The ABINIT Users Mailing List ( CLOSED )

[David Vanderbilt <dhv@physics.rutgers.edu>]

Walter,

> Hi David + list,
>
> I still have some questions about this quantum and spontaneous
> polarization.
>
> 1) As I understand it, strictly speaking only Delta P in an
> adiabatic trasnformation  is well defined but the Berry phase
> approach calculates an absolute P modulo a quantum 2eR/Omega with
> R a lattice vector or the generalization of it that you pointed
> out.So, I guess that quantum drops out somehow if we consider
> any physically measureable quantity like an interface charge
> induced at the interface between two materials with different
> P. Am I correct?

Not quite.  If a system is taken from state A to state
B adiabatically (meaning the change is slow AND the system
remains insulating along the path), then there is a definite
amount of charge that has been pumped, which we may call Delta P.
Another physical adiabatic path from A to B might, under unusual
circumstances, lead to a different Delta P that differs from
the first one by a quantum.  There really is a correct answer
for each given path, and the Berry phase formula that makes use
only of calculations at the end points is unable to tell you which
one to choose.

Usually the Delta P for most physical adiabatic paths of interest
is much less than the quantum, so if you guess that the smallest
Delta P is the correct one, you are probably correct.  But if
you want to check, you should pick a few intermediate points
along the path, do the Berry-phase calculations at these points,
and check how P is evolving along the path.

> So, for instance if we have a material with C2v symmetry with
> the 2 fold axis along z, then I can have a non-zero P along z
> but not along x and y because of the mirrorplanes according to
> the usual Landau Lifschitz definitions, but Berry phase could
> give us a non-zero Px, Py.

I again refer you to Sec. 3.4 of
http://www.physics.rutgers.edu/~dhv/pubs/pub_list.html#198 .  The
lattice of Berry-phase or "formal" (Px,Py) values must be invariant
under C2v symmetry.  I guess in this case there are four possible
results, namely, (Px,Py) contains (0,0), or else (1/2,0), or else
(0,1/2), or else (1/2,1/2) (in units of the quantum).

In any case, for comparison with experiment it is recommended to
convert to an "effective polarization" where P is measured relative
to a nearby high-symmetry reference structure.  So, for example,
if you do a calculation on your material with slight atomic
displacements along x, and find Px=0.534 of the quantum, and you
also noticed that the C2v reference had Px=1/2, you obviously
should report the polarization as 0.034 of the quantum, which is
what we call the "effective" polarization.

> 2) Just to feel safer, I would like to understand a little better
> the numbers that come out in the tutorial for AsAs. What is
> the exact relation between what you call Berry's phase and the
> spontaneous polarization?The program seems to calculate first
> a Berry phase "corresponding to a reciprocal vector. "Does this
> mean it calculates P.Gi  with Gi the reciprocal vector or a dot
> product of P with a unit vector along Gi? I thought P.Gi=2e/Omega
> phi with phi the Bery phase. But when I put in numbers for AlAs
> a=10.53 a.u.  this does not seem to come out?  This is probably
> something trivial but I'd like to understand.  Next, given the P.Gi
> along three reciprocal lattice vectors (or rather G/6 for 6 points
> long the string) it then can calculate the Cartesian components of
> P. Each of the components seems sqrt(3) times the value given for
> the P.G for example -1.569029714E-02 * sqrt(3)=-0.271763918E-01.
> The notes seem to say that this corresponds to a P which is
> (-3/4 -3/4 -3/4)a*2e/Omega.  But I don't seem to get the right
> number. Can somebody throw some light on this what exactly is
> calculated. That way I feel we could calculated by hand for some
> other case, we are interested in and check whether it really is
> just a quantum we need to eliminate.

I haven't looked at the tutorial for AlAs recently, so maybe I'll
wait and see if somebody else has a ready answer.

> 3) Finally, something on units. When converting from atomic units
> to SI units, the program simply multiplies by e=1.602 10^-19
> and divides by bohr^2=(0.529)^210^-20.  Good but in SI units one
> usually writes : D=epsilon0E+P (eqSI) , whereas in Gaussian units
> (usually adopted when using atomic units) one writes D=E+4piP (eqG)
> So, the values that come out or the program, do they correspond to
> eqSI or eqG?  Also, the value 0.029 C/m^2  for GaN given in the
> literature, does that correspond to eqG or eqSI? In other words,
> if I want to calculate surface charge induced at an interface,
> do I use 4pi Delta P.n or DeltaP.n ?

As I understand, Div P = - rho_bound(r) in both systems.  So surface
charge is DeltaP in both systems.  So a polarization of 0.029 C/m^2
always means that the surface charge would be 0.029 C/m^2.

Hope this helps,

David

------------------------------------------------------------------------
Prof. David Vanderbilt               Phone: (732) 445-2514
Department of Physics and Astronomy  Fax:   (732) 445-4400
Rutgers University                   Email:  dhv@physics.rutgers.edu
136 Frelinghuysen Road               http://www.physics.rutgers.edu/~dhv
Piscataway, NJ 08854-8019   USA
------------------------------------------------------------------------