# [Pw_forum] Is the phonon line-width gamma positive for a instable mode with negative freqency?

Dingfu Shao dingfu.shao at gmail.com
Fri Feb 13 04:58:22 CET 2015

```Dear Nicki and Sridhar,

Thanks a lot for your replies! I think you are right that the  imaginary
omega leads to the negative lambda.

Best regards,

Yours
Ding-Fu

2015-02-12 22:23 GMT+08:00 Sridhar Sadasivam <sridhu88 at gmail.com>:

> Dear Ding-Fu,
> As Nicki pointed out, omega(q,v) is imaginary for unstable modes. So
> omega^2 is negative and hence lambda is negative.
>
> If you look at the definition of electron-phonon matrix element g, it
> contains a sqrt(omega(q,v)) in the denominator. Thus |g|^2 contains
> omega(q,v) in the denominator. And this omega(q,v) cancels with the other
> omega(q,v) in the formula for gamma. I think this is the reason, that gamma
> is still positive for unstable modes.
>
> Sridhar
> Purdue University
>
> On Thu, Feb 12, 2015 at 8:46 AM, Dingfu Shao <dingfu.shao at gmail.com>
> wrote:
>
>> Dear Ion,
>>
>>
>> Best regards,
>>
>> Yours Ding-Fu Shao
>>
>>
>> 2015-02-12 15:44 GMT+08:00 Ion Errea <ion.errea at ehu.eus>:
>>
>>>  Dear Ding-Fu,
>>>
>>> Considering that the g contains a omega(q,v)^-1/2, gamma(q,v) is
>>> independent of the phonon frequency and, thus, always positive.
>>>
>>> On the contrary, calculating lambda for a mode with imaginary
>>> frequencies makes no sense, as the crystal is unstable in the harmonic
>>> approximation.
>>>
>>> Bests,
>>>
>>> Ion Errea
>>>
>>> Donostia International Physics Center (DIPC),
>>> Manuel de Lardizabal Pasealekua 4,
>>> 20018 Donostia-San Sebastián, Basque Country, Spain
>>>
>>> Email:  ion.errea at ehu.eus
>>> Tel:    +34 943 01 54 23
>>> Fax:    +34 943 01 56 00
>>> Web:    http://ionerrea.wordpress.com/
>>>
>>> 12/02/15 02:54(e)an, Dingfu Shao igorleak idatzi zuen:
>>>
>>>  Dear QE developers and users,
>>>
>>>        When calculating the electron-phonon coupling, the phonon
>>> line-width gamma is defined as
>>> gamma(q,v) = 2 * pi * omega(q,v) * sum( |g|^2) * delta(E_q) *
>>> delta(E_(k+q)).  (1)
>>>
>>>  And the electron-phonon coupling strength for mode v at q is
>>>
>>>  lambda(q,v) = gamma(q,v) / (pi * h * N(E_F) * (omega(q,v))^2)
>>> (2).
>>>
>>>  (See
>>> )
>>>
>>>  Accoding to these definitions, it seems that the  phonon line-width
>>> gamma(q,v), electron-phonon coupling strength  lambda(q,v) must be BOTH
>>> negative when a instable mode with negetive frequency omega(q,v) is
>>> calculated.
>>>
>>>  However, I checked the electron phonon coupling I calculated before, I
>>> found with a  negetive omega(q,v), there are negative lambda(q,v) and
>>> positive gamma(q,v). I found some similar report, for example, PHYSICAL
>>> REVIEW B 86, 155125 (2012), in which the positive gammas were calculated
>>> for the instable mode.
>>>
>>>  That  realy puzzles me. Can anybody explain it?
>>>
>>>  Thanks a lot!
>>>
>>>  Best regards,
>>>  yours
>>>  Ding-Fu Shao
>>>
>>>
>>>
>>>
>>>
>>>
>>> --
>>>
>>>
>>> *Ding-Fu Shao, Ph.D *
>>>
>>> *Institute of Solid State Physics*
>>>
>>>
>>> *P. O. Box 1129*
>>>
>>> *Hefei 230031*
>>>
>>> *Anhui Province*
>>>
>>> *P. R. China*
>>>
>>>
>>>
>>>
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>>>
>>>
>>>
>>
>>
>> --
>>
>>
>> *Ding-Fu Shao, Ph.D *
>>
>> *Institute of Solid State Physics*
>>
>>
>> *P. O. Box 1129*
>>
>> *Hefei 230031*
>>
>> *Anhui Province*
>>
>> *P. R. China*
>>
>> *Email: dingfu.shao at gmail.com <dingfu.shao at gmail.com>*
>>
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>
>
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--

*Dingfu Shao, Ph.D Candidate*

*Institute of Solid State Physics*

*P. O. Box 1129*

*Hefei 230031*

*Anhui Province*

*P. R. China*

*Email: dingfu.shao at gmail.com <dingfu.shao at gmail.com>*
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