Transport de chaleurvia les spins
Trans.Spins-1
Reference: Stephen R. Boona, Roberto C. Myers and Joseph P. Heremans, Spin Caloritronics, Energy Environ. Sci. 7 885-910 (2014) DOI:10.1039/C3EE43299H
Joseph P. HeremansOhio State UniversityDepartment of Mechanical and Aerospace EngineeringDepartment of [email protected]
Types of magnetismDiamagnetism
Uncompensated spin/angular momentum
Free electrons
Permanent atomic moments
Independent atomic moments:Paramagnetism
Cooperatingatomic moments
Ferromagnetism FerrimagnetismAntiferromagnetism
Pauli spinParamagnetism
Landau orbitalDiamagnetism
Band anti / ferromagnetism
T
ME 8194 Ch 7 Magnetism
Elemental excitationsElectrons
with or withoutspin
Phonons Magnons
Trans.Spins-3
Transport
Charge
Heat Magn. Moment
L g
E
Magnon Drag
M
Spin Caloritronics
?
gg
Structure of this lecture1. Longitudinal spin Seebeck2. Spin waves in the Ferromagnet:
Magnon thermal conductivityThermally driven spin/moment flux
3. Spin polarized electrons and spin/orbit interactions: Spin Hall effectInverse spin Hall effect
4. At the interface: spin mixing conductance5. Transverse Spin Seebeck effect
Non‐local version of longitudinal spin Seebeck effectGaMnAs: the role of phonons
6. Phonon‐magnon‐electron drag‐induced spin Seebeck effect: GaMnAs7. The giant spin Seebeck effect (InSb): Pure phonon‐electron drag, no magnons8. Phonon diamagnetism9. Applications
Trans.Spins-4
Trans.Spins-5
1. Longitudinal spin Seebeck
Heat Magn. Moment
Spin Caloritronics
Measurements rely on inverse spin-Hall effect:
Spin-Seebeck effect definition
σJE S ISHEISHE D
Grossly exaggerated for effect
TSCoefSeebeckSpin
x
xxy
.
Trans.Spins-6
Longitudinal Spin-Seebeck Effect (LSSE)
Transverse Spin-Seebeck Effect (TSSE)
COLD
HOT
VV1 V2 V3 V4
T, jq
zy
x
Hz
sjsjs
jsEihse
Eihse
Ferromagnet:
Metal: Pu (Uchida & al, Nature, 2008)Semiconductor: GaMnAs(Jaworski & al, Nat. Mater, 2010)Insulator: YIG(Uchida & al, Nat. Mater, 2010)
Pt
Ferromagnet: Can only be an Insulator: YIG(Uchida & al, APL2010) Trans.Spins-7
Theoretical understanding of the SSE1. T drives the magnons out of thermal equilibrium: 1.1 magnon thermal conductivity1.2 phonon‐drag
2. Spin‐torque results from magnons striving back to equilibrium
3. Net spin flux diffuses into Pt
4. Spin‐orbit interactions in Pt convert JS => EISHE
Ferromagnet
ISHE mediumPlatinum
Js
T
• Sinova, Finkelstein, Nat. Commun. 2013• Tserkovnyak, PRB, 2014 • H. Adachi, J.-I. Ohe, S. Takahashi, and S.
Maekawa, PRB 83, 094410 (2011).Trans.Spins-8
Trans.Spins-9
2. In the Ferromagnet:• Magnon thermal conductivity• Thermally driven spin/moment flux
Magnons
Heat Magn. MomentM
Burkard Hillebrands 12th Joint MMM/Intermag Conference Chicago, January 14, 2013
Ferromagnetic spin chain: magnonsMagnetic moments localized on core electrons on atoms (f or d‐levels)
wavevectorwavelength
2k π
Slide by Burkard Hillebrands, U. Kaiserslautern Trans.Spins-10
Spin waves, simple Heisenberg ferromagnet
N
pppJU
112 SS
The ground state has all spins parallel.
Exchange integralpS = angular momentum of spin at site pl
20 2NJSU
Treat spins as classical vectors of magnitude S => The ground state exchange energy is
The excited state has one spin reversed.This increases the excited state energy to
201 8JSUU Too expensive
The excited state has one spin reversed.Much lower energy state is all atoms share a little bit of the spin reversal, and precess.
Trans.Spins-11
Magnon dispersion, ferromagnets222 kJSagHzBk
Tb-YIG,295K & 83K
Plant, J. Phys. C 16 7035 (1983)
(THz)
k100k110
vMAGNON 8500 ms-1
Yttrium Iron Garnet (YIG)Y3Fe2(FeO4)3, or Y3Fe5O12
TCURIE = 550 K, TDEBYE = 538 & 567 K
Trans.Spins-12
Trans.Spins-13
Types of magnons in reality
J. S. Plant, J. Phys. C: Solid State Phys.. 16 (1983) 7037-7051
1 THz = 49 K
Magnons excited by microwaves: 6 GHz• Energies much below thermal magnons• Very high population density by FMR
Magnons below 700 GHz ~ 30 K• Parabolic dispersion (Kittel‐model)• Thermal magnons at T < 40 K• Subthermal magnons at T> 40 K
“Optical‐like” Magnons at 3‐4 THz (150 K) • Dispersion becomes cmplicated
Thermal magnons at room temperature• Dispersion? You judge.
