fermi surfaces of metals...novel materials and ground states fermi surfaces of metals –...

Novel Materials and Ground States

Fermi Surfaces of Metals –magnetoresistance and quantum oscillations

PHY 590B S19

Sergey L. Bud’ko

[part of the slides (blue) are taken from the presentation of Ilya Sheikin (Grenoble) – http://mfs-cargese.grenoble.cnrs.fr/Sheikin.pdf]

http://mfs-cargese.grenoble.cnrs.fr/Sheikin.pdf


Fermi surface - reminder


Magnetoresistance and Fermi surface topology

Requirements:- large magnetic fields (ωτ >> 1)-“normal” metal (no magnetic scattering or field-induced transitions)

ω – cyclotron frequency, ω = eH/m*cτ – relaxation time

“usually” magnetic scattering effects are stronger than FS effects

does not tell a whole

lot about the orbit



Weak magnetic field (ωτ << 1)

Δρ/ρ ~ H2

δ = 2rφ – 2rsin φ ≈ 1/3 rφ3

If we assume rφ ~ λ (mean free path),δ/ λ ~ (λ/r)2 => Δρ/ρ ~ (λ/r)2

r ~ v/ω; λ ~ vτ => Δρ/ρ ~ (ωτ)2 ~ H2

2φr

No help here…



Strong magnetic field (ωτ >> 1)

closed trajectories

open trajectories



Strong magnetic field (ωτ >> 1)


Magnetoresistance and Fermi surface topology Strong magnetic field (ωτ >> 1)

More formally for the tensor:

CLOSED ORBITS

OPEN ORBITS

(γ ~ 1/H)

ne ≠ nh ne = nh

Au


real and k-space

see e.g. Jeno Solyom, Fundamentals of Physics of Solids, v.II


Applications: Cd under pressure, mid 1960s

P=0H||c: closed orbits, Δρ/ρ ~ H2

all closed

open along c, closed in plane


Applications: Cd under pressure, mid 1960s

P=0 P~20 kbarH||c: closed orbits, Δρ/ρ ~ H2 H||c: open orbits, Δρ/ρ ~ const

Gaidukov, Voronovskii, I tskevich, … ~1965


Experimental studies of FS: - why?

(from Bruce Harmon’s lecture, PHY 509B S09)

Optimist would say:

And we want to keep theorists honest

Quantum oscillations are fascinatingly simple and (to a large extent) do not require a lot of assumptions to understand


Lev Shubnikov (1901-1937)

Shubnikov – de Haas effect

(quantum oscillations in resistivity, 1930, bismuth)

•Schubnikov, L.W.; de Haas, W.J. (1930). Proceedings of the Royal Netherlands Academy of Arts and Science 33: 130.•Schubnikov, L.W.; de Haas, W.J. (1930). Proceedings of the Royal Netherlands Academy of Arts and Science 33: 163.


cartoons: why one can see oscillations


measure of the FS curvature


per Lifshitz & Pogorelov (1954) if the FS is convex and has a center of symmetry there is an analytical procedure to calculate FS shape from oscillations’ frequencies and the velocities from the effective masses. (NEED ROTATION)


be careful with “large” field intervals and FFTs

with modern software should be able to fit


for a mundane material shouldn’t be very different for different FS sheets


need to know the effective mass


Quantum oscillations

HOW TO RECOGNIZE THEM:

Periodic in 1/H

Amplitude increases in higher fields

Amplitude decreases at higher temperatures


Quantum oscillationsNEED: low temperatures (in particular for large effective masses), high magnetic fields (in particular for large FS cross sections or not so clean samples), high quality single crystals, decent sensitivity and dynamic range of the measurement techniques

GET:precise areas of extremal orbits,effective masses,scattering time(s)

PARAMETERS (note: much more freedom than ARPES/2DACAR):temperature (e.g. oscillations through TN, TK),magnetic field (e.g. metamagnetic transitions, field dependent masses),angle,pressure/stress


Quantum oscillations

Cd3As2Cd3As2


Quantum oscillations in “everything”Magnetization/susceptibility (dHvA) (|dM/dH| = 2πF/H M/H)

Torque (Tq = -1/F dF/dθ MHV)Resistivity (Δρ/ρ ~ (m*F/H)2 dM/dH)Magnetostriction (Δli/l ~ dlnF/dpi MH)Magnetothermal oscillations (|ΔT| ~ T2H2/F2 |dM/dH|)Specific heatElastic constants and sound velocity…(oscillating amplitudes are written)

NOTE: different relative amplitudes of the oscillations in different techniques (can see in one but not in another…)Can learn something using more than one technique.


