Spins in few-electron quantum dots
R. Hanson
*
Center for Spintronics and Quantum Computation, University of California,
Santa Barbara, California 93106, USA and Kavli Institute of Nanoscience, Delft University
of Technology, P.O. Box 5046, 2600 GA Delft, The Netherlands
L. P. Kouwenhoven
Kavli Institute of Nanoscience, Delft University of Technology, P.O. Box 5046, 2600 GA
Delft, The Netherlands
J. R. Petta
Department of Physics, Princeton University, Princeton, New Jersey 08544, USA
and Department of Physics, Harvard University, Cambridge, Massachusetts 02138, USA
S. Tarucha
Department of Applied Physics and ICORP-JST, The University of Tokyo, Hongo, Bunkyo-
ku, Tokyo 113-8656, Japan
L. M. K. Vandersypen
Kavli Institute of NanoScience, Delft University of Technology, P.O. Box 5046, 2600 GA
Delft, The Netherlands
共Published 1 October 2007; publisher error corrected 4 October 2007兲
The canonical example of a quantum-mechanical two-level system is spin. The simplest picture of spin
is a magnetic moment pointing up or down. The full quantum properties of spin become apparent in
phenomena such as superpositions of spin states, entanglement among spins, and quantum
measurements. Many of these phenomena have been observed in experiments performed on
ensembles of particles with spin. Only in recent years have systems been realized in which individual
electrons can be trapped and their quantum properties can be studied, thus avoiding unnecessary
ensemble averaging. This review describes experiments performed with quantum dots, which are
nanometer-scale boxes defined in a semiconductor host material. Quantum dots can hold a precise but
tunable number of electron spins starting with 0, 1, 2, etc. Electrical contacts can be made for charge
transport measurements and electrostatic gates can be used for controlling the dot potential. This
system provides virtually full control over individual electrons. This new, enabling technology is
stimulating research on individual spins. This review describes the physics of spins in quantum dots
containing one or two electrons, from an experimentalist’s viewpoint. Various methods for extracting
spin properties from experiment are presented, restricted exclusively to electrical measurements.
Furthermore, experimental techniques are discussed that allow for 共1兲 the rotation of an electron spin
into a superposition of up and down, 共2兲 the measurement of the quantum state of an individual spin,
and 共3兲 the control of the interaction between two neighboring spins by the Heisenberg exchange
interaction. Finally, the physics of the relevant relaxation and dephasing mechanisms is reviewed and
experimental results are compared with theories for spin-orbit and hyperfine interactions. All these
subjects are directly relevant for the fields of quantum information processing and spintronics with
single spins 共i.e., single spintronics兲.
DOI: 10.1103/RevModPhys.79.1217 PACS number共s兲: 73.63.Kv, 03.67.Lx, 85.75.⫺d
CONTENTS
I. Introduction 1218
II. Basics of Quantum Dots 1219
A. Introduction to quantum dots 1219
B. Fabrication of gated quantum dots 1220
C. Measurement techniques 1221
D. The constant interaction model 1221
E. Low-bias regime 1222
F. High-bias regime 1223
III. Spin Spectroscopy Methods 1224
A. Spin filling derived from magnetospectroscopy 1224
B. Spin filling derived from excited-state spectroscopy 1225
C. Other methods 1226
IV. Spin States in a Single Dot 1226
A. One-electron spin states 1226
B. Two-electron spin states 1227
C. Quantum dot operated as a bipolar spin filter 1229
V. Charge Sensing Techniques 1229
VI. Single-Shot Readout of Electron Spins 1231
*
Electronic address: hanson@physics.ucsb.edu
REVIEWS OF MODERN PHYSICS, VOLUME 79, OCTOBER–DECEMBER 2007
0034-6861/2007/79共4兲/1217共49兲 ©2007 The American Physical Society1217
A. Spin-to-charge conversion 1231
B. Single-shot spin readout using a difference in energy 1231
C. Single-shot spin readout using a difference in tunnel
rate 1233
VII. Spin Interaction with the Environment 1234
A. Spin-orbit interaction 1235
1. Origin 1235
2. Spin-orbit interaction in bulk and two
dimensions 1235
3. Spin-orbit interaction in quantum dots 1236
4. Relaxation via the phonon bath 1237
5. Phase randomization due to the spin-orbit
interaction 1239
B. Hyperfine interaction 1240
1. Origin 1240
