“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