THERMOELECTRICS ( and some byproducts of its research ) Predrag Lazic, Ceder group talk.

THERMOELECTRICS(and some byproducts of its research)

Predrag Lazic, Ceder group talk

THERMOELECTRICS(and some byproducts of its research)

Predrag Lazic, Ceder group (farewell) talk

More general things in this talk

Picture says a thousand words



Well – then animations says millions



Well – then animations says millions

But – some things are just beyond any description – you just have to try them!

(thermoelectrics)

Mathematical tools are fine

But there is no substitute for intuition and imagination!

Thermoelectrics – what is that?

Basic idea :

Voltage heat gradientElectrical current

Phonons

Making a device (the one that you hold in your hands right now)

Because of the TOPOLOGY we need p- and n-type.

2 (3) crucial quantities:

Seebeck coefficient - S

Conductivity - σ

Thermal conductivity - κ

Power factor:

Intuition?!

2S

zT

2 (3) crucial quantities:

Seebeck coefficient - S

Conductivity - σ

Thermal conductivity - κ

Power factor:

Intuition?!

Huh, where do we start?

2S

zT

In principle we could just use DFT + BoltzTrap and start a search

Instead we have spent 1 year to get understanding and develop intuition

One typical thphys result

-Complicated to derive- looks cool and smart- most probably correct- practically it is almost useless

This function must be a Dirac delta function to maximize the figure of merit. Of course, this exact situation is not found in nature.However, our results indicate that we have to search for materials where the distribution of energy carriers is as narrow as possible, but with high carrier velocity in the direction of the applied electric field.

No insight from material perspective!

Let’s do one numerical example with the available code

We are trying to get insight.

k

iv i

kk ,1),(

u

iu kkiM

kk ,2

21 1

),(

),(),(),( ,2

, kkk k ivivei i

dTf

T

);()(

1);(

dTf

eTT

);())((

1);(

dTf

TeT

);())((

1);( 2

20

jiijS )( 1

It is all in the bandstructure!

Group velocity

Effective mass tensor

Conductivity (approximation of constant τ)

FD

Basically it boils down to:

2SzT */1 m

*mS This is a bit tricky

)1

(

zT

We chose a fairly good TE materialAnd the one that appears both in n and p type!So PbTe it is!

Notice – Pb, Te, Si, Ge, Sb

Graph looks fairly simple but to get insight we need much more understanding

PbTe from Boltztrap

300 K 800 K 1300 K

1. Temperature role in the integrals

Doping is crucial n-type p-type

Doping is crucial p-typeImportance of the gap!

Pure Pb – would be nice n-type Pure Te – would be nice p-type

Notice incredible similarity of Pb and Te bandstructure (just different Ef)!Why is this s-(p,d) hybridized band so important?

It is even more complicated that this – we need 3D to understand Snyder’s results

Snyder’s result Theory

You can already see hint of s-(p,d) hybridization importance(light and heavy band, it means effective mass)

We thought it is pure lattice constant thing – but it isn’t

We consider only lattice effect (PbTe expands with T)

300 K Lattice ~ 300 K

Lattice (volume) effect alone is rather small in band alignment

Crucial effect is just T (because of integration)

300 K

550 K

If we do the right thing (Wannier90)

1) How do we get such strange bandstructure? Is this even tractable?

Well it is almost trivial, and shows how simple DFT actually is (shocking! - it is disappointing and comforting at the same time).

1) How do we get such strange bandstructure? Is this even tracable?

cigars calzones

Pure Pb fermi surface

Well it is almost trivial, and shows how simple DFT actually is (shocking! - it is disappointing and comforting at the same time).

So this is not in the beginning of the textbook because it is the most simple thing, but because it is the most important!

Stolen from J.M. Ziman, Principles of the Theory of Solids

So this is not in the beginning of the textbook because it is the most simple thing, but because it is the most important!

Basically this is the DFT!

Stolen from J.M. Ziman, Principles of the Theory of Solids

Ok, but much more interesting question about that s-(p,d) hybridizaiton

We see this trick in every single compound that is at least decent TE.

Check for yourself!http://ceder.mit.edu/GROUPONLY/rar/THERMO/

What is so important in s-(p,d) hybridization for TE?

Stolen (again) from J.M. Ziman, Principles of the Theory of Solids

What is so important in s-(p,d) hybridization for TE?

Stolen (again) from J.M. Ziman, Principles of the Theory of Solids

Gap!

Well, gap is one thing, but still how does one understand the problem:

*/1 m*mS

Notice that s band represents – something fast, small m*, something itinerant

While p & d represent something – localized, heavy

How do we understand something that is heavy (slow) and light (fast) at the same time?!

