Introduction to Small-Angle Neutron Scattering and Neutron ...
Neutron scattering as a tool to study quantum magnetism · Neutron scattering as a tool to study...
Transcript of Neutron scattering as a tool to study quantum magnetism · Neutron scattering as a tool to study...
Neutron scattering as a tool to studyquantum magnetism
Radu Coldea
Oxford
Outline
- principles of (magnetic) neutron scattering
- spin waves in a Heisenberg ferromagnet
- spin waves in square-lattice AFM La2CuO4
- quantum renormalization, spinons and method to determine
Hamiltonian – triangular AFM S = 1/2 Cs2CuCl4
- quantum phase transition in the Ising chain CoNb2O6 in
transverse field
Neutron reactors(Institute Laue Langevin)- nuclear fission of Uranium
Three-axis neutron spectrometer
neutronguide
spectrometers
ki – kf = QEi – Ef = E
ki
kf
Q
CliffordShull
BertramBrockhouse
Spallation neutron sources (ISIS, SNS …)
proton synchrotronaccelerator ~800MeV
p+
- “evaporation” when fast protonshit a heavy nucleus (Ta)
neutronguide
time-of-flightspectrometer
~40,000 detector elementscount simultaneously (time-stamp each arriving neutronLET@ISIS
Magnetic neutron diffraction
Neutrons have- no charge- spin-1/2 moment
ki
dipole moment M
kf
- Intensity ~
Periodic magnetic order=> magnetic Bragg peaksat Q = t ± q
magneticunit cell
Fourier transform of magneticmoment density (perp toscattering wavevector Q)
q
magneticstructural
Bragg peak t
t
Inelastic magnetic neutron scattering
matrix element for transition
ground state
excitedstateki
kf
sample
polarization factorFourier transform ofmagnetic e- density
Fourier transform of magneticmoment density M a
Single crystals for inelastic neutron scattering
7 single-crystal mount ~50 g(flux growth)
2.5 cm
Cs2CuCl4(solution growth)
La2CuO4
4 cm
CoNb2O6
(mirror furnace growth)
floating-zone mirror furnace
S- =- neutrons flip over one spin
magnon
H = J Si·Sj - h Siz
Jij
ij ij
ij
i
= (Si Sj+
+ Si+Sj ) + Si
zSjz - h Si
z_2
- -ij
Jij i
B
hopping
coherent propagation of spin-flip states(if Hamiltonian conservs Sz)
One spin flipmanifold
Wavevector (q)
(q) = J(q) - J(0) + h
(q)
Ene
rgy
ZeemanenergyFourier transform
of magnetic couplings
S-
magnonenergy
0
Gap
exactresult
J(q) = ijJ
ije
iqrij1
2_
Dispersion images exchange Hamiltonian
|q> = e iqri | i>i
1
N
Spin waves in a Heisenberg ferromagnet
-
--
- (0) +
H = - J Si·Sjij ijif all Jij > 0 T=0 ground state is ↑↑↑↑↑↑
Wavevector k
L (rlu)
incidentscattered
Magnon
Sz = -1
ki
kf
k
neutron
S z = -1/2
S z = + 1/2
ki - kf = k
Ei - Ef = ħ w(k)
Neutron scattering by ferromagnetic magnons
Spin waves in the square-lattice anti-ferromagnetGround state has Neel order(<S> reduced by quantum fluctuations)
Keimer at al, PRB (1992)
Magnetic Bragg peak (1/2,1/2,0)
J antiferromagnetic Spin wave excitations(approximate eigenstates)
La2CuO4
- insulating parent ofhigh-TC cuprates
- square-lattice of CuO2
planes, Cu2+ S=1/2
Neutron scattering experiments on La2CuO4
7 single crystal mount, ~50 g
ki
kf
magnonS = 1
RC et al, PRL 86, 5377 (2001)
Magnetic excitations in La2CuO4Collect maps of magnetic scattering in the whole2D Brillouin zone (h,k) at increasing energies E
325 meV
275 meV
75 meV
Spin-wave dispersion surface
Energy
RC et al, PRL 86, 5377 (2001)
Dispersion relations
La2CuO4, T=293 K
AntiferromagneticBragg peak
position
RC et al, PRL 86, 5377 (2001)
Dispersion relation and interactions
- “wiggle” in high-energy dispersion is evidencefor a cyclic-exchange between the 4 spins on aplaquette in addition to the main exchange J
- dispersion shape is a direct fingerprint of the magnetic interactions
RC et al, PRL 86, 5377 (2001)
Ring exchange in the Hubbard model
Expand up to 4electron hops
A.H. MacDonald (1990),Takahashi (1977)
- dispersions and intensitieswell described by linear spin-wave theory
t = 0.30(2) eV, U = 2.2 (4) eV
U/t = 7.3 ± 1.3 (10 K)
Hubbard model parameters for La2CuO4
Intensity renormalization factorZc = 0.51 +/- 0.13 (predicted 0.61)
RC et al, PRL 86, 5377 (2001)
Magnetic excitations in S =1/2 triangular lattice AFM Cs2CuCl4
JJ' - sharp spin-wave mode only very small
weight
- dominant scattering continuum withstrongly-dispersive boundaries
=> Quantum fluctuations very strong, spin-wave theory inadequate
1.5 1.25 1(0,0.5,l)
Energy (meV)0.0 0.2 0.4 0.6 0.8 1.0 1.2
Inte
nsi
ty(a
rb.
