Investigating The Use Of Quadrupolar Nuclei for NMR-Based ...
Nuclear magnetism and the electron state in low ... · interacting electrons hyperfine interaction...
Transcript of Nuclear magnetism and the electron state in low ... · interacting electrons hyperfine interaction...
Nuclear magnetism and the electron state in low-dimensionalconductors
Bernd BrauneckerUniversidad Autónoma de Madrid, Spain
in collaboration with (mainly)
Daniel Loss (Basel)Pascal Simon (Orsay)
Motivation: Why nuclear magnetism?
Electron spin trappedin quantum dot
Main culprit: Ensemble of ~ 105 nuclear spins within envelope of electronwave function
Well-defined quantum state may be used as a qubit.
Problem: Decoherence due to interaction withenvironment
B. Braunecker, Varenna 2012
Motivation: Why nuclear magnetism?
Decoherence substantiallyslowed down if nuclearspins are fully polarized
How to achieve polarization?
through coupling to electrons
electrons should do it themselves
B. Braunecker, Varenna 2012
Motivation: Why nuclear magnetism?
Decoherence substantiallyslowed down if nuclearspins are fully polarized
How to achieve polarization?
through coupling to electrons
electrons should do it themselves
required: many electrons
B. Braunecker, Varenna 2012
Motivation: Why nuclear magnetism?
look atnuclear spins in a conductor
B. Braunecker, Varenna 2012
Motivation: Why nuclear magnetism?
Can we obtain nuclear magnetic order
intrinsically through a phase transition?
B. Braunecker, Varenna 2012
3D metals: nuclear ferromagnet; old story Weiss mean-field theory (Fröhlich & Nabarro, 1940)
Dimensionality matters: interactions become important through restriction of scattering phase space
3D
2D
1D
2D: RKKY interaction renormalized through electron-electron interactions can stabilize nuclear magnetic order
1D: Renormalization is essential; electrons and nuclear spins can form a combined ordered state of matter
Can we obtain nuclear magnetic order
intrinsically through a phase transition?
B. Braunecker, Varenna 2012
2D
Order possible only due to non-Fermi liquid corrections by electron-electron interactions.
Estimated transition temperatures up to 10 – 100 μK for strong interactions.
Nuclear ferromagnet
Simon, Loss PRL 2007Simon, Braunecker, Loss PRB 2008
B. Braunecker, Varenna 2012
Novel collective phase: combined state of
nuclear spin helixhelical electron spin density wave helical conductor
Transition temperatures up to 100 mK possible in GaAs wires.
1D
Ferromagnetic locking on cross-section
2kF spin rotation along wire
Nuclear helimagnet
relevantbackactionbetween electrons andnuclear spins
schematic wire
Braunecker, Simon, Loss PRL 2009, PRB 2009
B. Braunecker, Varenna 2012
Novel collective phase: combined state of
nuclear spin helixhelical electron spin density wave helical conductor
Transition temperatures up to 100 mK possible in GaAs wires
1D
Ferromagnetic locking on cross-section
2kF spin rotation along wire
Nuclear helimagnet
relevantbackactionbetween electrons andnuclear spins
a bit more realistic wire
Braunecker, Simon, Loss PRL 2009, PRB 2009
B. Braunecker, Varenna 2012
Microscopic model
Microscopic model
interactingelectrons
hyperfine interaction
A ~ 90 μeV (GaAs)A ~ 0.6 μeV (13C)
dipolar/quadrupolar interactions< 0.1 neV (smallest energy scales)
A priori a tremendously complicated mixture of 3D nuclear spinsand low-dimensional electrons.
→ reduce to treatable effective model
B. Braunecker, Varenna 2012
Reduction to an effective model
Reduction to effective model (I)
interactingelectrons
hyperfine interaction
A ~ 90 μeV (GaAs)A ~ 0.6 μeV (13C)
dipolar/quadrupolar interactions< 0.1 neV (smallest energy scales)
electron spin nuclear spin
B. Braunecker, Varenna 2012
Reduction to effective model (II)
Focus only on the nuclear spins within the support of the confined electron liquid.
unoccupiedprojection ontolowest subband nuclear spins in
transverse direction (cross-section)
with
projectedelectron spin
compositenuclear spinof length I
The hyperfine interaction reduces to the coupling between projected electron spins and composite nuclear spins
Expand electron operators in basis of transverse modes t0, t1, ...