First of all – effective mass tensor is a horrible name – it is just killing intuition

Try to imagine that your mass is different if you move left-right that when you moveBack-forth.

And we want large S2σ

Doesn’t work. But imagine this!

Trough!

With a ball in it.


Trough!

With a ball in it.

heavy

light


Trough!

With a ball in it.

d,p-character of potential (localized)

s-character (itinerant)

heavy

light

x

y

E

Now we are getting somewhere, we need fast transfer of heat (entropy)

What is heat?!

Now we can understand intuitively importance of s-d(p) bands and large zT.(plus we will get electron-phonon coupling for free, end of the movie)

Word of caution – keep in mind – spin orbit coupling for TE systems.

WITH SOC

NO SOC

And now – for one strange observation…..

Notice – Pb, Te, Si, Ge, Sb

Pb – 7.2 KTe – 7.5 KSi – 8.2 KGe – 5.35 KSb – 3.9 KSn – 3.7 KBi – 8.5 K

Largest Nb 9.5 K

Bad metals such as Pb, like to go SC!While Cu for example never does.

What is superconductivity?

Google The property of zero electrical resistance in some substances at very low absolute temperatures.

WikiSuperconductivity is a phenomenon of exactly zero electrical resistance

This is not very useful as we will see later. Those are consequences not causes.

Intuitive picture

Atomic orbitals – finite energy required for “interaction” (perturbation, scattering)

Now imagine such orbital in macroscopic dimension – that is SC in physics.

There is a finite energy gap for exciting (scattering) a particle carrying a current.

(this is not a perfect conductivity! – it is a deeply quantum effect)

Introduction to SC (briefest possible)

Discovered 1911 in Mercury (Hg) by

Heike Kamerlingh Onnes on April 8, 1911 in Leiden.

The BCS theory (conventional superconductivity)1957 – first microscopic theory of SC. (low Tc <10K)

Explains how a “glue” binds two electrons into a Cooper pair, creating a SC gap.

In BCS the ”glue” is electron-phonon coupling - difficult to explain intuitively.

Important is – one pair is not enough – all the pairs have to play the game perfectly together! – strong correlation, that is the SC state.

So let’s take purely electronic coupling as example of different “glue”

http://en.wikipedia.org/wiki/Heike_Kamerlingh_Onnes




http://en.wikipedia.org/wiki/Leiden

Fermi gas can not remain normal at low temperatures – at least this coupling has to occur!

Fermi gas

Screening – Friedel oscillations (because of finite kf) – overscreening –effectively attractive potential!

Purely electronic pairing

When Frӧlich suggested e-ph coupling Landau said – no way! You can not beat Coulomb force!

So let’s take purely electronic coupling as example of different “glue”

THE HTSC mechanism!

(High Temperatrure SuperConductivity)

HTSC MECHANISMSince 1986

For horrible crimes against (theoretical) physics community

Mostly – model Hamiltonians.

DFT is bad! It can not give you SC.

Common knowledge – state of the art (HTSC)

But we are talking about Tc~150 K

1meV ~ 11 KSo we are talking about differences of ~20 meV per unit cell!

After 2 years of Materials Science – I decide to use DFT for HTSC – electronic structureCan not be light years away from the real one (even in SC state!) – otherwise MS makes no sense at all, and physics as well.

Ole Eriksson - has a very good idea on this!

The method may best be described as data-filtering. The basic philosophy is to identify a known class of materials which has been well characterized with respect to a certain property (e.g. superconductivity). If these materials have conspicuous and unique similarities in the underlying electronic structure (the ’code’), one may make a comparison of the electronic structure of other materials, which may not have been subjected to a detailed experimental investigation of the relevant materials property.

The basic hypothesis of the present study is that whatever the mechanism or combination of mechanisms that cause the pairing of charge carriers, there is a crucial aspect in that this takes place in a unique electronic structure and crystal geometry.

Namely, that of a quasi two-dimensional/layered crystal structure, in which the d-shell of a transition metal atom hybridizes strongly with p-orbitals of ligand atoms. In the cuprates this is manifested in a band of primary dx2−y2 character that hybridizes with oxygen p-orbitals. We also suggest that it is important that in the normal state, only one single hybridized band cuts through the Fermi level (EF ),for each CuO2-plane. ….

Current state of the art how to make a good HTSC – Matthias Rules

1. Must have d electrons (not just s-p, nor f)

2. High symmetry is good, cubis is the best

3. Certain electron concentrations are favored

4. Peak in the density of states at Fermi level

So is this a coincidence? What is behind it?

Let’s take one of the best HTSCs – YBCO! (Tc~130K)

YBa2Cu3O7YBa2Cu3O8

Insulator HTSC

Anti-ferro by supper exachange

The problem Is that we have spins and in AF interactions

There is a frustration in the system!