un
its)
0
2
4
6
E (meV)
Inte
nsi
ty(a
.u.)
0.1
1
0.2 0.8
RC et al, PRB 68, 134424 (2003)
Experimental method to determine Hamiltonianvia spin waves in the fully-polarized state in high field
BC
AppliedField
Fully-polarizedstate↑↑↑↑↑
gappedmagnons
k
ω
Δ
J antiferromagnetic
B=0 B >> J
State with AFcorrelations
Dispersion is the Fouriertransform of exchange
couplings
(k) = J (k) - J (0) + h
J (k) = S Jd exp( i k · d)d
12
Magneticallyordered, spin
liquid etc.
Exact eigenstate ifHamiltonianconserves SzBose condensation
of magnons at Bc
RC et al, PRL 88,137203 (2002)
Fourier transform ofcouplings J(q)
J=0.374(5) meVJ’=0.128(5) meV
2D Hamiltonian
Excitations in the saturated ferromagnetic phase at B=12 T
[0,k,0]1.0 1.5 2.0
[0,2,l]
0.0 0.5 1.0
B = 12 T || aT < 0.2 K
[0,,-1]
1.01.52.0
Ene
rgy
(meV
)
0.0
0.5
1.0
1.5
(010) (020) (021)(021)
D
b
JJ'
c
J’’= 0.017(2) meV interlayer couplingDa = 0.020(0) meV DM anisotropy | bc plane
Energy (meV)0.4 0.5 0.6
0
10
20
30 (0,1.447,0)
Lineshapes show sharpmagnon excitations
J’J_ ~
13_
_
Field // a (Tesla)0 2 4 6 8 10
Mag
net
izat
ion
(
)
0.0
0.2
0.4
0.6
0.8
1.0T=60 mK
Bc = 8.42 T
Field // a (Tesla)0 2 4 6 8 10
Sus
cep
tib
ilit
yd
M/d
B(
/T
)
0.0
0.1
0.2
0.3
0.4 T=60 mK
RC et al, PRL 88,137203 (2002)
FERROCONE
B (T) || a0 2 4 6 8 10 12
Ene
rgy
(meV
)
0.00
0.25
0.50
0.75
1.00
1.25
1.50(020)
(010)
(0,.447,1)
-(0,.4
47,1)+
B (T) || a0 2 4 6 8 10 12
Pea
kin
tens
ity
(arb
.uni
ts)
0.00
0.01
0.02
0.03
0.04(b)
(a)
B>Bc magnondispersion
Gap
B=BcB<Bc
Q
Magnon condensation below critical field induces transverse order
[0,k,0]
1.00 1.25 1.50 1.75 2.00
En
erg
y(m
eV)
0.0
0.5
1.0
1.5
D
Bragg peaks appearwhere the gap closes
K (rlu)
1.40 1.45 1.50In
ten
sity
a.u
.)
