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Spins in few-electron quantum dots

TL;DR: In this article, the physics of spins in quantum dots containing one or two electrons, from an experimentalist's viewpoint, are described, and various methods for extracting spin properties from experiment are presented, restricted exclusively to electrical measurements.
Abstract: 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 1 the rotation of an electron spin into a superposition of up and down, 2 the measurement of the quantum state of an individual spin, and 3 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 i.e., single spintronics.

Summary (11 min read)

Jump to: [Introduction][II. BASICS OF QUANTUM DOTS][B. Fabrication of gated quantum dots][C. Measurement techniques][D. The constant interaction model][E. Low-bias regime][F. High-bias regime][III. SPIN SPECTROSCOPY METHODS][A. Spin filling derived from magnetospectroscopy][B. Spin filling derived from excited-state spectroscopy][C. Other methods][A. One-electron spin states][B. Two-electron spin states][C. Quantum dot operated as a bipolar spin filter][V. CHARGE SENSING TECHNIQUES][A. Spin-to-charge conversion][B. Single-shot spin readout using a difference in energy][C. Single-shot spin readout using a difference in tunnel rate][VII. SPIN INTERACTION WITH THE ENVIRONMENT][1. Origin][2. Spin-orbit interaction in bulk and two dimensions][3. Spin-orbit interaction in quantum dots][4. Relaxation via the phonon bath][5. Phase randomization due to the spin-orbit interaction][2. Effect of the Overhauser field on the electron-spin time evolution][3. Mechanisms and time scales of nuclear field fluctuations][4. Electron-spin decoherence in a fluctuating nuclear field][C. Summary of mechanisms and time scales][VIII. SPIN STATES IN DOUBLE QUANTUM DOTS][A. Electronic properties of electrons in double dots][1. Charge stability diagram][2. High bias regime: Bias triangles][B. Spin states in two-electron double dots][D. Hyperfine interaction in a double dot: Singlet-triplet mixing][A. Single-spin manipulation: ESR][B. Manipulation of coupled electron spins][X. PERSPECTIVES][ACKNOWLEDGMENTS][1. Sign of the spin ground states][2. Sign and magnitude of the thermal nuclear field] and [3. Sign of the dynamic nuclear field]

Introduction

  • The full quantum properties of spin become apparent in phenomena such as superpositions of spin states, entanglement among spins, and quantum measurements.
  • This review describes the physics of spins in quantum dots containing one or two electrons, from an experimentalist’s viewpoint.
  • In addition to applications, important scientific discoveries have been made in the field of spintronics Awschalom and Flatte, 2007 , including magnetic semiconductors Ohno, 1998 and the spin Hall effect Sih et al., 2005 .
  • It is now routinely possible to make small electron boxes in solid-state devices that contain an integer number of conduction electrons.
  • The authors therefore also describe experiments that probe spin-orbit and hyperfine interactions by measuring the dynamics of individual spins.

II. BASICS OF QUANTUM DOTS

  • The dot can be coupled via tunnel barriers to reservoirs, with which electrons can be exchanged see Fig. 1 .
  • By attaching current and voltage probes to these reservoirs, the authors can measure the electronic properties.
  • Therefore the authors focus on single and coupled quantum dots containing only one or two electrons.

B. Fabrication of gated quantum dots

  • The bulk of the experiments discussed in this review was performed on electrostatically defined quantum dots in GaAs.
  • 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.
  • All experiments that are discussed in this review are performed in dilution refrigerators with typical base temperatures of 20 mK.
  • The combination of the two gate types allows the dot potential and thereby electron number to be changed over a wide range while keeping the tunnel rates high enough for measuring electron transport through the dot.
  • This design has become the standard for lateral coupled quantum dots and is used with minor adaptions by several research groups Petta et al., 2004; Pioro-Ladrière et al., 2005 ; one noticeable improvement has been the electrical isolation of the charge sensing part of the circuit from the reservoirs that connect to the dot Hanson et al., 2005 .

C. Measurement techniques

  • Two all-electrical measurement techniques are discussed: i measurement of the current due to transport of electrons through the dot, and ii detection of changes in the number of electrons on the dot with a nearby electrometer, so-called charge sensing.
  • The QPC is the most widely used because of its ease of fabrication and experimental operation.
  • The authors briefly compare charge sensing to electrontransport 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 see, e.g., Vandersypen et al. 2004 for a discussion on noise sources .
  • In the next section the authors use this model to describe the physics of single dots and show how relevant spin parameters can be extracted from measurements.

