Nuclear magnetism in quantum dots

Alexander Tartakovskii

Department of Physics and Astronomy University of Sheffield, UK

Central spin problem in a III-V quantum dot

Behaviour of QD-spin-qubit is a central spin problem: electron or hole spin interacts with a bath of 105-106 nuclear spins

Nuclear B-fields up to 5T with fluctuating part of ~20 mT

Single electron or hole in a QD can be used as a spin-qubit

III-V semiconductor quantum dot (QD) - a complex solid state system with 105-106 atoms

20 nm

5 n

m

Context and motivation

Context: spin qubits in semiconductor quantum dots (QDs)

- Electron spin coherence in QDs: most spectacular results in gated unstrained GaAs QDs using electrical control at mK temperatures, electron spin T2~200 s (Bluhm Nat Physics) - Strained self-assembled QDs (SAQDs) are suitable for optical as well as electrical control (Press Nat Photonics, De Greve Nat Physics) - Nuclei are a major source of spin decoherence in InGaAs SAQDs (see eg Kuhlmann Nat Physics for work on SAQDs)

- In order to answer how the nuclear spin bath dynamics influence the central spin coherence, we need to understand properties of nuclear spins in QDs – for this we use NMR

Main theme of this talk

1. NMR basics

2. Magneto-spectroscopy of quantum dots

3. Dynamic nuclear polarisation in QDs

4. Optically detected NMR in QDs: nano-NMR

5. Coherent NMR spectroscopy in strained QDs

Talk outline

All experiments on single quantum dots

Conclusion of our recent studies: strain self-assembled QDs (or disordered/strained semiconductors in general) are overlooked for their potential for realisation of spin qubits

1. Basics of NMR

-1/2

+1/2

Basic ideas behind NMR

Bz=0 Bz≠0

MZ

BZ

Spin 1/2 example

zBE

gyromagnetic ratio

Brf

“Rotation” of magnetisation around oscillating in-plane Brf

BZ

)cos( tBBB zxrf

Conventional approach: time domain measurement

Conventional NMR experiment

MZ

polarize (T1)

BZ

precession (T2*)

RF waveform source

/2 pulse using Brf

Signal from the pick-up coil is Fourier-transformed and NMR spectrum is obtained

)cos( tBBB zxrf

Limitations of conventional NMR

Relatively low signal due to small difference in spin level populations The minimum number of nuclei studied in conventional NMR is around 1014-1015

1000 MHz spectrometer (23.4 Tesla)

-1/2

+1/2

BZ

In the following we consider nuclei with I1 in the solid state. NMR spectra for such nuclei are broader and require high sensitivity for the measurement

Nuclei with spin I1: Zeeman effect

Bz=0

Strong field Bz (~Tesla)

If a weak in-plane rf B-field is applied at resonant frequency L, adjacent spin states couple: 1 transitions lead to change of <I>

-3/2

-1/2

+1/2

+3/2

Example for spin 3/2 nuclei (e.g. Ga, As) with 1/2,3/2 projections: equidistant splittings

See eg Slichter “Principles of magnetic resonance”

Bz≠0

L

L

L

2

zL

B

)cos( tBBB zxrf

Nuclei with spin I1: Quadrupole moment

Nuclei with spin I1 (1, 3/2, 2, 5/2...) have non-spherical shapes and possess an electric quadrupole moment

Urbazsek Rev.Mod.Phys. (2013) Electric quadrupole moment couples to inhomogeneous electric fields with non-zero electric field gradients occurring in the solid state due to the strain or alloy disorder

50 nm QD

GaAs on Si

NMR in the presence of quadrupole effects

Important for 69Ga, 71Ga, 75As (all spin 3/2), 115In (spin 9/2)

Absent for 31P (spin 1/2)

)1(36

1ˆˆˆ 2 I-IIνIνHHH zQzLQZ

-3/2

-1/2

+1/2

+3/2

L

L

L L-Q

L

L+Q

Zeeman

Quadrupole

2I NMR lines in the case of homogeneous strain transform in broad satellite bands for inhomogeneous strain/alloy disorder

QD chemical composition and strain is revealed in a non-invasive fashion

What is NMR in QDs useful for?

Gives new insights in the spin properties of holes and electrons

Nuclear spin coherence time – an upper limit to Central Spin coherence time?