If there is order, energy is minimized if all spins in the composite spin are parallel: ferromagnetic on the cross-section
B. Braunecker, Varenna 2012
Reduction to effective model (III)
interactingelectrons
hyperfine interaction
A ~ 90 μeV (GaAs)A ~ 0.6 μeV (13C)
dipolar/quadrupolar interactions< 0.1 neV (smallest energy scales)
small number compared with EF ~ 10 – 100 meV
separation of time scales
• electrons move in static nuclear background (Overhauser field)• nuclear spins see an instantaneously reacting electron gas→ 2 coupled effective models
(similar to Born-Oppenheimer approx.)
electron spin nuclear spin
B. Braunecker, Varenna 2012
Reduction to effective model (III): RKKY
Schrieffer-Wolff transformation;integrate out electron degrees of freedom
static electron spin susceptibility
RKKY interaction Jij = J(ri-rj)long ranged
A / EF ~ 1/100 (or smaller): separation of time scales between electrons & nuclear spins
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Effective model
effective electron Hamiltonian
effective nuclear spin Hamiltonian
electrons in staticnuclear background(Overhauser field)
nuclear spins with RKKY interaction transmittedby electrons
B. Braunecker, Varenna 2012
Effective model
effective electron Hamiltonian
effective nuclear spin Hamiltonian
mutualdependence
novel physics
self-consistency
B. Braunecker, Varenna 2012
Mean-field analysis(naive but instructive)
Mean-field theory
similar to Fröhlich & Nabarro (1940)
Ground state determined by minimum of Jq
if at q = 0: ferromagnetif at q ≠ 0: helimagnet
Theory depends on single energy scale: TMF = |min(Jq)|, e.g. J0
B. Braunecker, Varenna 2012
Mean-field theory
similar to Fröhlich & Nabarro (1940)
If a system is characterizedby a single energy scale, thismean-field argument is valid
Thermodynamics controlled by single scale TMF
e.g., Curie temperaturefor ordering transition:
B. Braunecker, Varenna 2012
Order in : role of interactions2D
Order in 2D: role of interactions
susceptibility at T=0
Noninteracting electrons:Lindhard functionno unique minimum: no nuclear spin order possible
B. Braunecker, Varenna 2012
Simon, Loss PRL 2007Simon, Braunecker, Loss PRB 2008
Order in 2D: role of interactions
Interacting electrons:Non-Fermi-liquid corrections of self-energy lead to nonanalytic contribution, linear in |q|.
dominant self-energy renormalization of Cooper channel scattering amplitude
Chubukov, Maslov PRB 2003Aleiner, Efetov PRB 2006Saraga, Altshuler, Loss, Westervelt PRB 2005Shekhter, Finkel'stein PRB 06, PNAS 2006
susceptibility at T=0
Simon, Braunecker, Loss PRB 2008Chesi, Żak, Simon, Loss PRB 2009Żak, Maslov, Loss PRB 2010, 2012
B. Braunecker, Varenna 2012
Simon, Loss PRL 2007Simon, Braunecker, Loss PRB 2008
Order in 2D: role of interactions
Interacting electrons:Non-Fermi-liquid corrections of self-energy lead to nonanalytic contribution, linear in |q|.
susceptibility at T=0
However, the sign of the linear correction seems to be nonuniversal.
Possible other outcomes:
B. Braunecker, Varenna 2012
Simon, Loss PRL 2007Simon, Braunecker, Loss PRB 2008
Consequences
minimum at q = 0: nuclear ferromagnet MF scale: TMF ~ J0
interaction-induced scale
2 energy scales: pure MF theory not applicable
Which physics is described by this susceptibility?
B. Braunecker, Varenna 2012
Simon, Loss PRL 2007Simon, Braunecker, Loss PRB 2008
Fluctuations
MF scale: TMF ~ J0
interaction-induced scale
Spin-wave (magnon) analysis of fluctuations
T0 >> T* , independent of TMF
modified Bloch law
Calculate magnetization m per nuclear spin at T < T*
T0 provides an estimate of Tc, reaching up to the mK range - consistent with noninteracting limit: T* = 0 → T0 = 0 - TMF absent (but role not clear)
finite because ωq
is linear at small q
T0 ~ 1 mK for rs ~ 10
main message:m > 0 for 0 < T < T*
magnon spectrum: ωq = |J0 – Jq | ~ |q|
B. Braunecker, Varenna 2012 Simon, Loss PRL 2007; Simon, Braunecker, Loss PRB 2008
Generalized Mermin-Wagner theorem
Order in 2D: What about the Mermin-Wagner Theorem?