D. The constant interaction model

  • The authors briefly outline the main ingredients of the constant interaction model; for more extensive discussions, see van Houten et al. 1992 and Kouwenhoven et al.
  • This capacitance is the sum of the capacitances between the dot and the source CS, the drain CD, and the gate CG: C=CS+CD+CG.
  • It is this property that makes the electrochemical potential the most convenient quantity for describing electron tunneling.
  • Electron tunneling through the dot critically depends on the alignment of electrochemical potentials in the dot with respect to those of the source S and the drain D. The application of a bias voltage VSD=VS−VD between the source and drain reservoir opens up an energy window between S and D of S− D=− e VSD.
  • The size of the bias window then separates two regimes: the low-bias regime where at most one dot level is within the bias window − e VSD E ,Eadd , and the high-bias regime where multiple dot levels can be in the bias window − e VSD E and/or − e VSD Eadd .

E. Low-bias regime

  • If this condition is not met, the number of electrons on the dot remains fixed and no current flows through the dot.
  • After it has tunneled to the drain, another electron can tunnel onto the dot from the source.
  • At the positions of the peaks in IDOT, an electrochemical potential level corresponding to transport between successive ground states is aligned between the source and drain electrochemical potentials and a singleelectron tunneling current flows.
  • In the valleys between peaks, the number of electrons on the dot is fixed due to Coulomb blockade.

F. High-bias regime

  • The authors now look at the regime where the source-drain bias is so high that multiple dot levels can participate in electron tunneling.
  • Typically the electrochemical potential of only one of the reservoirs is changed in experiments, and the other one is kept fixed.
  • When VSD is increased further such that also a transition involving an excited state falls within the bias window, there are two paths available for electrons tunneling through the dot see Fig. 4 a .
  • The transition between the N-electron GS and the N+1 -electron GS dark solid line defines the regions of Coulomb blockade outside the V shape and tunneling within the V shape .

III. SPIN SPECTROSCOPY METHODS

  • These methods make use of various spin-dependent energy terms.
  • Each electron spin is first influenced directly by an external magnetic field via the Zeeman energy EZ=Szg BB, where Sz is the spin z component.
  • Moreover, the Pauli exclusion principle forbids two electrons with equal spin orientation to occupy the same orbital, thus forcing one electron into a different orbital.
  • Finally, the Coulomb interaction leads to an energy difference the exchange energy between states with symmetric and antisymmetric orbital wave functions.
  • Since the total wave function of the electrons is antisymmetric, the symmetry of the orbital part is linked to that of the spin.

A. Spin filling derived from magnetospectroscopy

  • The spin filling of a quantum dot can be derived from the Zeeman energy shift of the Coulomb peaks in a magnetic-field.
  • An in-plane magnetic-field orientation is favored to ensure minimum disturbance of the orbital levels.
  • This change in spin is reflected in the magnetic-field dependence of the electrochemical potential N via the Zeeman term 6Note that energy absorption from the environment can lead to exceptions: photon-or phonon-assisted tunneling can give rise to lines ending in the N=0 Coulomb blockade region.
  • Many experiments are performed at very low temperatures where the number of photons and phonons in thermal equilibrium is small.
  • In circularly symmetric few-electron vertical dots, spin states have been determined from the evolution of orbital states in a magnetic field perpendicular to the plane of the dots.

B. Spin filling derived from excited-state spectroscopy

  • Spin filling can also be deduced from excited-state spectroscopy without changing the magnetic field Cobden et al., 1998 , provided the Zeeman energy splitting EZ=2 EZ =g BB between spin-up and spin-down electrons can be resolved.
  • Therefore as the authors add one electron to a dot containing N electrons, there are only two scenarios possible: either the electron moves into an empty orbital, or it moves into an orbital that already holds one electron.
  • The two electrons need to have antiparallel spins, in order to satisfy the Pauli exclusion principle.
  • If the dot is in the ground state, the electron already present in this orbital has spin up.