NMR provides information about nuclear spin system and its interactions with the ‘Central Spin’ (spin-qubit)

Provides fast (s) control of large nuclear fields experienced by the Central Spin

2. Magneto-spectroscopy of quantum dots

Quantum dots: structural properties

50 nm

Self-assembled InP/GaInP and InGaAs/GaAs QDs

GaAs 9 ML

Interface GaAs/AlGaAs QDs

Crystal growth by MBE and MOVPE

Shchukin and Bimberg, Rev.Mod.Phys. (1999); Peter PRL (2005)

QD consists of ~105 atoms

Quantum dots: optical properties

Important scenario: uncharged (neutral) QD Optical properties tested using photoluminescence (PL)

photon

Uncharged QD

1.3600 1.3602 1.3604

Photon energy (eV)

Narrow PL lines of 20-80 eV in single QDs

laser

See eg Warburton Nature (2000) v. b.

c. b.

Magneto-optics experiments

Optical excitation and photoluminescence (PL) detection along Oz, magnetic field along Oz (Faraday geometry)

Oz

Ox

Sample with QDs

Oy

External magnetic field Bz

Optical excitation

PL detection

QD sample

3D piezo positioner

lens

laser beam

Micro-PL

Review on spins in QDs: Warburton Nature Materials (2013)

QD sample

3D piezo positioners (Attocube)

lens

laser beam

QD sample

Magneto-optics experiments

Micro-PL is needed to address individual dots

3. Dynamic nuclear polarisation in quantum dots

Overhauser field BN

electron

Btot=BextBN

nuclear depolarisation

Dynamic nuclear polarisation (DNP) in QDs

BN can be measured by all-optical methods through Overhauser shifts (max ~100-150 eV): BN up to 5T

Contact hyperfine interaction:

i

iiihf ISψAv

H ˆˆ)(8

20 R

totBeeZ BgE

1.8275 1.8280

EZ

PL energy (eV)

InP/GaInP

Bz=6T

1.3600 1.3605

EZ

+

-

PL

In

ten

sity (

arb

. u

nits)

InGaAs/GaAs

Bz=5.3T

PL hyperfine shifts in various QDs

Lai PRL (2006), Eble PRB (2006), Braun PRB (2006), Tartakovskii PRL (2007), Maletinsky PRB (2007), Urbaszek PRB (2007), Skiba-Szymanska PRB (2008), Latta Nature Physics (2009), Belhadj PRL (2009), Chekhovich PRL (2010), Klotz PRB (2010); Chekhovich PRB (2011)

Reviews: Urbaszek Rev. Mod. Phys. (2013), Chekhovich Nat. Materials (2013)

iiieHS IAE

Hyperfine shifts (EeHS) of the electron spin states with contributions from all nuclear subsets can be observed in photoluminescence:

DNP under non-resonant excitation

Braun PRB (2006), Maletinsky PRB (2006), Tartakovskii PRL (2007), Urbaszek PRB (2007), Skiba-Szymanska PRB (2008)

0 50 100 150 200

150

200

250

Bext

=2T

Exciton

Ze

em

an s

plit

ting

(

eV

)

Incident power (arb. units)

Btot=Bext-BN

Nucle

ar s

pin

bistability

1.3180 1.3184

PL

Bext

=2.5T

ExZ

(-)

Incid

en

t p

ow

er

Energy (eV)

Nuclear polarisation >50% routinely obtained in QDs Various non-linear phenomena occur

T1 of nuclear polarisation in SAQDs

Maletinsky PRL (2007), Belhajd (2008), Makhonin PRB (2008), Nikolaenko PRB (2009), Chekhovich PRB (2010), Latta PRL (2011)

Large nuclear polarisation in a dot may be created in a few ms to few s under optical pumping Nuclear polarisation in a self-assembled QD is extremely long-lived, 103-105 s

Indication of the effect of strain – suppression of spin diffusion

510 540 570

101

103

105

200 mK

4 K Bz=5T

D

eca

y t

ime

(s)

Vwait

(mV)

InGaAs QDs, Latta PRL (2011)

Summary on optics and DNP

Single-electron/hole states in individual QDs can be easily accessed in optics experiments Sharp optical resonance of individual QDs (eg in photoluminescence) makes them perfect sensors for nuclear polarisation Large degrees of nuclear polarisation (typically up to 50% or higher) can be easily created in a QD by optical pumping Nuclear polarisation in self-assembled QDs is extremely long-lived, 103-105 s – indication of the effect of strain

4. Optically detected NMR in QDs:

nano-NMR

ODNMR in unstrained GaAs dots

1.7074 1.7076 1.7078

PL

in

ten

sity RF off

RF on

Bext

=2T

Photon energy (eV)

laser

PL

Sample

Gammon Science (1997), Makhonin PRB (2010), Makhonin Nature Materials (2011)

