (in E-k space) (of course depending on the direction of applied stress, voltage, and current). Thus you would have a population redistribution from the higher energy valley to the lower energy valley causing 2 effects that Smith suggests are of equal importance: 1.) The mobilities in the different valleys are different so a population redistribution would cause a mobility weighted average to change and 2.) The population redistribution affects the scattering probabilities between these two valleys thus affecting mobility.

Finally, there is a paper that deals more with effective mass changes:
K. Suzuki etal "Origin of the linear and nonlinear piezoresistance effects in p-type silicon" Jpn. J. Appl. Phys. vol 23 pp L871-L874

These papers have plenty more references (for example Pikus and Bir have a much cited work) if you are interested.