Quantum oscillations in “everything”

NOTE: different relative amplitudes of the oscillations in different techniques (can see in one but not in another…)Can learn something using more than one technique.

PRB 60, 13371 (1999)


How to measure quantum oscillations?

torque, magnetization, resistivity (including TDR):

YBa2Cu3O6.51 crystal glued to the cantilever beam

[A. Audouard, C. Jaudet, D. Vignolles, R. Liang, A. Bonn, W. N. Hardy, L. Taillefer and C. Proust, Phys. Rev. Lett. 103 157003 (2009)]

High field Labs: -US, France, Germany, Netherlands, Japan, Poland(?)

up to ~100 T, pulse fieldor static fields up to ~ 45 T

(very) small samples, low temperatures, high fieldscan be mail-in experiment



in-house:

limited H and T

M(H)ρ(H), τ(H), ∆L(H), χ(H)



in-house:

τ(H)



Wheatstone bridge:

Balance and then measure small differences.(Partially) takes care of the temperature/magnetic field effects on the chip/cantilever.



So, you can do it “brute force”: - use your favorite expensive measurements system (like PPMS

or pulse field facility at Magnet Lab);- record as much data as possible;- do FFT; - live with the consequences

background – in post-processing [for dHvA – issues with magnetic materials, for SdH – with XMR materials]

GOOD: “limited” mental activity is required during the measurements, often getting absolute values of amplitude

Your freedom: sample quality and size, base temperature, maximum field, angle to the applied field



modulation technique

Proc. Roy. Soc. A281, 62 (1964)



modulation techniqueac voltage you measure

Bessel function

modulation field

phase (tune with lock-in)



modulation technique

Offers:- high signal to noise (use harmonics to eliminate background,

use of filters, phase-sensitive detectors – lock-ins);- possibility to (partially) suppress the dHvA frequencies using

combination of modulation amplitude, frequency, phase, and order of harmonics)

Requires tuning of measurements parameters

Hard to get absolute values


Magnetic breakdown

MgTunneling probability:H ~ U2/EF

U ~ 10-2 eV, EF ~ 1 eV => H ~ 104 Oe

Need to keep in mind if there are differences with band structure.


Exam

ples


Magnetic field dependent effective mass – systems with field induced quantum critical point


Exam

ples


Osc

illat

ions

thro

ugh

AFM

tran

sitio

n


Appendix: 2 ½ order phase transition

First-order phase transitions–exhibit a discontinuity in the first derivative of the free energy with a thermodynamic variable.

Second-order phase transitions–continuous in the first derivative–exhibit discontinuity in a second derivative of the free energy.

Paul Ehrenfest:


2 ½ order phase transitiona.k.a.: electronic topological transition (ETT), Lifshitz transition

also:


Lifshitz transition

electronic DOS

parameter:

thermodynamic potential

T=0, no scattering


low temperature, no scatteringALSO other control parameters

Li-Mg alloy

Lifshitz transition

resistivity

TEP


Lifshitz transition

δ

ε

H||[1120]

H||[0001]

Elemental Cd under pressure, dHvA

two Lifshitz transitions

~17 kbar


Lifshitz transition

Elemental Cd under pressure, TEP in magnetic field

H||[0001]


Temperature – induced Lifshitz transition

[1010]

[0001]

Elemental titanium


Temperature – induced Lifshitz transition


LSCO high-Tc SC

QCP in heavy fermions

New box – the same good old taste

Lifshitz transition


Lifshitz transition


Generalized Lifshitz transition

Qualitative change of FS without changes in topology

Would be detected in quantum oscillations and electronic transport


What Lifshitz transition is not

Change of the FS topology as a result of structural transitionChange of the FS topology as a result of magnetic (AFM) transition

Word of caution:Lifshitz transition is sometimes invoked to explain experimentally observed “anomalies” when other possibilities (structural, magnetic transitions, CEF effects, etc.) are ambiguous. Use your best judgment.