C. Other methods

  • If the tunnel rates for spin up and spin down are not equal, the amplitude of the current can be used to determine the spin filling.
  • This name is slightly misleading as the current is not actually blocked, but rather assumes a finite value that depends on the spin orientation of transported electrons.
  • Care must be taken when inferring spin filling from the amplitude of the current as other factors, such as the orbital spread of the wave function, can have a large, even dominating influence on the current amplitude.
  • As an example, in the transition from a one-electron S =1/2 state to a two-electron S=0 state, only a spin-up electron can tunnel onto the dot if the electron that is already on the dot is spin down, and vice versa.
  • Second, spin-filling measurements do not yield the absolute spin of the ground states, but only the change in ground-state spin.

A. One-electron spin states

  • The simplest spin system is that of a single electron, which can have one of only two orientations: spin up or spin down.
  • Let E↑,0 and E↓,0 E↑,1 and E↓,1 denote the one-electron energies for the two spin states in the lowest first excited orbital.
  • The orbital level spacing Eorb in device A is about 1.1 meV.
  • Comparison with Fig. 6 shows that a spin-up electron is added to the empty dot to form the one-electron ground state, as expected.
  • These measure- Rev. Mod. Phys., Vol. 79, No. 4, October–December 2007 ments allow a straightforward determination of the electron g factor.

B. Two-electron spin states

  • Figure 8 a shows the possible transitions between the one-electron spin-split orbital ground state and the twoelectron states.
  • In contrast, the onset of second-order cotunneling currents is governed by the energy difference between states with the same number of electrons.
  • The fact that EST is about half the single-particle level spacing Eorb=1 meV indicates the importance of Coulomb interactions.
  • Experiments and calculations indicating this double-dot-like behavior in asymmetric dots have been reported Zumbühl et al., 2004; Ellenberger et al., 2006 .

C. Quantum dot operated as a bipolar spin filter

  • If the Zeeman splitting exceeds the width of the energy levels which in most cases is set by the thermal energy , electron transport through the dot is for certain regimes spin polarized and the dot can be operated as a spin filter Recher et al., 2000; Hanson, Vandersypen, et al., 2004 .
  • Thus the polarization of the spin filter can be reversed electrically, by tuning the dot to the relevant transition.
  • Spectroscopy on dots containing more than two electrons has shown important deviations from an alternat- ing spin filling scheme.
  • Already for four electrons, a spin ground state with total spin S=1 in zero magnetic field has been observed in both vertical Kouwenhoven et al., 2001 and lateral dots Willems van Beveren et al., 2005 .

V. CHARGE SENSING TECHNIQUES

  • The use of local charge sensors to determine the number of electrons in single or double quantum dots is a recent technological improvement that has enabled a number of experiments that would have been difficult or impossible to perform using standard electrical transport measurements Dips in dGQPC/dVG coincide with the current peaks, demonstrating the validity of charge sensing.
  • The sensitivity of the charge sensor to changes in the dot charge can be optimized using an appropriate gate design Zhang et al., 2004 .
  • The authors should mention here that charge sensing fails when the tunnel time is much longer than the measurement time.
  • This means that the number of electrons fluctuates between N−1 and N at the pulse frequency.

A. Spin-to-charge conversion

  • The ability to measure individual quantum states in a single-shot mode is important both for fundamental science and for possible applications in quantum information processing.
  • By correlating the spin states to different charge states and subsequently measuring the charge on the dot, the spin state can be determined Loss and DiVincenzo, 1998 .
  • Two meth- ods, both outlined in Fig. 14, have been experimentally demonstrated.
  • At time t=0, the levels of both ES and GS are positioned far above res, so that one electron is energetically allowed to tunnel off the dot regardless of the spin state.
  • A conceptually similar scheme has allowed single-shot readout of a superconducting charge qubit Astafiev et al., 2004 .

B. Single-shot spin readout using a difference in energy

  • Single-shot readout of a single electron spin has first been demonstrated using the ERO technique Elzer- Rev. Mod.
  • To test the single-spin measurement technique, the following three-stage procedure is used: i empty the dot, ii inject one electron with unknown spin, and iii measure its spin state.
  • It is now energetically allowed for one electron to tunnel onto the dot, which will happen after a typical time −1.
  • Second, IQPC tracks the charge on the dot, i.e., it goes up whenever an electron tunnels off the dot, and it goes down by the same amount when an electron tunnels onto the dot.
  • Since the ERO relies on precise positioning of the spin levels with respect to the reservoir, it is very sensitive to fluctuations in the electrostatic potential.