Yttrium Iron Garnet (YIG)Y3Fe2(FeO4)3, or Y3Fe5O12
Magnon density of states D(), ferromagnets
222 kJSagH zBk Magnons have only a single polarization for each value of k
Number of magnon states of frequency between and +d
/
/)(
zB
zB
gHJSa
gH
, 24
1 , 0
3/2
22
D
zBgH
D
Magnetic fields can freeze out magnons, 10‐4eV/T or 1.3 K / T
Trans.Spins-14
Magnon specific heat
TkE
JSaTk
dTdUTC
dfTU
B
gMBMAGNON exp.)(
)()()(
3/2
2
00
21130
D
C (7T)C(0T)
C(0T) – C(7T)
Mostly phonon specific heat
Phonons + magnons
phonons only
Assuming phonons and magnons are independent, one can assume additivity
Approximate: magnons and phonons hybridize Trans.Spins-15
Magnon specific heat in YIG
TkE
JSaTk
dTdUTC
B
gMBMAGNON exp.)(
3/2
221130
CPHONON T3
CMAGNON T1.1CTOTAL
C(0T) – C(7T)
Mostly phonon specific heat
Trans.Spins-16
Magnon thermal conductivity in YIG
MAGNONMAGNONMAGNONMAGNON vC 31
TOTAL
PHONON
MAGNON
Same idea: freeze out magnons by applying magnetic field
along [100] axis
Trs.Spins-17
Magnon and phonon mean free paths (YIG)
PHONONMAGNON
MFPMAGNON
MFPPHONON
PHONONPHONONPHONONPHONON
MAGNONMAGNONMAGNONMAGNON
vC
vC
3131
5000 ms‐1
8500 ms‐1
crystalthickness
In YIG at low temperature the magnon mean free path 5 to 50 m
Magnons and phonons have comparable mfp’s Tr.Spins-18
Heat current implies spin current
Heat current of spin-wave implies a spin current =>
A temperature gradient implies a magnetization gradient.
MAGNONBM
MAGNONS
MAGNONBQ
MAGNONQ
jgjjj
jTkj
Tj
..
.
Trans.Spins-19
Pure spin current without charge current? YES
• Spin‐polarized charge carriers• No net charge transport• Net spin transport• Need a spin‐flip transformation at some interface
- p p p p
Power dissipated as heat (phonons) T increases
je > 0
je = 0
Charge current je: conjugate force = voltage difference x electron charge
Spin current jM:
- -
Conjugate force? Magnetization gradientTrans.Spins-20
Treatment of spin flux
• For magnetic systems, the most important thermodynamic variable is magnetization itself, whose conjugate force is just the Landau‐Lifshitz effective field Heff.
• The (M,Heff) pair enters Onsager symmetry on par with other thermoelectric quantities. In fact, any well‐defined thermodynamic quantity (along with its conjugate force) based on a physical observable would obey Onsager reciprocity.
• In general spin flux is not conserved. This does not create any obstacles for invoking Onsager reciprocity for the magnetic dynamics (couple to charge currents, energy currents, mechanical motion etc.).
Tserkovnyak et al, PRB DOI: 10.1103/PhysRevB.79.014402.• In some special cases, when the spin relaxation is weak, one could approximate
ferromagnetic metals by a two‐fluid model: spin‐up and spin‐down electrons, and use spin‐up and spin‐down densities as well‐defined thermodynamic quantities (which could be approximately conserved) that enter Onsager reciprocity relations. In general, spin‐orbit interaction makes this impossible.
Magnons
Heat Magn. MomentM g
S
Trans.Spins-21
Spin Onsager relations
TS
Te
LLLLLLLLL
jj
ejj
MMMTME
TMTTTE
EMETEE
q
ne
/Fj
LFH
j
M
0
Entropy production
• Reciprocity holds• Neither spin nor heat are conserved• Discussion in Bauer, Tserkovnyak et al., PRB DOI:
10.1103/PhysRevB.81.024427
Trans.Spins-22
Caution 1: Magnon and phonon temperatures are not equal
D. J. Sanders and D. Walton, Phys. Rev. B 15 1489 (1977) MPMP
MP
MP
MP
PM
QM
CCCCA
ALAAxx
jTxT
12
0
)cosh()sinh()(
Magnons don’t couple directly to heat reservoirs
=> they couple through the phonons
=> difference between phonon temperature and magnon temperature during heat conduction
Trans.Spins-23
Caution 2: Sometimes magnons hybridize with phonons
Optical phonon
Transverse Acoustic phonon (TA100)
The whole discussion above is oversimplified: it treated magnons phonons as different and independent particles.
Magnon + phonon hybridizedT < TN T > TN
YMnO3 S. Pailhès et al., Phys. Rev. B 79,134409 (2009)
Trans.Spins-24
3. Spin polarized electrons and spin/orbit interactions: Spin Hall effectInverse spin Hall effect Electrons
with or withoutspin
Trans.Spins-25
Charge
Magn. Moment
g
Initial theoretical suggestion: M. I. Dyakonov and V. I. Perel,; Perel' Sov. Phys. JETP Lett. 13: 467 (1971). Observation: Y. Kato; R. C. Myers, A. C. Gossard, D. D. Awschalom Science 306 (5703): 1910–1913 (2004). Inverse Spin‐Hall effect:S.O. Valenzuela; M. Tinkham Nature 442(7099): 176–9 (2006)E. Saitoh; M Ueda, H. Miyajima, and G. Tatara Applied Physics Letters 88 (18): 182509 (2006).
