Good Vibrations Studying phonons with momentum resolved

spectroscopy

D.J. Voneshen

20/6/2018

Overview

• What probe to use?

• Types of instruments.

• Single crystals example

• Powder example

• Thing I didn’t talk about because they were tricky.

Select your probe.

• Lets say we want to measure the phonons in Aluminium.

• We need ~1 meV resolution.

• We also want information at many wavevectors.

Select your probe.

Image pinched from:

http://www2.lbl.gov/MicroWorlds/ALSTool/EM

Spec/EMSpec2.html

So, x-rays it is.

• Good news, we can produce lots!

• Bad news, for 1 Å wavelength the energy is around 107 meV.

• So for 1 meV resolution we wont have many left.

• Anything else?

Image pinched from: http://www.esrf.eu/

Neutrons!

• Typical energy 10-100 meV.

• 1 Å wavelength.

• Hard to produce, but we wont have to waste many.

Image pinched from: http://www.esrf.eu/

Where can we find neutrons?

Nuclear reactors Spallation sources

Nuclear reactor, now what?

• Neutrons from reactor have many energies following some profile. Emitted continuously.

• Profile depends on what bit of the reactor you look at.

• For spectroscopy, we need to know how much energy was transferred so we need to know incident and final energies.

Image pinched from:

http://pd.chem.ucl.ac.uk/pdnn/inst3/reactors.htm

In diffraction we use a monochromator for Ei….

So why not use another!

The triple axis spectrometer

Spallation, what is different?

• Most spallation sources are pulsed.

• So time integrated flux is much lower.

• But, the pulsed nature gives a big advantage. We can use the neutron Time-Of-Flight (TOF) to determine either Ei or Ef.

Chopper spectrometers

• For phonons we normally use the TOF to give us Ef.

• To get Ei we use a neutron absorber with a hole cut in it.

• By spinning this we get a “chopper”.

Chopper

Sample

Detector

Chopper spectrometers

• For phonons we normally use the TOF to give us Ef.

• To get Ei we use a neutron absorber with a hole cut in it.

• By spinning this we get a “chopper”.

A chopper spectrometer

A return to x-rays

• We can do inelastic x-ray scattering.

• It’s basically a TAS.

• But, instead of varying angles, we vary the temperature of the monochromator!

Doing a single crystal measurement

Where do we look?

• We have two options.

– We could use diffuse scattering to tell us where the signal should be strong.

– We can think about the phonon scattering cross section.

• Lets go with the second.

One phonon scattering cross section

𝐼 𝑄, 𝐸 =𝑁ℏ

2

1

𝜔 𝑄, 𝜈𝜈

𝑏𝑗

𝑚𝑗1 2

𝑗

𝑄 ∙ 𝑒𝑗 𝑘, 𝜈 𝑒𝑖𝑄∙𝑅𝑗𝑇𝑗 𝑄

2

× 𝑛 𝜔 𝑄, 𝜈 , 𝑇 + 1 𝛿 𝐸 + ℏ𝜔(𝑄, 𝜈)

+ 𝑛 𝜔 𝑄, 𝜈 , 𝑇 𝛿 𝐸 − ℏ𝜔(𝑄, 𝜈) .

One phonon scattering cross section

𝐼 𝑄, 𝐸 =𝑁ℏ

2

1

𝜔 𝑄, 𝜈𝜈

𝑏𝑗

𝑚𝑗1 2

𝑗

𝑄 ∙ 𝑒𝑗 𝑘, 𝜈 𝑒𝑖𝑄∙𝑅𝑗𝑇𝑗 𝑄

2

× 𝑛 𝜔 𝑄, 𝜈 , 𝑇 + 1 𝛿 𝐸 + ℏ𝜔(𝑄, 𝜈)

+ 𝑛 𝜔 𝑄, 𝜈 , 𝑇 𝛿 𝐸 − ℏ𝜔(𝑄, 𝜈) .

We know this bit, its just the normal elastic scattering equation.