2. Effect of the Overhauser field on the
electron-spin time evolution 1241
3. Mechanisms and time scales of nuclear field
fluctuations 1242
4. Electron-spin decoherence in a fluctuating
nuclear field 1243
C. Summary of mechanisms and time scales 1244
VIII. Spin States in Double Quantum Dots 1244
A. Electronic properties of electrons in double dots 1244
1. Charge stability diagram 1244
2. High bias regime: Bias triangles 1246
B. Spin states in two-electron double dots 1248
C. Pauli spin blockade 1249
D. Hyperfine interaction in a double dot:
Singlet-triplet mixing 1251
IX. Coherent Spin Manipulation 1254
A. Single-spin manipulation: ESR 1254
B. Manipulation of coupled electron spins 1256
X. Perspectives 1260
Acknowledgments 1261
Appendix: Sign of the Ground-State Spin and the Nuclear
Fields in GaAs 1261
1. Sign of the spin ground states 1261
2. Sign and magnitude of the thermal nuclear field 1261
3. Sign of the dynamic nuclear field 1261
References 1261
I. INTRODUCTION
The spin of an electron remains a somewhat mysteri-
ous property. The first derivations in 1925 of the spin
magnetic moment, based on a rotating charge distribu-
tion of finite size, are in conflict with special relativity
theory. Pauli advised the young Ralph Kronig not to
publish his theory since “it has nothing to do with real-
ity.” More fortunate were Samuel Goudsmit and George
Uhlenbeck, who were supervised by Ehrenfest: “Pub-
lish, you are both young enough to be able to afford a
stupidity!”
1
It requires Dirac’s equation to find that the
spin eigenvalues correspond to one-half times Planck’s
constant ប while considering the electron as a point par-
ticle. The magnetic moment corresponding to spin is re-
ally very small and in most practical cases it can be ig-
nored. For instance, the most sensitive force sensor to
date has only recently been able to detect some effect
from the magnetic moment of a single-electron spin
共Rugar et al., 2004兲. In solids, spin can apparently lead to
strong effects, given the existence of permanent mag-
nets. Curiously, this has little to do with the strength of
the magnetic moment. Instead, the fact that spin is asso-
ciated with its own quantum number, combined with
Pauli’s exclusion principle that quantum states can at
most be occupied with one fermion, leads to the phe-
nomenon of exchange interaction. Because the exchange
interaction is a correction term to the strong Coulomb
interaction, it can be of much larger strength in solids
than the dipolar interaction between two spin magnetic
moments at an atomic distance of a few angstroms. It is
the exchange interaction that forces the electron spins in
a collective alignment, together yielding a macroscopic
magnetization 共Ashcroft and Mermin, 1974兲. It remains
striking that an abstract concept as 共anti兲symmetrization
in the end gives rise to magnets.
The magnetic state of solids has found important ap-
plications in electronics, in particular for memory de-
vices. An important field has emerged in the last two
decades known as spintronics. Phenomena like giant
magnetoresistance or tunneling magnetoresistance form
the basis for magnetic heads for reading out the mag-
netic state of a memory cell. Logic gates have been re-
alized based on magnetoresistance effects as well 共Wolf
et al., 2001; Zutic et al., 2004兲. In addition to applications,
important scientific discoveries have been made in the
field of spintronics 共Awschalom and Flatte, 2007兲, in-
cluding magnetic semiconductors 共Ohno, 1998兲 and the
spin Hall effect 共Sih et al., 2005兲. It is important to note
that all spintronics phenomena consider macroscopic
numbers of spins. Together these spins form things like
spin densities or a collective magnetization. Although
the origin of spin densities and magnetization is quan-
tum mechanical, these collective, macroscopic variables
behave entirely classically. For instance, the magnetiza-
tion of a micron-cubed piece of cobalt is a classical vec-
tor. The quantum state of this vector dephases so rapidly
that quantum superpositions or entanglement between
vectors is never observed. One has to go to systems with
a small number of spins, for instance in magnetic mol-
ecules, in order to find quantum effects in the behavior
of the collective magnetization 关for an overview, see,
e.g., Gunther and Barbara 共1994兲兴.