“No long-range order in 2D for sufficiently short-ranged interactions.”
● RKKY interactions are long ranged
● Is there some extension of the theorem?
Yes: General proof of absence of long-range order for"a wide class of models including any form of electron-electron and single-electron interactions that are independent of spin"
Loss, Pedrocchi, Leggett PRL 2011
Mermin, Wagner PRL 1966
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Generalized Mermin-Wagner theorem
Are we in trouble? — No!
1) Nonequilibrium situation:T=0 for electrons andT>0 for nuclear spins
2) Finite system size in μm rangeprovides cutoff (thermal lengthis larger)→ practically most important
3) Backaction of the nuclear spinscreates a small excitation gap→ fundamentally important
B. Braunecker, Varenna 2012
Backaction on electronsself-consistency
The ordered nuclear spins generate the Overhauser field, similar to a magnetic field
→ small electron polarization→ makes RKKY interaction spin-dependent and anisotropic→ does no longer fulfil the conditions of the theorem→ creates small excitation gap for the magnons
A very similar effect is obtained by spin-orbit interactions in the 2DEG.Żak, Maslov, Loss PRB 2012
True long-range order could actually be possible.
Simon, Braunecker,Loss PRB 2008
B. Braunecker, Varenna 2012
Summary of 2D
A nuclear ferromagnet in a 2DEG is possible and stable
– crucial is the non-Fermi liquid modification of the RKKY interaction by electron-electron interactions
– the self-consistent backaction between electrons and nuclear spins provides conditions that would in principle even allow for true long-range order
– estimates for transition temperatures are in the range 10 – 100 μK reaching for very strong interactions into the mK range
Simon, Loss PRL 2007Simon, Braunecker, Loss PRB 2008
B. Braunecker, Varenna 2012
Order in : spectacular results from strong non-Fermi liquid physics
1D
Electrons in 1D
Interacting electron system expressed by bosonic field theory
charge/spindensity fluctuations
conjugated fields (→ currents)
electron-electron interactions:strong renormalization of - velocities: compressibility/susceptibility- charge/spin fraction of density waves
electron Coulomb interaction
another electron
Model interacting electrons by Luttinger liquid
B. Braunecker, Varenna 2012
RKKY interaction from the Luttinger liquid
electron / nuclear spin density
thermal lengthinteraction-dependentexponent
for a Luttinger liquidthe electron spin susceptibilitycan be calculated explicitly
→ strongly susceptible at 2kF
→ determines RKKY interaction
cf. also Egger and Schoeller, PRB (1996)
B. Braunecker, Varenna 2012 Braunecker, Simon, Loss PRL 2009, PRB 2009
Nuclear spin ground state: Solve by inspection...
nuclear helimagnet
● minimize energy:all weight in Fourier modesq = ± 2kF
● ferromagnetic locking on cross-section: large spinof maximal length
stability?
minimum at ±2kF
B. Braunecker, Varenna 2012 Braunecker, Simon, Loss PRL 2009, PRB 2009
confirmed by magnon calculation;finite size cutoff is required (no long-range order),but transition temperature is independent of size
Stability of the helical order
thermal lengthdepth
depth & width determined by T
theory depends on a single scale:the mean-field argument applies
sets crossover temperature
width
B. Braunecker, Varenna 2012
Braunecker, Simon, Loss PRL 2009, PRB 2009
Nuclear helimagnet
Helical order of nuclear spins helimagnet along the wire
Order stable up to temperature T*
magnetization
generates spiral Overhauser fieldeffect on electrons?