One phonon scattering cross section

𝐼 𝑄, 𝐸 =𝑁ℏ

2

1

𝜔 𝑄, 𝜈𝜈

𝑏𝑗

𝑚𝑗1 2

𝑗

𝑄 ∙ 𝑒𝑗 𝑘, 𝜈 𝑒𝑖𝑄∙𝑅𝑗𝑇𝑗 𝑄

2

× 𝑛 𝜔 𝑄, 𝜈 , 𝑇 + 1 𝛿 𝐸 + ℏ𝜔(𝑄, 𝜈)

+ 𝑛 𝜔 𝑄, 𝜈 , 𝑇 𝛿 𝐸 − ℏ𝜔(𝑄, 𝜈) .

This looks horrible, but its just telling us that we have more phonons at

high temperatures (they follow Bose-Einstein statistics).

One phonon scattering cross section

𝐼 𝑄, 𝐸 =𝑁ℏ

2

1

𝜔 𝑄, 𝜈𝜈

𝑏𝑗

𝑚𝑗1 2

𝑗

𝑄 ∙ 𝑒𝑗 𝑘, 𝜈 𝑒𝑖𝑄∙𝑅𝑗𝑇𝑗 𝑄

2

× 𝑛 𝜔 𝑄, 𝜈 , 𝑇 + 1 𝛿 𝐸 + ℏ𝜔(𝑄, 𝜈)

+ 𝑛 𝜔 𝑄, 𝜈 , 𝑇 𝛿 𝐸 − ℏ𝜔(𝑄, 𝜈) .

Phonons at higher energy are weaker.

One phonon scattering cross section

𝐼 𝑄, 𝐸 =𝑁ℏ

2

1

𝜔 𝑄, 𝜈𝜈

𝑏𝑗

𝑚𝑗1 2

𝑗

𝑄 ∙ 𝑒𝑗 𝑘, 𝜈 𝑒𝑖𝑄∙𝑅𝑗𝑇𝑗 𝑄

2

× 𝑛 𝜔 𝑄, 𝜈 , 𝑇 + 1 𝛿 𝐸 + ℏ𝜔(𝑄, 𝜈)

+ 𝑛 𝜔 𝑄, 𝜈 , 𝑇 𝛿 𝐸 − ℏ𝜔(𝑄, 𝜈) .

This is the only “complicated” bit. 𝑒 is the phonon eigenvector,

physically, you can think of this as the direction atoms are

moving in.

Things to remember

• Phonons are strong when 𝑄 is parallel to direction of atomic motion.

• Phonon intensity goes up with 𝑄2.

• Phonons are weaker at high energy.

• Strong Bragg reflections often have strong phonons around them.

• Phonons can be stronger at high temperature (but care needed here).

Measuring a longitudinal acoustic phonon

We want to measure a longitudinal phonon along 00L.

Which peak would be best to measure around?

Measuring a longitudinal acoustic phonon

We want to measure a longitudinal phonon along 00L.

Which peak would be best to measure around?

𝑄

Measuring a transverse acoustic phonon

What about a transverse phonon?

Measuring a transverse acoustic phonon

What about a transverse phonon? 𝑄

Exploiting 𝑸 ∙ 𝒆, an example

• We were looking for a

phonon around 12

meV.

• However, the

background from the

cryostat/mount was

huge.

• Rotate sample 90°.

Suppresses, phonon

but background

unchanged

Exploiting 𝑸 ∙ 𝒆, an example

• We were looking for a

phonon around 12

meV.

• However, the

background from the

cryostat/mount was

huge.

• Rotate sample 90°.

Suppresses phonon

but background

unchanged

TOF, what do we get?

• With TOF we get an excellent survey.

• However, we are normally struggling for statistics.

• Data here was at 300 K counted for two days.

• Need at least 1 cm3 single crystal.

Comparison to DFT

A quick example of this with IXS and INS

• We were interested in Na0.8CoO2 as a thermoelectric.

• We wanted to show that the Na vacancy order has a big impact on the phonons.

• We also wanted to relate this to the thermal conductivity.

A quick example of this with IXS and INS

• We were interested in Na0.8CoO2 as a thermoelectric.

• We wanted to show that the Na vacancy order has a big impact on the phonons.

• We also wanted to relate this to the thermal conductivity.

Voneshen, D.J. et al. Nature Mater. 12, 1028 (2013).

