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