0
50
100
150 B > BcB < Bc
RC et al, PRL 88,137203 (2002)
Link magnetic order with magnon wavefunctions
(000)
[0 k 0]-0.5 0.0 0.5
Ene
rgy
(meV
)
0.0
0.5
1.0
+Q-Q
2 condensates at+Q and -Q
Alternating layerscounter-rotate
+D
Senseselected by
DM couplings
D (Si x Sj)
-D
B (T) || a0 1 2 6 8A
sym
met
ry<
Sb
>/<
Sc>
1.0
1.2
1.4
1.6
predicted asymmetry usingmagnon wavefunctions
[0,k,0]
1.00 1.25 1.50 1.75 2.00
Ene
rgy
(meV
)
0.0
0.5
1.0
1.5
RC et al, PRL 88,137203 (2002)
·
Quantum renormalization of incommensurate ordering wavevector
B=Bc
B<Bc
QBc= 8.44(1)T
c=0.0536(5)
B (T) || a0 2 4 6 8 10 12
Inco
mm
ensu
rati
on
(2
/b)
0.02
0.03
0.04
0.05
(0,1.5-,0)
(0,0.5-,1)
mean-fieldresult
Strong quantumrenormalization
e0/ecl=0.5620°10°
Classicalinstability
large-S
Wavevector k
QuantumSpin-1/2
Collinearspins
Energy
RC et al, PRL 88,137203 (2002)
Magnetic excitations at zero field: spin-waves fractionalizeinto pairs of S=1/2 spinons
JJ'
S=1/2Spinons
Kohno, Starykh, Balents, Nat. Phys. (07)
J’/J ~ 1/3, J=0.37 meV
S+ S- + S- S+i j j i
- anisotropic triangular lattice S=1/2 AFM Cs2CuCl4 hasspiral order coexisting with strong quantum fluctuations
- renormalization of Q-vector and zone-boundary energymeasured by quenching quantum fluctuations via field andrevealing “classical” behaviour
- dominant continuum scattering (spin-waves fractionalizeinto pairs of spinons)
Summary on anisotropic triangular AFM Cs2CuCl4
JJ'
La2CuO4
Headings, Hayden, RC …, PRL (2010)
Ising magnets and phase transitions
Temperature0 TC
<Sz>
add transverse field- B Sx
- B (S++S-) / 2
i,jH = - J Si Sj +
z z
quantum fluctuations
quantum tunneling
- 2D model Onsager exact solution (1944)
- classical Ising modelClassical
thermally-drivencontinuous phase
transition
B field0 BC
<Sz>Quantum fluctuations
driven continuousphase transition
T=0 “quantum melting of order”
An Ising ferromagnet in transverse field
- transverse field- B Sx = - B (S++S-) / 2
generates quantum fluctuationsthat “melt” the spontaneousmagnetic order at BC ~ J/2
iH = - J Si Si+1 +
z z
Field0
Ferromagnet
↑ ↑ ↑ ↑BC
TC Quantumparamagnet
z
T
continuous quantum phase transition
→→→→
Magnetization
Orderedmoment
- B S ix
x
Quantum criticality: experimental challengesand opportunities
-what is microscopic mechanism of transition, can one observe thequantum fluctuations that drive transition ?
- how quasiparticles evolve near critical point ?
- what are the fundamental symmetries that govern physics of QCP ?
- what are finite-T properties (interplay of thermal and quantumfluctuations, under what conditions universal scaling ?
Phase 1
TemperatureCritical
Phase 2
BC
0Tuning
parameter
iH = - J Si Si+1 +
z z
1D Ising chain in transverse field
↑↑↑↓↓↓↓↓↓↑↑• •
- transverse field B Sx ~ - B (S++S-)flips spins
=> propagating solitons(Jordan Wigner fermions)
Bx
2 ground states: or↑↑↑↑↑↑
domain walls (solitons)
↑↑↑↑↑↑
zJ
↑↑↑↑↑↑
↑↑↑↓↓↓↓↓↓↑↑• •
↑↑↑↓↑↑↑↑↑↑↑…
Ferro-magnet
QuantumparamagnetBC
Gap D
solitonsspin-flip
quasiparticle↑↑↑↓↓↓↓↓↓↑↑• •
classical soliton(large-S limit) →→←→→
Ising chain at criticality
- w/T scaling expected, special “conformal” symmetry
- different universality class from Luttinger liquids (1D Heisenberg and XYAFM chains)
gapless linear (Dirac) spectrum
w = c|k|
for critical solitons
1) good 1D character to see solitons2) low-exchange J ~ 1 meV to access critical field BC ~ J/2 < 10 T3) strong uniaxial anisotropy (Ising character) but not perfect to stillhave transverse g-factor