A quick example of this with IXS and INS

• We were interested in Na0.8CoO2 as a thermoelectric.

• We wanted to show that the Na vacancy order has a big impact on the phonons.

• We also wanted to relate this to the thermal conductivity.

Voneshen, D.J. et al. Nature Mater. 12, 1028 (2013).

A quick example of this with IXS and INS

• We were interested in Na0.8CoO2 as a thermoelectric.

• We wanted to show that the Na vacancy order has a big impact on the phonons.

• We also wanted to relate this to the thermal conductivity.

Voneshen, D.J. et al. Nature Mater. 12, 1028 (2013).

A quick example of this with IXS and INS

• We were interested in Na0.8CoO2 as a thermoelectric.

• We wanted to show that the Na vacancy order has a big impact on the phonons.

• We also wanted to relate this to the thermal conductivity.

Voneshen, D.J. et al. Nature Mater. 12, 1028 (2013).

Powders!

• Good news, powders are simpler!

• Much simpler.

• With them we can extract the neutron weighted phonon density of states.

• But, going beyond that is hard.

Powders, what are we doing? • We are averaging over a

sphere at some 𝑄 .

• So, for small values of 𝑄

we are covering just a few

(or even 1) Brillouin zones.

• But, for large 𝑄 we are

covering many zones,

essentially capturing

everything in 1 shot.

• This means for high 𝑄 we

can no longer see the effect

of 𝑸 ∙ 𝒆 and the signal is the

same as incoherent

scattering.

Powders, what are we doing? • We are averaging over a

sphere at some 𝑄 .

• So, for small values of 𝑄

we are covering just a few

(or even 1) Brillouin zones.

• But, for large 𝑄 we are

covering many zones,

essentially capturing

everything in 1 shot.

• This means for high 𝑄 we

can no longer see the effect

of 𝑸 ∙ 𝒆 and the signal is the

same as incoherent

scattering.

The neutron weighted phonon density of states

• We normally correct the data for the effects of Bose statistics, 1 𝜔 and 𝑄2.

• Then, in the incoherent approximation, the real Phonon Density of States (PDOS) is related to the PDOS we see via the above.

• This means we cannot obtain the true PDOS for anything other than a monoatomic system.

𝑃𝐷𝑂𝑆𝑛𝑒𝑢𝑡 𝐸 = 𝐴 𝑏𝑗2

𝑚𝑗𝑗

𝑃𝐷𝑂𝑆𝑗 𝐸 ,

A quick powder example

• Another thermoelectric.

• This time, the idea was that superionic diffusion would dramatically change the phonons, leading to a very low thermal conductivity.

• It doesn’t. Instead the low thermal conductivity is caused by regular anharmonic effects.

A quick powder example

• Another thermoelectric.

• This time, the idea was that superionic diffusion would dramatically change the phonons, leading to a very low thermal conductivity.

• It doesn’t. Instead the low thermal conductivity is caused by regular anharmonic effects.

A quick powder example

• Another thermoelectric.

• This time, the idea was that superionic diffusion would dramatically change the phonons, leading to a very low thermal conductivity.

• It doesn’t. Instead the low thermal conductivity is caused by regular anharmonic effects.

A quick powder example

• Another thermoelectric.

• This time, the idea was that superionic diffusion would dramatically change the phonons, leading to a very low thermal conductivity.

• It doesn’t. Instead the low thermal conductivity is caused by regular anharmonic effects.

Phys. Rev. Lett. 118, 145901

The thing I was scared to talk about

• Measuring phonon dispersions from time resolved diffuse scattering.

• Still very early, but really cool.

Nature Communications, 7, 12291

(2016)

The thing I was scared to talk about

• Measuring phonon dispersions from time resolved diffuse scattering.

• Still very early, but really cool.

Nature Communications, 7, 12291

(2016)

Summary

• Momentum resolved spectroscopy is great!

• We have options, TOF for surveys, TAS for focussed studies and x-rays for tiny samples.

• Remember that the intensity of phonons depends on

𝑄 ∙ 𝑒 2

.

• If you have questions, email me! David.voneshen@stfc.ac.uk