The technological drive to make electronic devices
continuously smaller has some interesting scientific con-
sequences. For instance, it is now routinely possible to
make small electron boxes in solid-state devices that
contain an integer number of conduction electrons. Such
devices are usually operated as transistors 共via field-
effect gates兲 and are therefore named single-electron
transistors. In semiconductor boxes the number of
trapped electrons can be reduced to 0, or 1, 2, etc. Such
semiconductor single-electron transistors are called
quantum dots 共Kouwenhoven et al., 2001兲. Electrons are
1
See URL: http://www.lorentz.leidenuniv.nl/history/spin/
goudsmit.html
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Hanson et al.: Spins in few-electron quantum dots
Rev. Mod. Phys., Vol. 79, No. 4, October–December 2007
trapped in a quantum dot by repelling electric fields im-
posed from all sides. The final region in which a small
number of electrons can still exist is typically at the scale
of tens of nanometers. The eigenenergies in such boxes
are discrete. Filling these states with electrons follows
the rules from atomic physics, including Hund’s rule,
shell filling, etc.
Studies with quantum dots have been performed dur-
ing the 1990s. By now it has become standard technol-
ogy to confine single-electron charges. Electrons can be
trapped as long as one desires. Changes in charge when
one electron tunnels out of the quantum dot can be
measured on a microsecond time scale. Compared to
this control of charge, it is very difficult to control indi-
vidual spins and measure the spin of an individual elec-
tron. Such techniques have been developed only over
the past few years.
In this review we describe experiments in which indi-
vidual spins are controlled and measured. This is mostly
an experimental review with explanations of the under-
lying physics. This review is limited to experiments that
involve one or two electrons strongly confined to single
or double quantum dot devices. The experiments show
that one or two electrons can be trapped in a quantum
dot; that the spin of an individual electron can be put in
a superposition of up and down states; that two spins can
be made to interact and form an entangled state such as
a spin singlet or triplet state; and that the result of such
manipulation can be measured on individual spins.
These abilities of almost full control over the spin of
individual electrons enable the investigation of a new
regime: single spin dynamics in a solid-state environ-
ment. The dynamics are fully quantum mechanical and
thus quantum coherence can be studied on an individual
electron spin. The exchange interaction is now also con-
trolled on the level of two particular spins that are
brought into contact simply by varying some voltage
knob.
In a solid the electron spins are not completely decou-
pled from other degrees of freedom. First of all, spins
and orbits are coupled by the spin-orbit interaction. Sec-
ond, the electron spins have an interaction with the spins
of the atomic nuclei, i.e., the hyperfine interaction. Both
interactions cause the lifetime of a quantum superposi-
tion of spin states to be finite. We therefore also describe
experiments that probe spin-orbit and hyperfine interac-
tions by measuring the dynamics of individual spins.
The study of individual spins is motivated by an inter-
est in fundamental physics, but also by possible applica-
tions. First of all, miniaturized spintronics is developing
towards single spins. In this context, this field can be
denoted as single spintronics
2
in analogy to single elec-
tronics. A second area of applications is quantum infor-
mation science. Here the spin states form the qubits.
The original proposal by Loss and DiVincenzo 共1998兲
has been the guide in this field. In the context of quan-
tum information, the experiments described in this re-
view demonstrate that the five DiVincenzo criteria for
universal quantum computation using single-electron
spins have been fulfilled to a large extent 共DiVincenzo,
2000兲: initialization, one- and two-qubit operations, long
coherence times, and readout. Currently, the state of the
art is at the level of single and double quantum dots and
much work is required to build larger systems.
In this review the system of choice is quantum dots in
GaAs semiconductors, simply because this has been
most successful. Nevertheless, the physics is entirely
general and can be fully applied to new material systems
such as silicon-based transistors, carbon nanotubes,
semiconductor nanowires, graphene devices, etc. These
other host materials may have advantageous spin prop-
erties. For instance, carbon-based devices can be puri-
fied with the isotope
12
C in which the nuclear spin is
zero, thus entirely suppressing spin dephasing by hyper-
fine interaction. This kind of hardware solution to engi-
neer a long-lived quantum system will be discussed at
the end of this review. Also, we here restrict ourselves
exclusively to electron-transport measurements of quan-
tum dots, leaving out optical spectroscopy of quantum
dots, which is a very active field in its own.
