Novel Materials and Ground States
Fermi Surfaces of Metals –magnetoresistance
and quantum oscillations
590B S09
Sergey L. Bud’ko(Сергей
Леокадьевич
Будько)
[part of the slides (blue) are taken from the presentation of Ilya
Sheikin
(Grenoble) –
http://mfs-cargese.grenoble.cnrs.fr/Sheikin.pdf]
Novel Materials and Ground States
Experimental studies of FS: -
why?
(from Bruce Harmon’s lecture, Feb. 23)
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
Novel Materials and Ground States
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*c
τ
– relaxation time
“usually”
magnetic scattering effects are stronger than FS effects
does not tell a w
hole
lot about th
e orbit
Novel Materials and Ground States
Magnetoresistance
and Fermi surface topology
Weak magnetic field (ωτ
<< 1)
Δρ/ρ
~ H2
δ
= 2rφ
– 2rsinφ ≈ 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…
Novel Materials and Ground States
Magnetoresistance
and Fermi surface topology
Strong magnetic field (ωτ
>> 1)
closed trajectories
open trajectories
Novel Materials and Ground States
Magnetoresistance
and Fermi surface topology
Strong magnetic field (ωτ
>> 1)
Novel Materials and Ground States
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
Novel Materials and Ground States
Applications: Cd
under pressure, mid 1960s
P=0 P~20 kbar
H||c: closed orbits, Δρ/ρ
~ H2 H||c: open orbits, Δρ/ρ
~ const
Gaidukov, Voronovskii, Itskevich, … ~1965
Novel Materials and Ground States
Novel Materials and Ground States
Novel Materials and Ground States
measure of the FS curvature
Novel Materials and Ground States
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)
Novel Materials and Ground States
Novel Materials and Ground States
be careful with “large”
field intervals and FFTs
Novel Materials and Ground States
for a mundane material shouldn’t be very different for different FS sheets
Novel Materials and Ground States
need to know the effective mass
Novel Materials and Ground States
Novel Materials and Ground States
Quantum oscillations
HOW TO RECOGNIZE THEM:
Periodic in 1/H
Amplitude increases in higher fields
Amplitude decreases at higher temperatures
Novel Materials and Ground States
Quantum oscillationsNEED: low temperatures, high magnetic fields, high quality single crystals, decent sensitivity of the measurement techniques
GET:precise areas of extremal
orbits,effective masses,scattering time
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
Novel Materials and Ground States
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 heat
Elastic 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 more using more than one technique.
Novel Materials and Ground States
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.
Novel Materials and Ground States
Exam
ples
Novel Materials and Ground States
Exam
ples
Novel Materials and Ground States
Novel Materials and Ground States
Osc
illat
ion
s th
rou
gh A
FM t
ran
siti
on
Novel Materials and Ground States
WHERE IS IT? WHAT IS IT?
Novel Materials and Ground States
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
Novel Materials and Ground States
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.
Novel Materials and Ground States
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)
Novel Materials and Ground States
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