“Listening” to the spin noise ofelectrons & holes in semiconductors
(What can we learn about spin dynamics & magneticresonance without ever perturbing the system?)
Scott CrookerLos Alamos National Lab
‘Listening’ to the spin noise of electrons & holesin semiconductors (& alkali atomic vapors…)
S. A. Crooker, D. L. Smith (Los Alamos Nat’l Lab)J. Brandt, C. Sandfort, A. Greilich, D. R. Yakovlev, M. Bayer (TU-Dortmund)
D. Reuter, A. D. Wieck (Uni. Bochum)
Outline• Physics goal: Measure intrinsic fluctuations of spins in thermal equilibrium• Fluctuation spectra reveal dynamical properties (g-factors, decoherence)• Spin noise spectroscopy of electrons in n-GaAs & holes in InGaAs QDs• Real-time spectral analysis with all-digital processing & FPGAs
Nature 431, 49 (2004)PRB 79, 035208 (2009)PRL 104, 036601 (2010)
0 100 200Frequency (MHz)
spin
noi
se
“Noise spectroscopy”: a simple exampleSimple mechanical system: Cantilever (diving board)
• What’s resonant frequency (ω0)?• What’s ringdown time (Q)?
Method 1Perturbative methods…
Measure dissipative response
F
timedisp
lace
men
t
Method 2“Listen” carefully to intrinsic thermal
fluctuations (vibration noise)Measure <δx(0) δx(t)>
δx(t)
• What’s resonant frequency (ω0)?• What’s ringdown time (Q)?
o sed a t
timedisp
lace
men
t
“Noise spectroscopy”: a simple exampleSimple mechanical system: Cantilever (diving board)
Method 2“Listen” carefully to intrinsic thermal
fluctuations (vibration noise)Measure <δx(0) δx(t)>
δx(t)
• What’s resonant frequency (ω0)?• What’s ringdown time (Q)?
“Noise spectroscopy”: a simple exampleSimple mechanical system: Cantilever (diving board)
ω0
Γ ~ 1/Q
Noi
se p
ower
(dis
p/H
z)Frequency (Hz)
Fluctuation-Dissipation Theorem: ‘Spectrum of fluctuations completely describes the driven response’
Nanometer-scale diving board (mechanical resonator)K. Schwab (Science, 2004)
** Noise signatures become an increasing fraction of “driven” signal as things get small **
Magnetic analogy: stochastic “spin noise”Normally, spin dynamics revealed with spin resonance, or pump-probe studies: Pump (drive) is necessarily perturbative
pump laser (circular pol.)
probe laser
sample
detector
…then measure (dissipative) response:• First inject, tip, pump, etc…
pump laser (circular pol.)
probe laser
sample
detector
Indu
ced
spin
pol
ariz
atio
n Measure dynamics:T2
*, g-factors, etc
Time (ps)
0
Magnetic analogy: stochastic “spin noise”Normally, spin dynamics revealed with spin resonance, or pump-probe studies: Pump (drive) is necessarily perturbative
z
Consider N uncorrelated spins in equilibrium… 0)( =tSzIn thermal equilibrium,
)0()( zz StSCorrelation function:
NtSz ~)]([ 2But fluctuations exist:
Fluctuation-dissipation theorem:“Linear response of a system to external perturbation (ie, the susceptibility) can be described by the fluctuation properties of the system while in thermal equilibrium”.
dteStS tizz
ωωχ −∞
∞−∫ )0()(~)("
• In principle: spin noise alone completely describes dynamics
Dynamics also available via stochastic “spin noise”
time
Spi
n pr
ojec
tion
0
+N
-N
on page 2…
In 1946…
39 years later…
Proc. Nat. Acad. Sci. 103, 6790 (2006)
PNAS 106, 1313 (2009)
• Using optics, fluctuations in N↑ - N↓ are readily measurable in alkali vapors (Rb, K)- E.B. Aleksandrov (Sov. Phys. JETP, 1981)- T. Mitsui (PRL, 2000)
• Spin noise imposes a fundamental limit on accurate measurement of ensemble spin(e.g., quantum non-demolition measurements, femtoTesla magnetometers)
- Sorensen (PRL, 1998); Kuzmich (PRA, 1999)-Romalis (PRL 2010)
Measuring electron spin noise in atomic alkali vapors
• Here: frequency spectrum of spin noise for non-perturbative magnetic resonance(in accord with fluctuation-dissipation theorem).