Spin polarized band electrons: Stoner model
= EF = EF
Trans.Elec-26
Spin‐Hall and inverse spin‐Hall
Sadamichi Maekawa, Nat. Mater. VOL 8 p777, OCTOBER 2009
Trans.Spins-27
Spin‐Hall effectSpin Hall Effect (SHE) : the appearance of spin accumulation on the lateral surfaces of an electric current‐carrying sample.• Result of spin‐orbit interaction, does NOT require
magnetization• The signs of the spin directions is opposite on the opposing
boundaries. • In a cylindrical wire, the current‐induced surface spins will
wind around the wire. • When the current direction is reversed, the directions of spin
orientation is also reversed. awsch‐web.physics.ucsb.edu
Experiment:• Application electric field• Measured Kerr rotation as a function of
magnetic field at two positions: left and right edges
• Observe Lorentzian curves=>out‐of‐plane spin polarization (Hanle effect).
Trans.Spins-28
Spin‐Hall effect in GaAs
Trans.Spins-29
T. Seki et al., Nature Mater. 7, 125 (2008).Guang-Yu Guo, Sadamichi Maekawa, and Naoto Nagaosa Phys. Rev. Lett. 102, 036401 (2009)
Spin‐Hall effect in Au/Fe
Charge injection
Charge consists of spin-up and spin-down polarized electrons
Scattering depends on spin-polarization
Trans.Spins-30
Inverse Spin‐Hall effectInverse Spin Hall Effect: an electrical current is induced by a spin flow. Due to a space-dependent spin polarization[7]
The existence of both direct and inverse effects is demonstrated in metals and semiconductors.[
σJE S ISHEISHE D
Trans.Spins-31
4. Spin transfer at the interface
Electronswith or without
spin
Magnons
Trans.Spins-32
Charge
Magn. Moment
g
gg
Spin transfer torque and spin pumping
Spin Polarizedmatter
Normal Metal
Spin Polarized
matter
Normal Metal
x
M
x
M
The extreme case of the study of the effect of M
Trans.Spins-33
Spin transfer across interfaces
• Magnon transport in YIG (FM insulator)– by heat current or FMR pumping
• Spin transfer YIGPt at the interface: exchange interaction between free electrons in
metal (“s”) and “d”‐electrons on Fe atoms in YIG
e‐
Reversible effect follows Onsager reciprocity• Spin accumulation in Pt (e.g. by spin Hall effect)• Spin transfer PtYIG, generating magnons in YIG• Exponential decay of magnon current in YIG
Efficient spin transfer across interface is essential:• FM insulator with very low damping• High interfacial spin mixing conductance g YIG with
• Excellent structural and magnetic uniformity in bulk and at surface• Clean, atomically sharp YIG/Pt interface: strong interfacial exchange coupling
PtYIG
Trans.Spins-34
Electrical spin pumping: 1. InjectionM. Johnson and R. H. Silsbee, Phys. Rev. 37 5312 (1988)
Partially spin Polarized matter
Normal Metal
Fully spin Polarized matter
Normal Metal
Polarization ratio:
nnn
Trans.Spins-35
Current injection from ferromagnet to normal metal
PMFM
Trans.Spins-36
Current injection
M. Johnson and R. H. Silsbee, Phys. Rev. 37 5312 (1988)
Fully spin Polarized
Normal Metal
V0Apply electrical potential difference V0
Drive electrical current je across interface
Magnetization is injected at a rate proportional to the electric current.
The transport of magnetization is proportional to the magnetization of each electron, B
The injected magnetization current is then:
eB
M je
j
For non-fully polarized FM’s: eB
M je
j
Trans.Spins-37
Interface conductanceDefine interface conductance per unit area
TvEeVjG xFe )(D
2
2
0
DOS Average velocity across interface
transmission probability
Do this for spin up and spin-down electrons separately
TvEeG
TvEeG
xF
xF
)(
)(
D
D
2
22
2
Weak coupling between & subbands =>
In effect, an electrical current is related to a difference in “magnetization potential” across the interface
HMH EFF
Thermodynamic formalism:M. Johnson and R. H. Silsbee, Phys. Rev. 35 4959 (1987) Trans.Spins-38
Onsager formalism for spin dependent transconductance
S
C
BM
q
GGGGGGGG
ejj
GG
jj
21
00
GGGGGG
jje
j
jjj
C
C
BM
q
21
charge current
magnetization current
charge chemical potential
“spin accumulation potential”
charge conductance
“spin mixing conductance”
Ferromagnet Normal metal
GG
ejj B
e
M
Spin injection efficiency
Trans.Spins-39
Spins decay in the normal metal
Spin Polarized matter
Normal Metal
nMnnn
BnM ,
x
Spin diffusion length L
)/exp()( Lxnxn 0
FE
x
L
FE
SDL
Electron diffusion constant
spin lifetime
Trans.Spins-40
Inverse injectionSpin Polarized
matterNormal Metal with spin-polarized electrons
If a normal metal in which there is an imbalance between spin‐up and spin‐down electron is put in contact with a metal in which there is a difference between the spin‐up and spin‐down DOS, apply the reverse argument for G separately.