- best Ising magnets are based on Co2+ 3d7
lowest Kramers doublet effective spin-1/2 Ising-like
2D Ising AF K2CoF4 (Birgeneau ‘73, Cowley ‘84)1D Ising AF CsCoCl3 (Goff ’95) also CsCoBr3 (Nagler)
J ~ 12 meV BC > 50 T not accessible
Experimental requirements
Strong Crystal field + Spin Orbitl L•S
Quasi-1D Ising ferromagnet CoNb2O6
Single crystalof CoNb2O6
(Oxfordimage
furnace)
4 cm
Co2+
Oxygen
Ferromagneticsuperexchange
~ 90° bond Co-O-Co~ 20K ~ 2meV
Ferromagnetic order along chainStrong easy-axis (Ising) in ac plane
zig-zag Co2+ spin chain along c
T= 5 K, 1D phase above TN
Magnetic excitations in 1D phase seen byneutron scattering
↑↑↑↑↑↑↑↑↑↑↑
↑↑↑↑↑↓↑↑↑↑↑
↑↑↑↑↓↓↓↑↑↑↑
incident ↑↓
••
••- gapped continua characteristic of
2-soliton excitations
k1 k2
scattered
k1 k2RC et al, Science 327, 177 (2010)
Magnetic excitations in zero field
3D phase1D phase
T= 5 K, 1D phase above TN T= 0.04 K, deep in 3D phase
0
ordered↑ ↑ ↑ ↑
BC
TC
z
- rich structure : continuum as characteristic of 2-soliton excitations
+ sharp modes (bound states)
RC et al, Science 327, 177 (2010)
Excitations have 1D character –no measurable dispersion ┴ chains
1.1 1.2 1.3 1.4 1.50
0.1
0.2
0.3
0.4
Energy (meV)
Inte
nsi
ty(a
rb,u
nits
)
H=-2.5
H=-1.5
H=-2.0
|| chain direction
┴ chain direction
Energy (meV)1.0 1.2 1.4 1.6
Inte
nsit
y(a
rbun
its)
0.0
0.1
0.2
0.3
0.04 K
5 K
L = 0.00(5)
1
2
34
5
Zeeman ladder of bound states in 3D ordered phase
Bound states inconfining potential
↑ ↑ ↑ ↑ ↑ ↑ ↑ ↑ ↑
↑ ↑ ↓ ↓ ↓ ↓ ↑ ↑ ↑x x x xJ’
E
Zeeman ladder of bound statesT =40 mKT = 5 K
12
… V(x) = l x
Continuum offree 2-soliton
statesSoliton separation
costs energy
l ~ J’ Sz
Longitudinal mean-field -hSz , l=2 h Sz
Energy (meV)1.0 1.2 1.4 1.6
Inte
nsit
y(a
rbun
its)
0.0
0.1
0.2
0.3
0.04 K
5 K
L = 0.00(5)
1
2
34
5
Zeeman ladder of bound states in 3D ordered phase
Bound states inconfining potential
↑ ↑ ↑ ↑ ↑ ↑ ↑ ↑ ↑
↑ ↑ ↓ ↓ ↓ ↓ ↓ ↑ ↑x x x xJ’
E
Zeeman ladder of bound statesT =40 mKT = 5 K
12
…
Continuum offree 2-soliton
statesx V(x) = l x
Soliton separationcosts energy
l ~ J’ Sz
Longitudinal mean-field -hSz , l=2 h Sz
Energy (meV)1.0 1.2 1.4 1.6
Inte
nsit
y(a
rbun
its)
0.0
0.1
0.2
0.3
0.04 K
5 K
L = 0.00(5)
1
2
34
5
Zeeman ladder of bound states in 3D ordered phase
Bound states inconfining potential
↑ ↑ ↑ ↑ ↑ ↑ ↑ ↑ ↑
↑ ↑ ↓ ↓ ↓ ↓ ↓ ↓ ↑ x x x xJ’
E
Zeeman ladder of bound statesT =40 mKT = 5 K
12
…
Continuum offree 2-soliton
statesx x V(x) = l x
Soliton separationcosts energy
l ~ J’ Sz
Longitudinal mean-field -hSz , l=2 h Sz
-10 -5 0 5
-0.5
0
0.5
z
Ai(
z)Soliton confinement
Schrödinger’s equation m m
McCoy&Wu (‘78)
=> bound states
kinetic energy string tension
zn = 2.33, 4.08, 5.52, 6.78…Airy function
Energy (meV)1.0 1.2 1.4 1.6
Inte
nsit
y(a
rbu
nits
)
0.0
0.1
0.2
0.3
0.04 K
5 K
L = 0.00(5)
1
2
34
5
Phenomenological model of soliton gas
J
↑↑↑↑↑↑
↑↑↑↓↓↓↑↑• •
- work perturbatively around the Ising limit
(spin clusters)2-soliton states
a 2 soliton state↑↑↑↓↓↓↓↓↓↑↑
= J ↑↑↑↓↓↓↓↓↓↑↑
-a ↑↑↑↓↓↓↓↓↓↓↑ +…
-b d n,1
from Ising J Sz Sz gap
soliton hopping
kinetic bound state
- Jxy ( Sx Sx + Sy Sy )i i+1
i i+1
from XY term↑↑↓↑↑ ↑↑↓↑↑+i+1 i-1
i i+1~ S+ S- + S- S+
i i+1 i i+1
H
RC et al, Science 327, 177 (2010)
Phenomenological model of soliton gas describes full spectrum
↑↑↑↓↓↓↓↓↓↑↑
T = 40 mK
Gap: J ~ 1.94 meV from Ising zz exchange
Bandwidth a = 0.12 J domain-wall hopping term[ microscopic origin SzSx … ?]