• Inverse scaling with interaction volume – potential for systems w/ few spins.
z
Mz
time0
+N
-N
Spin || k : RCP
Spin || -k : LCP
• Alkali atoms (Rb, K, Cs): one unpaired electron in outermost S-shell• Spin-orbit splitting of atomic P levels (P1/2, P3/2)• Angular momentum selection rules in alkali atoms:
Measuring magnetization through optical Faraday rotation
5P1/2
5P3/2
Rubidium D1 line (~794 nm)
F=I+½
F=I-½5S1/2
Rubidium D2 line (~780 nm)
n+
n−
Faraday rotation: ΘF(t) ∝ (n+-n-) ∝ N+ - N - ∝ Mz(t)
Energy
α−
α+↓↑ ≠ NNIf ,
“up”
“down”
…absorption α+≠α-
…& indices n+≠n-
• Laser tuned near -- but not on – resonance: No absorption.
“Listening” to magnetization fluctuations (spin noise)Paramagnetic alkali vapors – a well-understood, classical ensemble of N uncorrelated spins
• Random magnetization fluctuations δMz(t) generate noise in Faraday rotation δθF(t).
• Alkali vapor (Rb, K) in buffer gas, in thermal equilibrium (T~350K). <Mz(t)>=0.
• Measure spin correlation function, S(t)=<Mz(0) Mz(t)>, without perturbing system• Spin ensemble always remains in thermal equilibrium
(in contrast with conventional magnetic resonance)
spectrumanalyzer
δθF(t)
rubidium orpotassiumN~109 mm-3
I+45
x
zI-45
B (<10 gauss)
cw Ti:S laserδV(t)
Nature 431, 49 (2004)
Small transverse magnetic field…
spectrumanalyzer
δθF(t)
rubidium orpotassiumN~109 mm-3
I+45
x
zI-45
B (<10 Gauss)
cw Ti:S laserδV(t)
δMz(t) δMz(t)z
• Shifts peak of the noise from 0 Hz to MHz frequencies• Precession corresponds to coherence between Zeeman sublevels
F=I+J
F=I-J
2S1/2
-101
mF
2
-2
01
-1
ωL = Larmour frequency geμBB/ћAny spin fluctuationis forced to precess
85Rb 87Rb21=Fg3
1=FgF=I+1/2
F=I-1/2
Δhf
-101
mF2
-2
01
-1
12 +=
Ig J
200
240
0.75 1 1.25 1.5Frequency (MHz)
Fara
day
rota
tion
(nan
orad
/Hz1/
2 )
25
30
B=1.85 G
Spin
noi
se (n
V/H
z1/2 )
5 2S1/2
5 2P1/2
Rb D1 transition~794.6 nm
Bhg
BμΩ
=
Spectral density of spin noise
• Tune laser near D1 transition & “listen”…• Peaks in noise spectrum due to random, precessing fluctuations of ground-state spin
Nature 431, 49 (2004)
~13 kHz (Τ2
∗ ~ 100 μs)
Rubidium
0
10
20
30
40
-100 -50 0 50 100Laser detuning (GHz)
D2
~780
.0 n
m
Inte
grat
ed sp
in n
oise
(μV
)
D1
~794
.8 n
m
52S1/2
52P1/2
52P3/2
( )( )eJmfeNn βνπ ±Δ≅−± 14/1 20
cnnLF /)( −+ −= πνθFaraday rotation:RCP, LCP refraction indices:
ALN
mcfe
F0
22 1
4 Δ=
βδθr.m.s fluctuations:
Off-resonant Faraday rotation passively probes <M(0) M(t)>
Confirm 1/Δ (detuning)
dependence
170
190
2.6 2.7 2.8
Spin
noi
se (n
V/H
z1/2 )
56789
10
20
108 109 1010In
tegr
ated
spin
noi
se (μ
V)
0~ N
85Rb density (mm-3)
…integrated spin noise scales with sqrt[N]
364.0 K
359.1 K
354.3 K
349.4 K
339.6 K
MHz
Tune atomic density N0with temperature…
Spin noise scales with square root of particle number
Fluctuations from N uncorrelated, precessing spins ~sqrt[N]
85Rb5.8 G
170
190
2.6 2.7 2.8
0.238 mm20.105 mm2
0.060 mm2
0.040 mm2
0.030 mm2
0.024 mm2
MHz
56789
10
20
10-2 10-1 100
Beam area (mm2)
V1~A
Spin noise increases when probe area shrinks
Vary beam area, keeping laser power constant
Potential for small systems, with small N
Spin
noi
se (n
V/H
z1/2 )
85Rb5.8 G
230
240
250
260
26 26.5 27Frequency (MHz)
nuclear Zeemanquadratic Zeeman1
2
0
1
0
1
-1
0
-2
-1
-1
0
Spin
noi
se (n
V/H
z1/2 )
• At higher BT (~40 G), electron & nuclear spin decouple: noise peak splits into discrete peaks.• Noise coherences within F=2, F=1 hyperfine levels are no longer exactly degenerate
Spin noise reveals complex magnetic ground states: Hyperfine splittings (Δhf) & nuclear magnetism (μI, gI)
87Rb, I=3/2
F=2
F=1
5 2S1/2
Δhf=6835 MHz
-101
mF
2
-2
01
-1
87Rb38 G
BgIBE NIInuclear μμδ 2/2 ==
Fine structure in noise reveals:Hyperfine energy:
Nuclear moment:
ΩΩ≅Δ δ/2 20hf
450 460 470
Spin
noi
se (a
.u.)