B. Braunecker, Varenna 2012
Braunecker, Simon, Loss PRL 2009, PRB 2009
Electrons are susceptible at q = 2kF
Nuclear magnetic (Overhauser) field
spatial frequency q = 2kF
at which electrons are extremely susceptible
Overhauser field drives instability that exists in any 1D system
→ Peierls transition
self-consistency
B. Braunecker, Varenna 2012
Braunecker, Simon, Loss PRL 2009, PRB 2009
The electronic Peierls transition
one-dimensional conductor with external periodic potentialof spatial frequency 2kF
scattering between Fermi pointsopens gap
system becomes insulating
B. Braunecker, Varenna 2012
The spin-selective Peierls transition
external periodic potential:spiral magnetic fieldof spatial frequency 2kF
spin-selective scattering between Fermi pointsopens gap for 1/2 of the modes
1/2 of the system becomes insulating and forms a spiral density wave1/2 of the system remains conducting and forms a helical conductor
Δ
B. Braunecker, Varenna 2012
Braunecker, Klinovaja, Japaridze,Loss PRB 2010
Enhancement by electron-electron interactions
Feedback of Overhauser field on electrons
Bosonization treatment: relevant sine-Gordon interaction
Gap for
strongly enhanced
spiral electron spindensity wave;combines spinand charge fields
B. Braunecker, Varenna 2012 Braunecker, Simon, Loss PRL 2009, PRB 2009
Flow to strong coupling
effective Overhauserfield flows to strongcoupling limit
ξ : correlation lengthprecise form depends on material
renormalization absent in Fermi liquids (Kc = Ks = 1)
coupling constantRG equation for dimensionless
B. Braunecker, Varenna 2012
Braunecker, Simon, Loss PRL 2009, PRB 2009
Changed system properties
1) enhanced gap Δ* µ A* for mode
2) remaining gapless mode
• renormalized Luttinger liquid• spin-filtered; helical conductor• RKKY interaction maintains its
singular shape at 2kF
• Jq becomes deeper
R ↑
L ↓
Δ*
B. Braunecker, Varenna 2012
Braunecker, Simon, Loss PRL 2009, PRB 2009
Renormalized RKKY interaction
- same shape- much deeper (modified exponents)- boosts T*
RKKY interaction determined by remaining gapless modes
modified exponents / prefactors
B. Braunecker, Varenna 2012
Braunecker, Simon, Loss PRL 2009, PRB 2009
Renormalized RKKY interaction
- same shape- much deeper (modified exponents)- boosts T*
RKKY interaction determined by remaining gapless modes
T* still lower than renormalized A*: order vanishes by melting of nuclear spin alignment
T* = 10 – 100 mK GaAs quantum wires (depending on interaction strength)
T* ~ 10 mK carbon nanotubes
B. Braunecker, Varenna 2012
Braunecker, Simon, Loss PRL 2009, PRB 2009
Combined electron / nuclear spin order
pinned electron density wave (electron spin helix)
nuclear helimagnet
Below T* the ordered phases depend on each other:
- T* ~ 10 – 100 mK - huge renormalization due to electron-electron interactions- gapless modes: helical conductor
Phase of tightly bound electron & nuclear spin degrees of freedom
B. Braunecker, Varenna 2012 Braunecker, Simon, Loss PRL 2009, PRB 2009
Further consequences
Reduced conductance, helical conductor
Through backaction: Pinning of channels
Blocking of ½ of the conducting channels
Universal reduction of conductance by factor 2
Remaining channels are helical
R ↑
L ↓
spin-filter
s-wave superconductor
Majorana bound states
Still requires a complete self-consistent stability analysis!
B. Braunecker, Varenna 2012
Anisotropy in electron spin susceptibility
Overhauser field defines spin (x,y) plane
Anisotropy between spin (x,y) and z directions
B. Braunecker, Varenna 2012
Braunecker, Simon, Loss PRL 2009, PRB 2009
Unusual "irregular" density of states
Braunecker, Bena, Simon PRB 2012Schuricht PRB 2012
Local (tunneling) density of states has an "irregular" contributionarising from coupling to the gapped sector
Perfect helical LL: regular LL behavior
Helical LL in combined ordered phase(spiral LL): regular LL behavior + irregular contribution (pseudogap)
Possible to use local DOS to prove existence of nuclear orderor of helical states in nanowires
regular behaviorbelow gap
irregularbehaviordominatesgap edge
irregular exponentchanges signfor strong interactions
B. Braunecker, Varenna 2012
Summary of 1D
A nuclear helimagnet in a 1D wire is possible and stable
– crucial is the backaction between electrons and nuclear spins: it restructures the electron state and the resulting order consists of
– nuclear spin helix
– helical electron spin density wave
– helical electron conductor
– electron interactions strongly renormalize and stabilize the state
and transition temperatures up to 100 mK in GaAs are possible
Braunecker, Simon, Loss PRL 2009, PRB 2009
B. Braunecker, Varenna 2012
Global Conclusions
3D
2D
1D
2D: RKKY interaction renormalized through electron-electron interactions cana) overrule generalized Mermin-Wagner Theoremb) stabilize nuclear magnetic order
1D: Renormalization is essential; electrons and nuclear spins can form a combined state of matter
Simon, Loss PRL 2007Simon, Braunecker, Loss PRB 2008Braunecker, Simon, Loss PRL 2009, PRB 2009
B. Braunecker, Varenna 2012