There is a current across the interface proportional to the non‐equilibrium polarized electron population in the normal metal.
The difference in number of non‐equilibrium spins drives an electron current. Trans.Spins-41
Voltage bias due to inverse injectionSpin Polarized
matterNormal Metal with spin-polarized electrons
Unbalence in electrons:
Solve for VD:
)(
)()()(/
FPAULI
PAULI
Bd
eVE
EB
E
eMV
dEEfEEMn
B
DF
F
D
DD
2
2
Trans.Spins-42
Electrical spin pumping: ExperimentM. Johnson and R. H. Silsbee, Phys. Rev. 37 5326 (1988)
PermalloyPermalloy Aluminum
Trans.Spins-43
Silas Homan, Koji Sato, and Yaroslav Tserkovnyak, Landau-Lifshitz theory of the spin Seebeck effect, arXiv:1304.7295 (2013)
Transfer of spin flux across the YIG/normal metal interface
Spin transfer across Pt/YIG interface through the electron‐magnon exchange interaction at the interface.
Loosely speaking an "s‐d“ interaction with "s" referring to electrons in Pt and "d" to local Fe moments in YIG
Use inverse spin Hall in Pt for detectionSpin “sink”
js1 = 0 js2 = js, FERROMAGNET
(Pt)YIG
Trans.Spins-44
Spin‐mixing conductance formalism applies to LSSE
YIG/Pt
M. Weiler, …, S. Goennenwein, arXiv:1306.5012v1 (2013)
“Our experimental results support present, exclusively spin current based, theoretical models using a single set of plausible parameters for spin mixing conductance, spin Hall angle and spin diffusion length
Note: thermal excitation gives highest flux
Trans.Spins-45
Thermal and electrical pumping
TTLLL
LLLLLLLLLL
qj
jJ S
C
TTSTET
TS
TE
S
q
21
Generalize:
• Spin is not conserved, but neither is heat
• Spin Seebeck coefficient is function of LTS / (L+L),…• Spin transport is dissipative, because FJQ
.
TTLLL
LLLL
qjj
TTETET
TE
TE
00
• Spin‐Dependent Thermoelectric coefficients: LTE …• Bart van Wees, Ron Janssens
Or generalize differently:
Trans.Spins-46
K.Uchida & al. Appl. Phys. Lett. 97, 172505 (2010)
Longitudinal Spin‐Seebeck Effect YIG/Pt
Inverse spin‐Hall effect in the Pt
SSISHEISHE DE σj
Trans.Spins-47
x
z
y-x
K.Uchida & al. Appl. Phys. Lett. 97, 172505 (2010)
Longitudinal Spin‐Seebeck Effect YIG/Pt
Trans.Spins-48
Uchida, et al., JAP 111, 103903 (2012)
H. Jin, et al., to be pblished
0
40
80
120
0 100 200 300Spin Seebe
ckCo
efficient (n
V/K)
T (K)
Bulk YIG (1 mm YIG + 15nm Pt)• Peak in SSE at 50K • Does not match with phonon peak
GGG|YIG|Pt (0.5mm GGG + 4μm YIG + 10nm Pt)• No low temperature enhancement• More similar to polycrystalline YIG
Normalized results from our LSSE thin film measurements
LSSE YIG/Pt Film versus single crystal
Trans.Spins-49
0.1
1
10
1
10
100
1000
1 10 100 1000
SSE ratio
in YIG (from Uchida, et a
l.)
Spin Seebe
ck coe
fficient (n
V/K)
Temperature (K)
Thermal con
ductivity
(W/m
‐K)
0.1
1
10
1
10
100
1000
1 10 100 1000
SSE ratio
in YIG (from Uchida, et a
l.)
Spin Seebe
ck coe
fficient (n
V/K)
Temperature (K)
0.1
1
10
1
10
100
1000
1 10 100 1000
SSE ratio
in YIG (from Uchida, et a
l.)
Spin Seebe
ck coe
fficient (n
V/K)
Temperature (K)
Thermal conductivity in monocrystalline
YIG slab
SSE(T)/SSE(290K) for monocrystalline YIG slab(Diffusive + phonon drag)
(Mostly phonons)SSE in 4μm YIG film
(Diffusive only)
LSSE YIG/Pt Relation to thermal conductivity?
Trans.Spins-50
Slide from Gerrit Bauer Trans.Spins-51
Caution 3: What can go wrong experimentally?Thermomagnetic & Galvanomagnetic Effects
Longitudinal Spin‐Seebeck Effect ONLY on YIG/Pt
The planar Nernst effect has exactly the same symmetry as the longitudinal spin‐Seebeck effect
LSSE measurements impossible on electrically conducting ferromagnets
Trans.Spins-52
5. Transverse Spin Seebeck Effect
• Local versus non‐local spin injection• LSSE = local• TSSE = non‐local
• GaMnAs• Phonon Drag (phonon‐magnon, phonon‐
electron)
Trans.Spins-53
Local vs non‐local spin injection
Non‐local electrical spin injection:• spin‐polarized charge current is driven by
an applied electric field.• spin current parallel to a charge current • => spin current diffuses from the
ferromagnet into the normal metal.