Frequency (MHz)
1
2
-1
-2
0
-1
-1
0
1
00
1
39K
5.27 G
2.88 G
0.81 G
F=2
F=1
4 2S1/2
39K (I=3/2)
Δhf
-101
mF2
-2
01
-1
Δhf=461.7 MHz
Spin fluctuations generate high-frequency inter-hyperfine coherences
Spontaneous spin coherences also exist between hyperfine levels
What about spin noise in condensed-matter?Start with semiconductors…
pump laser (circular pol.)
probe laser
sample
detector
Indu
ced
spin
pol
ariz
atio
n Measure dynamics:T2
*, g-factors, etc
Time (ps)
0
Normally, spin dynamics revealed with pump-probe optics, or electron spin resonance: Pump is necessarily perturbative
• Laser tuned below 820 nm GaAs bandgap (840-860 nm): No absorption• Random spin fluctuations δSz(t) generate noise in Faraday rotation δθF(t).
• Bulk n-type (Si-doped) GaAs in thermal equilibrium (1.5–100K). <Sz(t)>=0
• Measure power spectra: Spins remain in thermal equilibrium (in contrast with conventional pump-probe studies of spin dynamics)
spectrumanalyzer
δθF(t) I+45
xz
I-45n-type GaAs
(ne~1016-1017 cm-3)
cw Ti:S laserδV(t)
“Listening” to electron spin fluctuations in n-type GaAsSpin noise magnetometer based on optical Faraday rotation
• Free electrons in the conduction band of n-type GaAs have spin ±1/2
n+
n-
…& indices n+≠n-
Measuring electron spin noise in n-GaAs with Faraday rotation
Spin || k : RCP
Spin || -k : LCP
“Up” spins couple to σ+ light
“Down” spins couple to σ- light σ+ σ-
C.B.
V.B.
Faraday rotation: θF(t)∝ n+-n- ∝ N↑ - N↓ ∝ Mz(t)Probe laser can be tuned far from absorption, but still measure spin via n+-n-
In this regard, it is “non-perturbing” probe
hhlh
split-off
PRB 79, 035208 (2009)• Lightly dope GaAs with silicon (~1016-17 / cm3)
energy
If N↑≠N↓, …absorption α+≠α-
α−
α+
• Angular momentum selection rules for RCP/LCP light:
Bx (0-300 G)
δSz(t) δSz(t)z
ωL = Larmour frequency geμBB/ћ
• Shifts peak of the noise from 0 Hz to MHz frequencies
Also: small transverse magnetic field…
spectrumanalyzer
I+45
x
zI-45
cw Ti:S laserδV(t)
ћωL
Any spin fluctuationis forced to precess
Spin noise of conduction electrons in bulk n-GaAsPRB 79, 035208 (2009)
0 50 100 150Frequency (MHz)
Fara
day
rota
tion
nois
e po
wer
(nra
d2/H
z) spin noise
“white” photon shot noise
• Spin noise signals are small! 10-1000x less than photon shot noise (nanoradians/√Hz)• Significant signal averaging required (hours): Efficient use of available data stream
0
Spin noise of conduction electrons in bulk n-GaAsPRB 79, 035208 (2009)
•Frequency gives g-factor•Width gives spin dephasing•Area gives # spins involved-not all spins, b/c of Fermi stats
0 50 100 150 200Frequency (MHz)Fa
rada
y ro
tatio
n no
ise
pow
er (n
rad2 /H
z)
ne=3.7x1016
T=10KBx=175G
Γ=(πτs)-1
Noise power is Lorentzian10-17
0
ωL
Increasing Bx: 0, 50, 100… 300 G
Spin noise of conduction electrons in bulk n-GaAsPRB 79, 035208 (2009)
Measure dynamical information (g-factor, spin lifetime) without perturbing spins
Inverse scaling with probed volume – fewer spins give more noise
Noise signal is a larger fractionof saturated signal when probingfewer spins.