Fermiology – grain of salt

In 60’s and 70’s scientists had a “Fermiology – induced” euphoria

Romanticism changed to realism (for some of us)

Sculpture – Tony Smith


Fermiology – grain of salt

M.I. Kaganov and Yu. V. Gribkova, Fiz. Nizkih Temp. 17, 907 (1991) [English – Low Temp. Phys.] + personal thoughts/experience

By combination of experiment (quantum oscillations, ARPES, …) and band structure calculations we know Fermi surfaces of many metals in intimate details.However the community is not able to use this knowledge to calculate precise physical properties. The knowledge is left unclaimed / wasted.It seems that after scientists understood the difference between different metals they returned to the concept of “general metal”.


Fermiology in the Physics Department

David Lynch remembers

I arrived in Ames in the Fall of 1960. The Fermi-surface people arrived starting the next year: Allan Mackintosh 1960, Andrew Gold 1961?, Bob Young 1963?, all from Cambridge. Then Terry Loucks, a theorist, and some postdocs and visitors: Bob Chambers (Cambridge or Bristol), John Collins (Australia, ex-Cambridge), George Dheer (Cambridge), and Sunny Sinha (Cambridge). The latter came as Allan?s postdoc, but stayed to run much of the CMP research on the new Ames-Lab reactor.

Allan set up a spectrometer for correlation of gammas from positron annihilation and he also measured transport property, especially of Cr doped with V and with Mn. Andrew built a pulsed dHvA system. One of the things done on it is still cited a lot: the Fs of Pb, which did not fit a free-electron model at all until spin-orbit splittings were added. Another was the FS of Fe and Co. One result from that was that the force on electrons in these metals was v x B, not v x H. Bob Young measured magnetoresistance, looking for open orbits.

Ames and Cambridge were the centers of Fermi surface research in the 1960s.

I think the sculpture seen from the department office was made by Dick Brown, the foreman of the department machine shop, but I don’ t know who designed it or got him to do it. It looks like a Fermi surface in the extended zone scheme, but I think it was not intended to be that of any metal whose Fs was known at that time. Maybe something has been determined since then for a real metal.


The sculpture made of intersecting rings of rusted Fe, was machined in the Physics Shop by Dick Brown based on a calculation by Terry Loucks for the iron actinides and it was called "Iron-Actin"

Gordon Danielson, our Semiconductor guy in the early 1960s spent a sabbatical at Cambridge UK and hired three students from the Shoenberg Fermi Surface group to start a new group in Ames. Andrew Gold was first and he worked mostly on the Fermi Surface of Pb in the early going doing de Haas van Alphen with pulsed fields and a giant capacitor bank in Tringides current lab. Allan Mackintosh came a year later and he was an idea guy more than a nuts and bolts guy. Allan collaborated with experimentalists all over the world doing phenomenology to sort out their data. Eventually Allan started a positron annihilation program for Fermi surface work doing a lot with the rare earths. Bob Young was the third Cambridge guy, I have forgotten his technique. Terry Loucks from Penn State was a theorist calculating everything in sight and he wrote a book. John Stanford started doing radio frequency size effect, the John went into atmospheric physics looking for Tornados. Sunny Sinha was the last of the Cambridge group. Sunny did mostly neutron scattering, but he did Fermi surface work as well. My guess is that Gold came in 1958 or 1959, Mackintosh in 1960, and Young in 1961. All three were here when I came in 1962. Loucks came in about 1964 and Stanford from Maryland in 1965. Between 1975 and 1980, Gold went to British Columbia, Mackintosh to Riso, Denmark, Young to Birmingham, Loucks to North American in California, and Stanford to tornados.

Fermiology in the Physics Department

Doug Finnemore remembers


Fermiology in the Physics Department – these days (in addition to the world class ARPES): 1999 - now

dHvA-SdH: resistivity, magnetization, torque, TDR, magnetostriction, magneto-TEP(more curiosity driven)


WHERE IS IT? WHAT IS IT?


Reading materials

Electron Theory of Metals Authors: I.M. Lifshits, M.Ya. Azbel’, M.I. Kaganov; Consultants Bureau, 1973

Fundamentals of the theory of metals Author: A.A. Abrikosov; North-Holland, 1988

Electrons at the Fermi surfaceEditor: M. Springford; Cambridge University Press, 1980

Magnetic oscillations in metals Author: D. Shoenberg; Cambridge University Press, 1984

Band theory and electronic properties of solidsAuthor: John Singleton; Oxford University Press, 2001

fermi surfaces of metals...novel materials and ground states fermi surfaces of metals –...

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