3
Again, much
of the physics discussed in this review also applies to
optically measured quantum dots.
Section II starts with an introduction on quantum dots
including the basic model of Coulomb blockade to de-
scribe the relevant energies. These energies can be visu-
alized in transport experiments and the relation between
experimental spectroscopic lines and underlying ener-
gies are explained in Sec. III. This spectroscopy is spe-
cifically applied to spin states in single quantum dots in
Sec. IV. Section V introduces a charge-sensing technique
that is used in Sec. VI to read out the spin state of indi-
vidual electrons. Section VII provides a description of
spin-orbit and hyperfine interactions. In Sec. VIII, spin
states in double quantum dots are introduced and the
concept of Pauli spin blockade is discussed. Quantum
coherent manipulations of spins in double dots are dis-
cussed in Sec. IX. Finally, a perspective is outlined in
Sec. X.
II. BASICS OF QUANTUM DOTS
A. Introduction to quantum dots
A quantum dot is an artificially structured system that
can be filled with electrons 共or holes兲. The dot can be
coupled via tunnel barriers to reservoirs, with which
electrons can be exchanged 共see Fig. 1兲. By attaching
current and voltage probes to these reservoirs, we can
measure the electronic properties. The dot is also
coupled capacitively to one or more gate electrodes,
which can be used to tune the electrostatic potential of
the dot with respect to the reservoirs.
2
Name coined by Wolf 共2005兲.
3
See, e.g., Atature et al. 共2006兲, Berezovsky et al. 共2006兲,
Greilich, Oulton, et al. 共2006兲, Krenner et al. 共2006兲, and refer-
ences therein.
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Because a quantum dot is such a general kind of sys-
tem, there exist quantum dots of many different sizes
and materials: for instance, single molecules trapped be-
tween electrodes 共Park et al., 2002兲, normal metal 共Petta
and Ralph, 2001兲, superconducting 共Ralph et al., 1995;
von Delft and Ralph, 2001兲, or ferromagnetic nanopar-
ticles 共Guéron et al., 1999兲, self-assembled quantum dots
共Klein et al., 1996兲, semiconductor lateral 共Kouwen-
hoven et al., 1997兲 or vertical dots 共Kouwenhoven et al.,
2001兲, and also semiconducting nanowires or carbon
nanotubes 共Dekker, 1999; McEuen, 2000; Björk et al.,
2004兲.
The electronic properties of quantum dots are domi-
nated by two effects. First, the Coulomb repulsion be-
tween electrons on the dot leads to an energy cost for
adding an extra electron to the dot. Due to this charging
energy tunneling of electrons to or from the reservoirs
can be suppressed at low temperatures; this phenom-
enon is called Coulomb blockade 共van Houten et al.,
1992兲. Second, the confinement in all three directions
leads to quantum effects that influence the electron dy-
namics. Due to the resulting discrete energy spectrum,
quantum dots behave in many ways as artificial atoms
共Kouwenhoven et al., 2001兲.
The physics of dots containing more than two elec-
trons has been previously reviewed 共Kouwenhoven et
al., 1997; Reimann and Manninen, 2002兲. Therefore we
focus on single and coupled quantum dots containing
only one or two electrons. These systems are particularly
important as they constitute the building blocks of pro-
posed electron spin-based quantum information proces-
sors 共Loss and DiVincenzo, 1998; DiVincenzo et al.,
2000; Byrd and Lidar, 2002; Levy, 2002; Wu and Lidar,
2002a
, 2002b; Meier et al., 2003; Kyriakidis and Penney,
2005; Taylor et al., 2005; Hanson and Burkard, 2007兲.
B. Fabrication of gated quantum dots
The bulk of the experiments discussed in this review
was performed on electrostatically defined quantum
dots in GaAs. These devices are sometimes referred to
as lateral dots because of the lateral gate geometry.