jC, jS, jQ
j C, j
S, j Q
Equivalent for electrical spin injection:
F. J. Jedema, A. T. Filip & B. J. van Wees, Nature 410 345 (2001)
Trans.Spins-54
CASE 1: In‐plane or cubicMn content 7-18%
[-110]
[110]
[100] [010]
-75 -50 -25 0 25 50 75
-40
-20
0
20
40
50KM (e
mu/
cm3 )
B (Oe)
[100]
[-110]
[110]16% Mn
[-1,1,0] & [100] are easy axes
Two possible experiments:Case 1a: B//T//easy axesCase 1b: B//T at 45o from easy axes
Trans.Spins-55
CASE 1a: B // T // easy axes
Trans.Spins-56
Dependences on temperature gradient
1. Dependence on temperature gradient is linear
=> can assign a "Seebeck coefficient" to the slope
2. Dependence on strip position is totally unusual for transport coefficient
Contrast with charge‐Seebeck between strips X
X
X
XXX
X
Y
X
YXY
TV
TE
wL
TV
TES
2
Trans.Spins-57
Position dependence of Spin‐Seebeck Sxy
A skewed and off-center sinh(x)function, with a characteristic length scale of 4 -6 mm
)]2
(sinh[)( 0xLxxS XY
Remember D. J. Sanders and D. Walton, Phys. Rev. B 15 1489 (1977) ? Trans.Spins-58
Temperature dependence and position dependence
T and x dependence normalize
Trans.Spins-59
Scratch 2
Scratch 1
Scratch the sample => spin current is vertical!=> What causes the long range of the effect?
x
Pt1Pt2
Pt3Pt4
Scratch 1Scratch 2
z
y
• If spin current were horizontal scratching the sample in half would result in 2 independent samples,
• There would be a negative signal above the scratch and positive signal below the scratch.
0.3 mm wide scratches
Trans.Spins-60
• No change in signal
• Spin-Seebeck does not result from a macroscopic spin-current.
• The substrate, not the film, carries the mm-range information
• Substrate has ONLY PHONONS
Spin Seebeck is NOT due to spin current along
2L
2L
0
Scratch #1Scratch #2
Vy
Hotend
Coldend
Crack
40 60 80 100 120 140Tavg (K)
-1.2
-0.8
-0.4
0
Sxy
V y/
T x (V
/K)
Intact
B // [110]
40 60 80 100 120 140Tavg (K)
-1.2
-0.8
-0.4
0
Sxy
V y/
T x (V
/K)
Intact1 Scratch
B // [110]
40 60 80 100 120 140Tavg (K)
-1.2
-0.8
-0.4
0
Sxy
V y/
T x (V
/K)
Intact1 Scratch
Both Scratches
B // [110]
Scratch me twice!
Trans.Spins-61
Electronswith or without
spin
Phonons Magnons
Trans.Spins-62
6. Phonon Drag
Charge
Heat Magn. Moment
L g
Magnon Drag
Spin Caloritronics
g
Extended T‐dependence in GaMnAs
0
0.3
0.6
0.9
|Sxy
(V
K-1
)|
0
0.5
1
1.5
(G
aAs)
(W
cm
-1K-
1 )
0
25
50
75
M (e
mu
cm-3
)
0 40 80 120 160T (K)
0
10
20
xx
(V
K-1
)
1 10 100T (K)
0.01
1
100
Cp
(J k
g-1 K
-1)
a+bT-1
a+bT1.5
bT-1
a+bT3
(a)
(b)
(c)
b(Tc-T)-
b(Tc-T)-
0
2
4
6
| Sxy
(V
K-1
) |
5
10
15
(G
aAs)
(W
cm
-1 K
-1)
10 100T (K)
0
0.25
0.5
0.75
xx
(mV
K-1
)
0
25
50
M (e
mu
cm-3)
(a)
(b)
Thermal conductivity of substrate
High Low
SpinSeebeck
Phonon- Drag
Thermo-power
Big Small
Big Small
Jaworski et al., Phys. Rev. Lett. 106186601 (2011)
63
Understanding so far
C o ldH o tT P
T (K
) T M
TM (K
)
x
Mx Mx Mx
x
M
x
Vy
Hx
Vy Vy
Hx Hx
(a)
(b)
(c)
(d)
GaMnAs
GaAs
ESHE
ESHE
Magnetization
+
_
xy
2L
2L
+
_
0x z
Vy
Two mechanisms can push magnon fluxes:1. Magnon thermal conductivity2. Phonon-magnon drag
Hot end: Tmagnon < Tphonon => TM>0Drag heats magnonsDrag decreases average Mx=> Mx<0
Cold end: inverse, TM<0, Drag cools magnonsDrag increases average Mx=> Mx>0
CHANGE in Mx => CHANGE in spin-polarization of whatever carries the spin
Inverse spin-Hall =>
Sxy Vy Mx TM
Trans.Spins-64
Adachi et al. Applied Physics Letters 97 252506 (2010)
Trans.Spins-65
Phonon Drag
0
2
4
6
| Sxy
(V
K-1
) |
5
10
15
(G
aAs)
(W
cm
-1 K
-1)
10 100T (K)
0
0.25
0.5
0.75
xx
(mV
K-1
)
0
25
50
M (e
mu
cm-3)
(a)
(b)
A = oil dropletsB = air molecules
A = phononsB = electrons
A = phonons,B = magnons
When: 1. A‐B collisions dominate
both A‐scattering and B‐scattering
2. A‐particles have drift velocity
Then:1. A‐particles impel B‐
particles with momentum IN ONE DIRECTION
2. Out‐of‐thermal equilibrium3. Very intense
A – wall collisions dominate
A – B collisions dominate
How do “phonon‐drag” curves get to have a maximum in temperature?