There is something very special about thisFrustration – its topology!

It can be resolved by running two currents!Such a scenario would, on average lower the Energy of the system.

Again this is better seen in animation.

Everything is there – even the strong correlation (which DFT can not give )

There are “2” currents shifted in phase that on average satisfy all the bonds but they avoid frustration. Which lowers the energy of the system – tradeoff with entropy in spin-flips.

SC current is entropyless! (it is a SINGLE state)

Which J determines Tc?

You can try other Fe/Cu HTSC – always this is to be found.Here frustration is not so nice – it is competition of a happy and frustrated bondSo avoiding both system saves only the difference in the energy –> low Tc.

If there is doping dependence – typically it is TOPOLOGICAL one½, 1/4, 1/3, 1/8 etc.

Now we can give some “rules” how to make HTSC

-Introduce a spin frustration which can be resolved dynamically ( spin is the best choice because system has least mechanisms to avoid it)

-Quasi 2d material is good because spin frustration can not be resolved through relaxation in 3rd dimension (layers are decoupled)

-anti-ferro scaffold is the easiest way to construct such frustration

-coupling of spin and current is natural

-AF scaffold in which spin frustration is resolved dynamically requires BOTH local and Itinerant spins (magnetism) – this is what makes similar demands on bandstructure as in TE!

Why was this not found since 1986?

Logic – following BCS:

1) Find the glue (excitation that provides coupling, currently AF spin ex. is the usual suspect)

2) Once the glue is found – ground state should be obvious

Coming from the DFT – total energy perspective led to that – find the ground state!And you are done. The only difference is that now my GS contains current. And that is the whole point! If I want someone to reconstruct this quickly – I would just say:

SUPERCONDUCTIVITY IS A GROUND STATE THAT CONTAINS CURRENT!

(reason for current then is of course – lowering the energy of the system)

Also currently there are no tools to calculate this – except one – imagination.

Also the reason the “glue” is not found yet


Is – there is no glue!

The whole trick is to avoid repulsive interactions rather than to try to make glue out of them.




And P. W. Anderson actually suspected this in 2007! (Science)





I argue here that this need for a bosonic glue is folklore rather than the result of scientific logic . It comes from the inappropriate Assumption that superconductivity in these materials is described by a mathematical framework called the Eliashberg formalism (10), which is an extension of the original ideas of Bardeen, Cooper, and Schrieffer.





I argue here that this need for a bosonic glue is folklore rather than the result of scientific logic . It comes from the inappropriate Assumption that superconductivity in these materials is described by a mathematical framework called the Eliashberg formalism (10), which is an extension of the original ideas of Bardeen, Cooper, and Schrieffer.

Once again – we are talking about P. W. Anderson here!

THE ANALOGY

Schrieffer’s analogy of BSC with dancing couples is legendary.

http://www.aip.org/history/mod/superconductivity/03.html

The wave function is just... symbols which record the dance the electrons are making.

http://www.aip.org/history/mod/superconductivity/03.html

HTSC - a pebble in a shoe

HTSC - a pebble in a shoe

Also you can see BCS as frustration avoiding if you want to.

And finally the question I wanted to answer, connection of TE and HTSC

TE(mixing of local and itinerant “mass” via s-d hybridizaition – as seen in the Trough movie )

BCS(s-d hybridization leads to strong el-ph coupling via localized d character of conductive electrons, therefore those materials are good for making TE, but not alone – because they lack the gap.

HTSC(mixing of itinerant and local magnetism is required for that scenario of frustration avoiding to take place. Obviously this is almost impossible to do without getting at the same time mixing of the local and itinerantmass – i.e. from there comes the similarity in bandstructure with TEs)

k-sp

ace

phen

omen

a

real and spin space phenomenon

anisotropic

Heads up warning – finding room temperature SC

THANK YOU!

The GROUP

Beginner's Mind

"In the beginner's mind there are many possibilities, but in the expert's there are few."

- Shunryu Suzuki-Roshi

Beginner's Mind

"In the beginner's mind there are many possibilities, but in the expert's there are few."

- Shunryu Suzuki-Roshi

Personal history of my engagement with cuprate superconductivity, 1986-2010Philip W Anderson

http://arxiv.org/abs/1011.2736v1

…., I do not find many colleagues and competitors who have given up on their first insight, gone back to the beginning and started over. This gives me an enormous advantage, and I’m not above using it.

http://arxiv.org/find/cond-mat/1/au:+Anderson_P/0/1/0/all/0/1

http://arxiv.org/abs/1011.2736v1

THE SCIENTIST?

DFTSom

ething

interesting

THERMOELECTRICS ( and some byproducts of its research ) Predrag Lazic, Ceder group talk.

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