PRB 79, 035208 (2009)
~106 spins)]()([)( λλ
λπλθ −+ −= nnL
F
)]()([1 λλλ
π ↓↑ −Δ
∝ ee NNL
ALNN e ××=
spin densities
Total rms noise:
Spin noise in a Fermi sea: Temperature dependence
• Electrons in n-GaAs obey Fermi-Dirac statistics• Only electrons within kBT of EF can fluctuate
It does not. There is an offset at T=0
PRB 79, 035208 (2009)
Possibly due to the embedded Si donors:Localized electrons won’t obey Fermi-Dirac stats
• Noise power should increase linearly with T,and vanish as T->0.
kBT
Energy
f(ε)0
1
EF
∝ T
fraction of electrons that can fluctuate
ne=7.1 x 1016
ne=3.7 x 1016
ne=1.4 x 1016
0 50 100 150Frequency (MHz)
Fara
day
rota
tion
nois
e po
wer
(nra
d2/H
z)
Real-time digital spectral analysis: on-board FPGA processing
• Spin noise signals are small! 10-1000x less than photon shot noise (nanoradians/√Hz)• Significant signal averaging required (hours): Efficient use of available data stream
• Conventional “sweeping” RF spectrum analyzers ignore ~99.9% of data streamAsk the radio astronomers:
Real-time FFT analysis with Field-Programmable Gate Arrays (FPGA)
“Configurable hardware”
spin noise
“white” photon shot noise
Real-time digital spectral analysis: on-board FPGA processing
2 GS/seconddigitizer
Parallel 32kptpipelined FFTs
On-boardaccumulation
• 1 GHz real-time spectral bandwidth (no experimental “dead time”)• ~1000 times faster than conventional RF spectrometer• 1 terabyte of data processed every 8.3 minutes• picoradian/root-Hz Faraday rotation noise sensitivity from 0-1 GHz
Phys. Rev. Lett.104, 036601 (2010)
Real-time digital spectral analysis: on-board FPGA processingPRL 104, 036601 (2010)
Noise spectroscopy of localized spins confined in quantum dots
Problem with n-type GaAs: Itinerant (free) electrons move.
Use quantum dots to localize single electrons & holes-localized electrons/ holes have long spin lifetimes
GaAsInAs
conduction band
valence band
+ + + + + + + + + + + + + + + Remote doping (n- or p-type)
GaAs / InAs / GaAs
Spin noise in (nominally) undoped InGaAs/GaAs quantum dots
InGaAs QDs Tune probe laser within PL band
& “listen”…
g-factor corresponds to holes
Inhomogeneous(ensemble)
broadening ->
Spin noise peak that evolves with Bx…
Phys. Rev. Lett.104, 036601 (2010)
Marked anisotropy of in-plane hole g-factor, g⊥
[110]
[1-1
0]Bx
Anisotropy ~45%
Hole g-factors: much more sensitive to local confinement potential than electronsE.g., Pryor & Flatte, PRL (2006)
Sheng & Hawrylak, PRB Rapid Comm (2008)
PRL 104, 036601 (2010)
Electron spin noise in InGaAs/GaAs quantum dotsElectron spin noise appears when dots are also weakly illuminated with above gap radiation (at 1.58 eV, or 785 nm)
PRL 104, 036601 (2010)
Precedent for n-type doping w/ additional above-gap illumination:-GaAs: Zhukov et al, PRB 79, 155318 (2009)-CdTe: Syperek et al, PRL 99, 187401 (2007)
Spin noise in ferromagnetic systems A real-world problem in hard disk drive read-heads
• Magnetization fluctuations also present in ferromagnets (spontaneous FMR)
• Magnetization noise > Johnson noisein small (<1 μm2) read-heads
• Scales inversely with head volume• Peaked at FMR frequency• Imposes fundamental limit on S:N ratios
in next-generation hard drives.
- Detailed lineshape of calculated spin noise probes correlations in condensed phase- gaps in spectrum reveal binding energy- tails reveal spectrum of excitations
B. Mihaila, S. Crooker, P. Littlewood, D. Smith, PRA (2006)
Spi
n no
ise
pow
er (a
.u.)
- Probes the correlated quantum systems in non-perturbative way
BEC-like
BCS-like
Spin noise in the presence of correlations: Application to ultracold fermionic atomic gases
- Ultracold fermions (40K, 6Li) pair up to form bosons, which form BEC or BCS state.
(kFa)-1 =
Summary
• General result: Most properties within kBT of ground state revealed in the noise.-Don’t have to perturb your system in order to extract dynamics- g-factors, coherence times, etc- in accord with fluctuation-dissipation theorem
• Ensemble remains unperturbed & in thermal equilibrium- a route towards “sourceless” magnetic resonance?
0 100 200Frequency (MHz)
spin
noi
se
• Inverse scaling with interaction volume- fewer spins, more noise! - intrinsic sqrt[N] fluctuations are increasingly important as N->1
• Noise spectroscopy as a useful probe of solid-state systems with few spins?-Eg, only ~100 holes being probed in these InGaAs dots
… spin noise of a quantum mechanical origin
•Nanometer-scale ferritin particles (antiferromagnetic)•Néel vector tunnels between two equivalent energy minima noise
susceptibility