?
Trans.Spins-66
The MacDonald Formalism, low temperaturePhonons exert pressure on electrons feel a pressure because of electron‐phonon collisions.
)(TUp 31
U(T) = phonon internal energy density
Phonon Pressure is then given by
TApply T‐gradient to a sample:More phonons on hot side than on cold side=> pressure gradient=> force per unit volume
dxdTC
dxdT
dTTdU
dxTdU
dxdpF Vx 3
1)(31)(
31
neC
dxdTE
dxdT
neC
neFE
FneE
VxPED
Vxx
xx
3
3
0
• Assume that phonons interact ONLY with electrons
• Fx pushes electrons toward cold side of sample.
• Electrostatic force neEx balances:
• Phonon‐electron drag thermopower
Trans.Spins-67
Electronswith or without
spin
Phonons
Trans.Spins-68
Charge
Heat Magn. Moment
L
Spin Caloritronics
g
7. InSb• No exchange coupling: spin
polarization from Landau levels
• No magnon conductivity: phonon drag
• Giant spin‐Seebeck‐like effect
InSb and its Landau levels
PauliLandau 31
R. Peierls, “Quantum Theory of Solids”, pp 144-149 (Oxford 1955)
Ferromagnetism not necessary
Spin‐Polarization necessary
InSb is a very narrow‐gap semiconductor Eg 0.2 eV
Very strong spin/orbit interactions, effective Landé factor g* ‐50
Separate energy levels into Landau levels
Landau diamagnetism and Pauli paramagnetism for free electrons
Trans.Spins-69
InSb and its Landau levelsLandau level Orbital and Zeeman splitting
**
**
22
221
2)(
mHe
mHe
Hsgnmk
EE
x
c
xc
xBCc
x
0 1 2 3 4 5 6 7H (T)
0
0.02
0.04
0.06
0.08
0.1
E (e
V)
n=2.82x1015cm-3
EF
(0,)
(0,)
(1,)
(1,)
(2,)(2,)
(3,)(4,)
From this field on up, most electrons are on the last Landau level(ultra‐quantum limit), spin‐polarized by Zeeman splitting
- 2 - 1 1 2
- 100
- 50
50
100Polarization (%)
B (T)5K
Trans.Spins-70
-8 -4 0 4 8Bx (T)
Sxy
(V
K-1)
4.3K2000V K-1
InSb Spin‐Seebeck data
xHx // xT
Hx(T)
1. Signal is very large, 8 mV/K
2. Even‐symmetric and small odd‐symmetric portions as function of magnetic field
3. Even‐in‐field part: 1. large2. ultra‐quantum‐field region
Jaworski et al., Nature 487 213 (2012) Trans.Spins-71
Sxy
(V
K-1)
6.3K
-8 -4 0 4 8Bx (T)
Sxy
(V
K-1)
10.4K
2000V K-1
1000V K-1
Sxy
(V
K-1)
2.75K
-8 -4 0 4 8Bx (T)
Sxy
(V
K-1)
4.3K
2500V K-1
2000V K-1
0 10 20 30 40T (K)
10
100
1000
10000
Sxy
,max
(|V
K-1|)
-8 -4 0 4 8Bx (T)
Sxy
(V
K-1)
25.1K
150V K-1
Sxy~e-b*T+A
T‐dependent Spin‐Seebeck Data
Blue = hotter end; red = colder end (experimentally some variation)Signal many mV/K below 10 K, much larger than any parasitic Trans.Spins-72
Temperature‐dependence of amplitudesignature of Zeeman splitting
0 10 20 30 40T (K)
10
100
1000
10000
|Sxy
|max
(V
K-1) RT )sinh(
BgTk
BgTk
R
B
B
B
B
T
2
2
2
2
Ratio between thermal energy (kBT) and Zeeman energy (gBB) for electrons on helical orbitsOnly adjustable parameter = amplitude
Shoenberg D. Magnetic Oscillations in Metals, Cambridge 1984
• RT decays slower than SdH oscillations in resistivity
0 5 10T (K)
0.1
1
10
Am
plitu
de (A
rb. U
nits
)
• Spin‐Seebeck effect exists even when orbital quantization is no longer resolved
Trans.Spins-73
The physics:
1. Temperature gradient creates phonon fluxChange in phonon momenta:
2. Strong phonon-drag impels additional momentum k to electrons:
3. Strong spin-orbit interactions transform k into a change in Zeeman splitting energy:
We actually can estimate from published electron‐concentration‐dependence of g‐factor
=> no adjustable parameters, for T=5K, T=40 mK:
cTkq B
qkx
xk k
TkeV B of 25120 %k Trans.Spins-74
kx+qx
xT
kx-qx xHx // xT
Explanation based on the two-momentum model (electrons versus phonons)
- 2 - 1 1 2
- 100
- 50
50
100
Polarization is odd in field
H(T)
5KPhonon‐drag DIFFERENCE
vis‐a‐visMiddle plane
Phonons are warmer on hot side => accelerate electrons more
Phonons are colder on cold side => slow down electrons more
Trans.Spins-75
+qx-qx Hx // xT
Phonon-drag driven, spin-orbit induced spin splitting
E
x
gBH
0
E =
k =
gBH‐qx gBH+qx
S+x <0 =0 >0
Spin‐polarized electron population is not at equilibrium spin‐pumping into Pt Non‐uniformly across sample (spatial dependence) Trans.Spins-76
+qx-qx
Phonon-drag driven, spin-orbit induced spin splitting
E
x
gBH
0
E =
k =
gBH‐qx gBH+qx
S+x <0 =0 >0-Hx // xT
S+x <0 =0 >0Therefore, S+x is even symmetric with magnetic field.