Kinetic bound state : transverse couplings for nn bond SxSx + SySy, J
┴/ J z = 0.24
and 2-nd neighbour AFM along chain J z' = - 0.15 J z
Weak confinement term: hz ~ 0.02 J longitudinal field includes interchain mean-field
numerical calculation agrees with exact analytic solution of effective HamiltonianS.B. Rutkevich, J. Stat. Phys (2010)
Experiments in applied transverse field
t
4 T3.25 T
↑↑↑↓↓↓↓↓↓↑↑• • BxSx = (S++S-)/2
Place crystal in metallic cage toprevent movement under high torque
CoNb2O6
crystal
Bz
torque
0 T
Gap decreaseswith field
wavevector
En
erg
y
- field tunes quasiparticle dispersion
Field ~ kinetic energy
Energy(meV)0.5 1.0 1.5 2.0
Inte
nsi
ty(a
.u.)
0
200
400
600
800
1000
Excitations as a function of transverse field
Single sharpmode
6 T4 T
Two-soliton states
7 T
→→←→→
Ordered Paramagnetic
Excitations changecharacter above
critical field
↑↑↑↓↓↓↓↓↓↑↑
Magnetic 3D LROBragg peak
Excitations intransverse field
2-soliton continuumRC et al, Science 327, 177 (2010)
Summary
- observed transmutation of quasiparticles at critical point
spin-flip2-soliton
states
- realized experimentally field-tuned quantum phase transition inquasi 1D Ising magnet CoNb2O6
ordered paramagnet
→←→
- neutron scattering is a very versatile probe of magnetic ordering anddynamics (meV->eV, magnetic field (15 T->25 T), low T (mK)- quantitative: probe dispersions of excitations through well-understood matrix element (quantitative comparison with theories)- sample size limited, large crystals needed (advanced crystal growth)- new neutron sources (ISIS 2nd target station, USA + JAPAN, ESS) willbring new opportunities: higher flux, higher resolution, wider coverage
Conclusions
50 detectors,2D data set
40,000 detectors,4D data set, 4energies measuredsimultaneously,
- complementary to resonant x-ray diffraction and inelastic (RIXS)(samples 10’s of mm, resolution >30 meV, T > few K – beam heating)
D.A. Tennant, (HZB Berlin)
Elisa Wheeler (Oxford)
Ewa Wawrzynska (Bristol)
M. Telling (ISIS)
K. Habicht, P. Smeibidl, K.Kieffer (HZB)
D. Prabhakaran (Oxford)
Ivelisse Cabrera (Oxford)
Jordan Thompson (Oxford)
R. Bewley, T. Guidi (ISIS)
C. Stock, J. Rodriguez-Rivera (NIST)
F. H.L. Essler, N. Robinson (Oxford)
Collaborators
CoNb2O6
Cs2CuCl4
S.M. Hayden (Bristol),
G. Aeppli (UCL)
T.G. Perring(ISIS)
C.D. Frost (ISIS)
T.E. Mason (Oak Ridge)
S.W. Cheong (Rutgers)
Z. Fisk (Florida)
La2CuO4
D. A. Tennant (HZB)
A.M. Tsvelik (Brookhaven)
K. Habicht, P. Smeibidl (HZB)
Z. Tylczynski (Poland)
Y. Tokiwa, F. Steglich (Dresden)