Power generation of a photo cell as a function of the electronic band gap

Following the post on

http://xiaoshanxu.blogspot.com/2010/07/blog-post.html

we calculated the power generation of a photo cell as a function of band gap, assuming the filling factor is unity.

As one can see that 1.1 eV is the optimal gap in terms of power generation.

Change of supersaturation

Systematic extinction rules

Systematic extinction rules

The structural factor in the crystallographic diffraction is:

$F=\Sigma { f_j exp(j \vec{r} \vec{k})}$,

where $j$ runs through all the atoms in the unit cell.

When the unit cell has more than one primitive cell in it, there might be some cases that $F$ can be zero. This is called systematic extinction (absence).

For example in BCC, for every atom at position $r$, there is another identical one at $\vec{r}+(1/2,1/2,1/2)$. Then the structural factor becomes:

$F=1+exp[i\pi (h+k+l)]$,

where $(h,k,l)$ are the reciprocal indices.

Obviously, $F$ is zero when $h+k+l=2n+1$.

TABLE I, Systematic absence because of lattice type

Lattice type	Symbol	Absence
a-center (100)	A	k+l=2n+1
b-center (010)	B	h+l=2n+1
c-center (001)	C	h+k=2n+1
face center	F	h,k,l not all odd and not all even
body center	I	h+k+l=2n+1
rhomohedral	R	-h+k+l=3n+1
hexagonal	H (triple unit cell)	h-k=3n+1 or h-k=3n+2
primitive	P	no absence

Above is only the systematic absence caused by typical symmetry of lattice types. There are other systematic absences coming from existence of screw axis and glide planes. In the end, for every space group the systematic absence is more complicated than the cases shown in the above table.

To get complete information or to find out the systematic absence for every space group, one can use Bilbao server : http://www.cryst.ehu.es/. Just click on " HKLCONDReflection conditions of Space Groups" and input the space group number.

Long wavelength dispersion relation in a solid

Long wavelength dispersion relation in a solid

In solid state physics, perhaps the most visited two elementary excitations are phonons (vibration) and magnons (spin wave). It is interesting to notice that the long wavelength dispersion relation are different for them, i.e.
phonon:
magnon: ,
where is the frequency and is the wave vector.

One can go back to the derivation of those two excitations and find the mathematical reason. On the other hand, the origin can be resorted to the fundamental difference between the two equation of motions.
phonon:
magnon: ,
where x is the displacement, is the angular momentum, is the spring constant and is the torque.

In solid, the restoring force for an oscillator comes from the imbalance of the interaction between neighbors. For long sinusoidal wave, it is always propotional to .

Hence, the final dispersion relation will be decided by the equation of motion. Since for phonon, it involves second derivative, one gets ; for magnon, the first derivative gives .

Supersaturation

Change of supersaturation

In gas-solid phase transition, the phase boundary as a function of  divides the two phases. In equilibrium, if a  combination falls on the solid side, the matter is in its solid phase. On the other hand, at inequilibrium, for a finite amount of time, a system can be with a  combination on the solid side but still remain in gas phase. This is called supersaturation. As shown in the Fig below for the points  and if they are in gas phase.
The quantitative definition of supersaturation is
,
where is the pressure corresponding to the that on the phase boundary line for the temperature . This is actually the chemical potential difference between state and state  in gas phase.

It is useful to know the dependence of on and . In other words, . Here we will start with the differential form of the relation, i.e. and . The second partial differential can be found from the definition:
,
while /Tneeds some effort to find.

To find , we look at  


From Gibbs-Duham equation:

The first term
.
Or

The second term is the chemical potential change along the phase boundary.
Since on the phase boundary, one has

where is the enthalpy change between solid and gas phase per mole, which does not vary too much with the temperature.
Again using Gibbs-Duham equation,
.

Therefore,

Or
.
Finally:
.
Or

$d\Delta\mu=RTdlnP-hdlnT$

Differential pumping

Differential pumping

Problem:

The governing equation of the problem is the following:
P2=G/S (1)
C(P1-P2)=P2S (2)

Note that P1>>P2, so the equation (2) can be written as
CP1=P2S.

In fact, if we believe these equation are true, the quantity P1 is not affected by the pumping speed Sas long as G is kept constant and P1>>P2. So changing the speed of the pump (e.g. by closing the gate valve upstream of pump to some extend) does not affect the pressure P1, at lest to the first order.