57.75 58.00

645

650

655

S

pe

ctr

al sp

littin

g (e

V)

75As

81.00 81.25 Frequency (MHz)

69Ga

103.00 103.25

71Ga

Bz=8T

NMR in InP/GaInP QDs: first attempts 31P spin ½, not sensitive to strain

115In has spin 9/2 and very large quadrupole moment: very sensitive to inhomogeneities of strain

48.0 48.1 48.2 48.3

255

260

265

31P

Spectr

al split

ting (eV

)

Radio frequency (MHz)

Bext

=2.8T

4.6x107

4.8x107

5.0x107

5.2x107

-4

-2

0

OD

NM

R S

ign

al (

eV

)

Radio frequency (Hz)

Bext

=5.508 T

115In tabulated

frequency

115In

For ODNMR in QD ensembles see work by Dortmund: Cherbunin PRB (2011), Flisinski PRB (2010), Kuznetsova PRB (2014)

Data in both figures for an individual InP QD

Special NMR method for strained QDs

Chekhovich Nature Nanotechnology (2012)

44 48 52 56

0

10

20

30

40

rf frequency, (MHz)

69GaN

MR

Sig

nal (

eV

)

115In

“Inverse” NMR, 280 kHz resolution: quadrupole isotopes spectra with rich structure Conventional “saturation” NMR, 450 kHz resolution: only weak line from 115In

InP/GaInP single QD

ODNMR in strained InP/GaInP dots

Chekhovich Nature Nanotechnology (2012)

Inverse method leads to signal enhancement by (I+1/2)3 compared with “saturation” NMR

Structural analysis using ODNMR

44 48 52 56

0

10

20

30

40

68 72

91.5

991.6

2

36 40 44 48 52 5668 72

wgap

=250kHz

InP/GaInP

-

69Ga

Bz~5.3 T

NM

R S

ign

al (

eV

)

115In

+

71Ga

31P

rf frequency, (MHz)

InGaAs/GaAs

75As

69Ga

115In

71Ga

Spectral widths: 115In ~20 MHz, 31P 3-10 KHz

Composition and strain distribution measured in the volume occupied by the electron

35% Ga, 65% In in an InP/Ga0.5In0.5P QD

Average quadrupole shifts,νQ: 1.2 MHz for InP

From this, average strain inside the dot, |εb|: 2.2% for InP

ODNMR in strained InGaAs/GaAs dots

44 48 52 56

0

10

20

30

40

68 72

91.59

91.62

36 40 44 48 52 56

0

5

10

15

68 72

NM

R S

ign

al (

eV

)

69Ga

Bz~5.3 T

NM

R S

ign

al (

eV

)

115In

+

71Ga

31P

rf frequency, (MHz)

InGaAs/GaAs

75As

69Ga

115In

71Ga

80% Ga, 20% In in an InGaAs/GaAs QD (nominally grown as InAs/GaAs QD)

Average quadrupole shifts,νQ: 1.4 MHz for InGaAs

Average strain inside the dot, |εb|: 2.7% for InGaAs

High resolution NMR in strained QDs

Central transition -1/2+1/2 affected by 2nd order quadrupole interaction

38.6 38.8 49.6 49.7 49.8 69.10

69.15

0

2

4

rf frequency, (MHz)

75As

115In

71Ga

NM

R S

ignal (

eV

)

71Ga: narrow (8 kHz) line 115In: broadening due to large quadrupole moment 75As: effect of random configurations of In and Ga

Shape (asymmetries) and width of CTs carry wealth of structural information: see Bulutay arxiv (2014)

Theory: high sensitivity to QD shape and distribution of In

Bulutay arxiv (2014)

Example of NMR ‘application’: element-sensitive hole hyperfine interaction

-40 -20 0 20 40

-4

0

4

-60 -40 -20 0 20 40 60

-10

-5

0

5

Ga71

Ga69

&InAs

As

Ho

le h

yp

erf

ine

sh

ift

E(+

2/-

1)

(e

V)

GaAs/AlGaAs

Ga

InGaAs/GaAs

Electron hyperfine shift

E(+2/+1) (eV)

Sign of the hole hyperfine constant has been determined for each element by selectively ‘erasing’ nuclear polarisation It was found that the sign of the hole hyperfine interaction has different signs for anions and cations

see Chekhovich Nature Physics (2013)