Lateral GaAs quantum dots are fabricated from het-
erostructures of GaAs and AlGaAs grown by molecular-
beam epitaxy 共see Fig. 2兲. By doping the AlGaAs layer
with Si, free electrons are introduced. These accumulate
at the GaAs/AlGaAs interface, typically 50–100 nm be-
low the surface, forming a two-dimensional electron gas
共2DEG兲—a thin 共⬃10 nm兲 sheet of electrons that can
only move along the interface. The 2DEG can have high
mobility and relatively low electron density 关typically
10
5
−10
7
cm
2
/V s and ⬃共1−5兲⫻ 10
15
m
−2
, respectively兴.
The low electron-density results in a large Fermi wave-
length 共⬃40 nm兲 and a large screening length, which al-
lows us to locally deplete the 2DEG with an electric
field. This electric field is created by applying negative
voltages to metal gate electrodes on top of the hetero-
structure 关see Fig. 2共a兲兴.
Electron-beam lithography enables fabrication of gate
structures with dimensions down to a few tens of na-
nometers 共Fig. 2兲, yielding local control over the deple-
tion of the 2DEG with roughly the same spatial resolu-
tion. Small islands of electrons can be isolated from the
rest of the 2DEG by choosing a suitable design of the
gate structure, thus creating quantum dots. Finally, low-
FIG. 1. Schematic picture of a quantum dot in 共a兲 a lateral
geometry and 共b兲 in a vertical geometry. The quantum dot
共represented by a disk兲 is connected to source and drain reser-
voirs via tunnel barriers, allowing the current through the de-
vice I to be measured in response to a bias voltage V
SD
and a
gate voltage V
G
.
FIG. 2. Lateral quantum dot device defined by metal surface
electrodes. 共a兲 Schematic view. Negative voltages applied to
metal gate electrodes 共dark gray兲 lead to depleted regions
共white兲 in the 2DEG 共light gray兲. Ohmic contacts 共light gray
columns兲 enable bonding wires 共not shown兲 to make electrical
contact to the 2DEG reservoirs. 共b兲, 共c兲 Scanning electron mi-
crographs of 共b兲 a few-electron single-dot device and 共c兲 a
double dot device, showing the gate electrodes 共light gray兲 on
top of the surface 共dark gray兲. White dots indicate the location
of the quantum dots. Ohmic contacts are shown in the corners.
White arrows outline the path of current I
DOT
from one reser-
voir through the dot共s兲 to the other reservoir. For the device in
共c兲, the two gates on the side can be used to create two quan-
tum point contacts, which can serve as electrometers by pass-
ing a current I
QPC
. Note that this device can also be used to
define a single dot. Image in 共b兲 courtesy of A. Sachrajda.
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resistance 共Ohmic兲 contacts are made to the 2DEG res-
ervoirs. To access the quantum phenomena in GaAs
gated quantum dots, they have to be cooled down to
well below 1 K. All experiments that are discussed in
this review are performed in dilution refrigerators with
typical base temperatures of 20 mK.
In so-called vertical quantum dots, control over the
number of electrons down to zero was already achieved
in the 1990s 共Kouwenhoven et al., 2001兲. In lateral gated
dots this proved to be more difficult, since reducing the
electron number by driving the gate voltage to more
negative values tends to decrease the tunnel coupling to
the leads. The resulting current through the dot can then
become unmeasurably small before the few-electron re-
gime is reached. However, by proper design of the sur-
face gate geometry the decrease of the tunnel coupling
can be compensated for.
In 2000, Ciorga et al. reported measurements on the
first lateral few-electron quantum dot 共Ciorga et al.,
2000兲. Their device, shown in Fig. 2共b兲, makes use of two
types of gates specifically designed to have different
functionalities. The gates of one type are big and largely
enclose the quantum dot. The voltages on these gates
mainly determine the dot potential. The other type of
gate is thin and just reaches up to the barrier region. The
voltage on this gate has a very small effect on the dot
potential but it can be used to set the tunnel barrier. The
combination of the two gate types allows the dot poten-
tial 共and thereby electron number兲 to be changed over a
wide range while keeping the tunnel rates high enough
for measuring electron transport through the dot.