On the other hand, if all the quantities except P2are unknown, there is not way to know the pressure P1.

gas mix

Surface thermodynamics

Steady gas flow

Experimental Thermodynamics II: Open System

Experimental Thermodynamics I: Closed System

Vapor pressure

Chemical thermodynamics, chemical equilibrium, and chemical reaction rate

Band gap of periodic potential

Puzzle

Puzzle

Thermaldynamic functions

As we know, combining the first and second law of thermaldynamics, one gets

dU=TdS-PdV, (1)

where T,S,P and V are temperature, entropy, pressure and volume.

In addition, we also know that S and V are extended functions, meaning:

U(nS,nV)=nU(S,V),

which leads to Euler’s equation (the deviation can be found on the Internet):

U=TS-PV.

On the other hand, we know there are other three energy related functions, say H (enthalpy), F(free energy) and G(Gibbs free energy). Their relation to internal energy U is the following:

H=U+PV
F=U-TS
G=U-TS+PV.

Therefore, with those functions, one can describe four different process, adiabatic+isometric, adiabatic+isobaric, isothermal+isometric, isothermal+isobaric.

However, if we consider Euler’s equation, we get
H=TS
F=-PV
G=0
what’s wrong here?

Periodic potential problem

When we study solid state physics, we often deal with periodic potential, which may make you think that it is trivial. Normally, we don’t really know the function form of the potential. So approximation is made and perturbation theory is invoked, e.g. in free electron approximation and tight binding approximation.

I was surprised when I tried to solve this Schodinger equation:

d2Ψ/dx2+cos(Kx)Ψ=EΨ, here hbar, m are omitted for simplicity.

We know from Bloch theorem that the wave function takes the form

Ψ=exp(-ikx)u(x),

where u(x)=u(x+2n/K).

But to it takes me only so far. I can not find the exact solution.

Does the analytical solution exist?

RHEED simulation

Periodic function and Fourier expansion

Periodic function and Fourier expansion

The theorem is that if a function f is periodic with frequency w (period a), the function can be expanded as Fourier polynomials. i.e.

If f(x+na)=f(x)

Then f(x)=\sum_{k=nK}f(k)e^{-ikx}

where K=\frac{2\pi}{a}

.

The proof actually relies on common sense.

define:

 f(k)=\int f(x)e^{-ikx}dx

One can see that both f(x) and exp(-ikx) are perodic, where average of exp(-ikx) is zero. So if the two periods are not commensurate, f(k) will be zero.

The only possible nozero f(k) occurs when kx=2npi, where k=nK.

pydao0.979 released

Pydao, a new software for data organize and analysis is released as a trial version.

Data organization is based on hierachical data file (HDF) structure.

Data analysis is based on the plugins designed for special usage.

Data visualization is based on matplotlib and mayavi pakage.

Now I have lattice dynamics as a useful plugin.

There are some build in analysis and visualization tools, not as much as the xpy1.xx. But we will make pydao more and more complete in the future.

Right now, only source code is provided. Win32 compiled will come soon.

Surface preparation of MgO

Surface preparation of MgO

What's the use of MgO

1) Good material for hight temperature processing 2800 C melting temperature

2) low dielectric constant ~10, low loss

Problem of MgO

1) surface reacts with H2O and CO2 after exposing to air

2) impurities like Ca

Preparation steps

1) Cleaving

Normally with charged surface, difficult for scanning

2)Polishing

sub-micron polishing is normally used.

3) Acid etching

This is to remove the additional material left from the polishing. So basically, the sample as received is already etched.

agent: phosphoric and nitric-acid

4) Annealing

Most important part. This is to solve problem 1) and to make atomically flat surface.

Surface orientation	Duration	Temperature	Step height	Terrace width	Comments	Reference
(100)	2 hours	Tanneal>1000 C, the surface changes remarkably The annealing is better for higher temperature until  Tanneal>1350 C		200-300 nm	Step comes from the Ca atoms diffuse to the surface	Ahmed1996
(100)	12 hours	> 700 C can get rid of Mg(OH)2 and MgCO3 >1100C, one can get atomically smooth surface	4 nm	700 nm		Aswal2002
(100) 2 degree miscut	360 min	1000	3-7 nm	120 nm		Benedetti2007
(110)	10 min	1000 in 10-7 torr			facet created because of liquid solid interface in the etching	Giese2000
(111)	30 min	1700 to heal the facet			The surface may reorganize into (332) 1700 C to heal the facet

Reference

[Ahmed1996] Ahmed F. et al J. of Low Temperature v105, p1343 (1996)

[Plass1998] Plass R. Surface Science, v414, p 268 (1998)

[Giese2000] Giese D.R. Surface Science, v 457, p 326 (2000)

[Aswal2002] Aswal D.K et al. Journal of Crystal Growth, v236, p.661. (2002)

[Benedetti2007] Benedetti S et al. Surface Science v601, p 2636. (2007)