In both field directions,
Trans.Spins-77
Phonons
Trans.Spins-78
8. Phonon Diamagnetism
Heat Magn. MomentL
Spin Caloritronics
new effect
Anharmonicity & Grüneisen
xU
Anharmonic Potential
)(2
2
xkdx
Ud
kBT
x0( kBT)
Apply to a solid
The phonon frequencies
with anharmonic bonds
Grüneisen parameterAnharmonicity is characterized by)ln(
)ln(),(
),,(
0
0
Vdd
xxkkmkx atoms
Cstdx
UddxdFk
xkFdxU
xxkF
2
2
20
2
)(
Harmonic Potential
Anharmonicity governs the phonon‐phonon interaction Hamiltonian
Trans.Spins-79
Anharmonicity and lattice thermal conductivity
0.1
1
10
100
1 10 100 1000
Ther
mal
Con
duct
ivity
(W c
m-1
K-1
)
Temperature (K)
model
data
Phonons scattered by crystal boundaries
Phonons scattered by defects (isotopes, …)
Morelli, Heremans &
Slack Physical Review B 66, 195304 (2002)
Tn
VMA atomL
322
3/13
Average Mass of Atoms
Debye Temp Volume per atom
# atoms in cell
Grüneisen Parameter
Smattering of
Constants
Phonons scattered by other phonons
Silicon
Trans.Spins-80
Size effect on boundary scattering phonons (Ballistic phonons in small arm) Difference in thermal conductivity Small arm serves as reference for phonon
scattering in large arm
Q
H
Experimental setup: “tuning fork” geometry
th
wL wS
Cernox
Tdiff
QL QS
LTS
H
TL
1.4x1015 cm‐3 dopedn‐type InSb single crystal
Q // H // [100]
wL (3mm) ~ 3wS (1mm) AL ~ 3AS
Geometry from T. H. Geballe and G. W. Hull, “La physique des basses temperatures”, p. 460 (1955)
30 mm
Trans.Spins-81
Principle of measurement: thermal potentiometer
0 1000 2000 3000Time (s)
-40
-20
0
20
40
Tdi
ff (m
K)
no heat heater on heater off
T = 30KH = 0T
Constant QLStep QS
4 12 20QS (mW)
-40
-20
0
20
40
T d
iff (m
K)
T = 30KH = 0T
QL = 44.5mW
Typical random error in ∆Tdiff ~ 10µK
When Tdiff= 0 (TS = TL ):
L S L
S L S
A Q
A Q
Tdiff
QL QS
TSTL
Advantages:1. Tdiff = 0 , heat flux only measurement2. Eliminates calibration errors on thermometers3. Eliminates magnetic field sensitivity of
thermometers4. Minimizes heat losses between arms5. Enables accuracy of 1:104 or even 105
L S L
S L S
A Q
A Q
Trans.Spins-82
Temperature & Magnetic field dependence
Magnetic field dependent lattice thermal conductivity• Electronic < 10‐5 times smaller than lattice
conductivity• No d or f‐electrons in system• Effect is even in field
-8 -4 0 4 8H (T)
1.44
1.48
1.52
1.56
1.6
L / S
T = 3K
T = 4.4K
Trans.Spins-83
Physical meaning of L/ S1. Effect occurs where the transport is still ballistic, but
starts picking up a phonon scattering component
11
11
111231
111231
111
231
31
BL
BS
BSV
BLV
S
L
B
VV
vC
vC
vCvC
2. Thermal potentiometer eliminates specific heat and sound velocities from the physics of the problem.
3. If we consider the limiting case where
4. For long‐wavelength phonons (T<<D)
111 BLBS 1
1
11
11
BS
BL
BS
S
L
Klemens’ model
aTA 21
Proportionality constant
Grüneisen a=3 for cubic crystals (1)a=2 for trigonal crystals (2)
21
1 1
BS
S
L
1 10T (K)
100
1000
S (W
/ m K
)
TS
Trans.Spins-84
What is “magnetic” in here?