Applying the same gate design principle to a double
quantum dot, Elzerman et al. demonstrated control over
the electron number in both dots while maintaining tun-
able tunnel coupling to the reservoir 共Elzerman et al.,
2003兲. Their design is shown in Fig. 2共c兲 关for more details
on design considerations and related versions of this
gate design, see Hanson 共2005兲兴. In addition to the
coupled dots, two quantum point contacts 共QPCs兲 are
incorporated in this device to serve as charge sensors.
The QPCs are placed close to the dots, thus ensuring a
good charge sensitivity. This design has become the stan-
dard for lateral coupled quantum dots and is used with
minor adaptions by several research groups 共Petta et al.,
2004; Pioro-Ladrière et al., 2005兲; one noticeable im-
provement has been the electrical isolation of the charge
sensing part of the circuit from the reservoirs that con-
nect to the dot 共Hanson et al., 2005兲.
C. Measurement techniques
In this review, two all-electrical measurement tech-
niques are discussed: 共i兲 measurement of the current due
to transport of electrons through the dot, and 共ii兲 detec-
tion of changes in the number of electrons on the dot
with a nearby electrometer, so-called charge sensing.
With the latter technique, the dot can be probed nonin-
vasively in the sense that no current needs to be sent
through the dot.
The potential of charge sensing was first demonstrated
by Ashoori et al. 共1992兲 and Field et al. 共1993兲. But
whereas current measurements were already used exten-
sively in the first experiments on quantum dots 共Kou-
wenhoven et al., 1997兲, charge sensing has only recently
been fully developed as a spectroscopic tool 共Elzerman,
Hanson, Willems van Beveren, Vandersypen, et al., 2004;
Johnson, Marcus, et al., 2005兲. Several implementations
of electrometers coupled to a quantum dot have been
demonstrated: a single-electron transistor fabricated on
top of the heterostructure 共Ashoori et al., 1992; Lu et al.,
2003兲, a second electrostatically defined quantum dot
共Hofmann et al., 1995; Fujisawa et al., 2004兲, and a quan-
tum point contact 共QPC兲共Field et al., 1993; Sprinzak et
al., 2002兲. The QPC is the most widely used because of
its ease of fabrication and experimental operation. We
discuss the QPC operation and charge sensing tech-
niques in more detail in Sec. V.
We briefly compare charge sensing to electron-
transport measurements. The smallest currents that can
be resolved in optimized setups and devices are roughly
10 fA, which sets a lower bound of order 10 fA/e
⬇100 kHz on the tunnel rate to the reservoir ⌫ for
which transport experiments are possible 关see, e.g.,
Vandersypen et al. 共2004兲 for a discussion on noise
sources
兴. For ⌫⬍100 kHz the charge detection tech-
nique can be used to resolve electron tunneling in real
time. Because the coupling to the leads is a source of
decoherence and relaxation 共most notably via cotunnel-
ing兲, charge detection is preferred for quantum informa-
tion purposes since it still functions for very small cou-
plings to a 共single兲 reservoir.
Measurements using either technique are conve-
niently understood with the constant interaction model.
In the next section we use this model to describe the
physics of single dots and show how relevant spin pa-
rameters can be extracted from measurements.
D. The constant interaction model
We briefly outline the main ingredients of the constant
interaction model; for more extensive discussions, see
van Houten et al. 共1992兲 and Kouwenhoven et al. 共1997,
2001兲. The model is based on two assumptions. First, the
Coulomb interactions among electrons in the dot, and
between electrons in the dot and those in the environ-
ment, are parametrized by a single, constant capacitance
C. This capacitance is the sum of the capacitances be-
tween the dot and the source C
S
, the drain C
D
, and the
gate C
G
: C = C
S
+C
D
+C
G
. 共In general, capacitances to
multiple gates and other parts of the 2DEG will also
play a role; they can simply be added to C.兲 The second
assumption is that the single-particle energy-level spec-
trum is independent of these interactions and therefore
of the number of electrons. Under these assumptions,
the total energy U共N兲 of a dot with N electrons in the
ground state, with voltages V
S
, V
D
, and V
G
applied to
the source, drain, and gate, respectively, is given by
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