More on NMR ‘application’: nano-MRI in QD

Collaboration with Toshiba (Cambridge)

Electron wavefunction can be moved along the growth direction inside QD, while PL is not quenched

V

QD layer

QD

V1

V2

QD V2

QD V1

Mapping of composition and strain inside QD

Work in progress

Summary on cw DNP

Special techniques are required to enhance the NMR signal in strained nano-structures Detailed information on QD structure can be extracted using NMR spectroscopy Important for qubits: QD is a ‘disordered’ object, with inhomogeneous strain leading to all nuclei having differing energy spectra Strain may lead to suppression of nuclear-nuclear spin flip-flops, as already evidenced by very long T1

5. Coherent NMR spectroscopy in strained QDs

Effect of spectral broadening on nuclear spin dynamics

Nuclear spin T1 in self-assembled QDs can be up to 30 hours for B-field of a few Tesla and T~4K (measured via decay of optically induced polarisation, see eg Latta PRL 2011)

Long depolarisation time indirectly indicates suppression of nuclear flip-flops. Flip-flop dynamics can be understood from spin-echo measurements

Nuclear flip-flops cause decoherence of the central spin (via so-called ‘spectral diffusion’, see eg de Sousa PRB 2003)

-3/2

-1/2

+1/2

+3/2

-3/2

-1/2

+1/2

+3/2

Selective pulsed NMR on central transitions

38.6 38.8 69.10 69.15

0

2

4

rf frequency, (MHz)

75As

NM

R S

ignal (

eV

)

f (MHz)

Brf on - Rabi flops

71Ga

For unstrained GaAs QDs see Makhonin Nature Materials 2011

71Ga CT in an InGaAs QD

Two spin-3/2 nuclei in a strained QD

Chekhovich arxiv 2014

Measurement of spin-coherence

If spins having different resonant frequencies are flipped into xy-plane, we measure T2* decay which tells nothing about coherence.

Figure from “Magnetic Resonance imaging” E. M. Haake et al

Figure from “Magnetic Resonance imaging” E. M. Haake et al

Need to reverse T2* decay to measure true decoherence caused by nuclear dipole-dipole interaction – T2 decay

Use of -pulse for spin-echo

Spin-echo on central transitions in a SAQD

75As

Bz=8T

Chekhovich arxiv 2014

0.0 0.2 0.4 0.6 0.8

0

3

6

NM

R s

ignal (

eV

)

Time delay, (ms)

Time-evolution of spin-echo: T2 decay

Bz=8T

1 10 100 1000315

320

325

Delay time (s)

Exciton

Zeem

an s

plit

ting

(m

eV

)

69Ga, B

ext=3.5T

Strained InGaAs QDs (top): for 71Ga T2=1.2 ms, for 75As T2=4.3 ms

Lattice-matched GaAs QDs (bottom): 69Ga T2=0.3 ms

GaAs QDs, Makhonin Nature Materials 2011

Chekhovich arxiv 2014

T2*

T2

Further analysis shows that T2 increase in SAQD is due to flip-flop suppression/freezing

Note: nuclear spin dephasing in strained QDs due to 1/2 states

Summary

Nuclear spins in strained quantum dots:

Broad NMR spectra (up to 20 MHZ) with narrow central transitions (CT) carry wealth of structural information

Chemical composition and strain distribution are determined in a non-invasive fashion

Central transitions in NMR (75As in particular) are sensitive probes of local nuclear environment and reveal disorder in In/Ga composition in QDs

Spin-echo measurements show strong suppression and complete freezing of flip-flops: opens ways to engineer nuclear spin bath, eg using InGaAsSb(P) QDs

Recent reviews on nuclear magentism in QDs: Urbaszek Rev. Mod. Phys. (2013), Chekhovich et al Nature Materials (2013)

Our recent NMR work: Chekhovich et al arxiv 1403.1510, Nature Physics (2013), Nature Nanotechnology (2012), Bulutay et al arxiv 1408.0373, Makhonin et al Nature Materials (2011), PRB (2010)

Sheffield:

Evgeny Chekhovich Maxim Makhonin Maurice Skolnick Andrey Krysa Mark Hopkinson

Contributions

Bilkent: Ceyhun Bulutay

Ioffe: M. Glazov, K. Kavokin

LPN-CNRS: P. Senellart, A. Lemaître

Toshiba Research Europe: J. Nilsson, A. Bennett, R. M. Stevenson, A. Shields

Konstanz: G. Burkard, E. Welander

Warwick: R. Beanland, A. Sanchez