32
11TBSS
L
What is most likely to depend on magnetic field?1. Electronic thermal conductivity? No: L 103 W m‐1K‐1;E 10‐2 W m‐1K‐1 10‐5 L
2. Electron‐phonon scattering? No: would go the other way. Electrons freeze out at low T and high H
3. Specific heat? No: we checked experimentally
4. Phonon‐phonon scattering? Yes: T-3 => phonon‐phonon interactions.
Trans.Spins-85
Theory of phonon diamagnetismConcept: frozen phonon
In
Sb
[010]
[101][101]
Frozen phonon
Valence band structure frozen phonon
In
Sb
M (
B/Å3)
H=7 T // [010]
Sb [010]
[101]
0
‐5x10‐6
Magnetization around frozen phonon
)()()( 2 rHHrMrFM
• Local moments weak• Gradients very sharp• Magnetic force on atoms:
• Magnetic force is anharmonic alters thermal conductivity
Trans.Spins-86
Origin of the diamagnetic moment-0.006 B/Å3
In
Sb
“Diamagnetic susceptibility of the
phonon”
A
Orbital magnetism in valence band
-0.03 B/Å3
Logarithmic mesh, 0.18 Å In displacement
Of course, net magnetic moment integrated over (a) the solid (b) time, is zeroTrans.Spins-87
-35
-30
-25
-20
-15
-10
-5
0
0 0.1 0.2 0.3
Sus
cept
ibilit
y (m
B/T
)
Displacement (Å)
Orbital diamagnetism(semiconductor)
Diamagnetic Moment vs. DisplacementMaterial has bandgap > 0:
What type of diamagnetism?
Landau diamagnetism(metal)
Diamagnetic χ function of displacement => Induced χ affects atom displacementTrans.Spins-88
Orbital phonon diamagnetism (insulators): Localized electrons in valence band
Magnetic field (HAPPL) induces electron precession generating magnetic moments
General expression for magnetic moment:
22
4r
mBZeme
2r = mean distance from all electrons from atom
Phonon‐induced Larmor‐like behavior of the valence electrons
Langevin diamagnetsm
Trans.Spins-89
Field‐induced anharmonicity: how it works
)( 0xxkF
Phonon “Magnetic force”
CstHkFk
HxxkF
APPL
APPL
2
1
201 ...)(
HAPPL MLOCAL
xLocal
displacement coordinate
0)(
)()(
2
APPLMAGN
APPLLOCAL
MAGN
HxF
HxMF HM
Interatomic harmonic force
Total:
)()(
)(11
APPLLL
APPL
APPL
HH
H
)( APPLHV
V
m
Hkmk APPL
21
Diamagnetic anharmonicity causes
Phonon frequency:
Grüneisen parameter
Need to make a mode and frequency average Trans.Spins-90
Comparison theory‐experiment (no adjustable parameter)
0 4 8 12 16T (K)
-5
0
5
10
15
20
-2
/ -2
(%)
-5
0
5
10
15
20
-(
L/
S) /
(L/
S) (
%) Theory
Experiment
Jin, Restrepo, Antonin, Boona, Windl, Myers, Heremans, under review
21
1 1
BS
S
L =>
20
20
27
0
07
TH
THTH
THS
L
THS
L
THS
L
APPL
APPLAPPL
APPL
APPLAPPL
No adjustable parameters
Trans.Spins-91
9. Applications
Heat dissipation by spin fluxThermal spin pumping efficiencySolid‐state heat enginesKovalev / Tserkovnyak designKirihara / LSSE design
Trans.Spins-92
Dissipation from spin currents?
Theory and assorted statements in literature:“pure spin currents allow for dissipationless information
transfer”. If Jc = 0, then Joule Heat = 0
- p p p p
Power dissipated as heat (phonons) T increases
Jc > 0
Jc = 0
Joule heating
Pure spin current
- -
Phonons are still producedPhonon‐spin interactions important in spin‐Seebeck
Trans.Spins-93
Spin‐Caloritronic engines based on domain wall motionAlexey A. Kovalev and Yaroslav Tserkovnyak
Solid State Communications 150 500-504 (2010)
Mechanisation: heat pushes domain wall
Heat pump/cooler
Power generator
Efficiency equations similar to thermoelectrics
Connect two of those with different pinning energy
T. Spin-94
Akihiro Kirihara & al., Nat. Mater. 11 686-689 (2012)
E‐based spin caloritronic energy‐recovering cloth, NEC corp.
Trans.Spins-95
ZT of LSSE process
Efficiency
hZ
ZhS
hS
Z
Z
TT
LlTz
Tz
Ll
2
2
CARNOT
11121
Ferromagnet
TSTz XYS
YIG
Pt2
Involves two materials• Ferromagnet• Inverse spin Hall
Involves two materials• Ferromagnet• Inverse spin Hall
7
86
322
32
Pt2
1010010
1010
mnm
LlS
S
Z
Z
TeBiTeBi
InSb
...
H. Adachi, Spin Caloritronic V, Ohio State, 2013 Trans.Spins-96
Spin
ChargeHeat
Spintronics
Thermoelectrics
SpinCaloritronics
Conclusions: spin caloritronics• Physics in its infancy
• Spin‐Seebeck effect is getting
fairly well understood
• Spin‐mixing conductance good concept for spin fluxes in the linear transport regime
• Phonons induce diamagnetic moments
Trans.Spins-97
Electrons Magnons PhononsHeat kT kT kTMagnetism Spin, Orbit B 2nd order
Landau/OrbitalCharge e ‐ ‐