C. Single-shot spin readout using a difference in tunnel rate

  • The main ingredient necessary for TR-RO is a spin dependence in the tunnel rates.
  • Figure 18 a shows several traces of IQPC, from the last part 0.3 ms of the pulse to the end of the readout stage see inset , for a waiting time of 0.8 ms.
  • In other traces, the tunneling occurs faster than the filter bandwidth.
  • A major advantage of the TR-RO scheme is that it does not rely on a large energy splitting between spin states.
  • By making the electron tunnel not to a reservoir, but to a second dot Engel et al., 2004; Engel and Loss, 2005 , the electron can be preserved and QND measurements are in principle possible.

VII. SPIN INTERACTION WITH THE ENVIRONMENT

  • As a result, electron spin states are only weakly perturbed by their magnetic environment.
  • For electron spins in semiconductor quantum dots, the most important interactions with the environment occur via the spin-orbit coupling, the hyperfine coupling with the nuclear spins of the host material, and virtual Rev. Mod.
  • Phys., Vol. 79, No. 4, October–December 2007 exchange processes with electrons in the reservoirs.
  • First, the spin eigenstates are redefined and the energy splittings are renormalized.
  • Finally, electron spins can also be flipped by fluctuations in the environment, thereby exchanging energy with degrees of freedom in the environment.

1. Origin

  • The spin of an electron in an atom can interact with the spin of “its” atomic nucleus through the hyperfine coupling.
  • Since the electron wave function is inhomogeneous, the coupling strength However, the authors note that the full quantum description is required to analyze correlations between microscopic nuclear spin states and the single electron spin state, as, e.g., in a study of the entanglement between electron and nuclear spins.
  • For any given host material, this value is independent of the number of nuclei N that the electron overlaps with—for larger numbers of nuclei, the contribution from each nuclear spin to BN is smaller the typical value for Ak is proportional to 1/N .
  • Similar values were obtained earlier for electrons bound to shallow donors in GaAs Dzhioev et al., 2002 .

2. Spin-orbit interaction in bulk and two dimensions

  • In order to obtain the spin-orbit Hamiltonian in twodimensional 2D systems, the authors integrate over the growth direction.
  • Usually the cubic terms are much smaller than the linear terms, since pz 2 px 2 ,py 2 due to the strong confinement along z.
  • Similarly, the authors now write down the spin-orbit Hamiltonian for the Rashba contribution.
  • From Fig. 19, the authors see that the Rashba and Dresselhaus contributions add up for motion along the 110 direction and oppose each other along 1̄10 , i.e., the spin-orbit interaction is anisotropic Könemann et al., 2005 .
  • The Dyakonov-Perel mechanism Dyakonov and Perel, 1972; Wrinkler, 2003 refers to spin randomization that occurs when the electron follows randomly oriented ballistic trajectories between scattering events for each trajectory, the internal magnetic field is differently oriented .

3. Spin-orbit interaction in quantum dots

  • From the semiclassical picture of the spin-orbit interaction, the authors expect that in 2D quantum dots with dimensions much smaller than the spin-orbit length lSO, the electron spin states will be hardly affected by the spinorbit interaction.
  • The spinorbit Hamiltonian does couple states that contain both different orbital and different spin parts Khaetskii and Nazarov, 2000 .
  • Here EZ refers to the unperturbed spin splitting in the remainder of the review, EZ refers to the actual spin splitting, including all perturbations .
  • It can be seen from inspection of the spin-orbit Hamiltonian and the form of the wave functions that many of the matrix elements in these expressions are zero.

4. Relaxation via the phonon bath

  • Electric fields cannot cause transitions between pure spin states.
  • The authors have seen that the spin-orbit interaction perturbs the spin states and the eigenstates become admixtures of spin and orbital states, see Eqs. 17 – 21 .
  • The electric field associated with a single phonon scales as 1/ q for piezoelectric phonons and as q for deformation potential phonons, where q is the phonon wave number.
  • The authors can similarly work out the 1/T1 dependence on the dot size l or, equivalently, on the orbital level spacing Eorb l −2 in single dots, Eorb can only be tuned over a small range, but in double dots, the splitting between bonding and antibonding orbitals can be modified over several orders of magnitudes Wang and Wu, 2006 .

5. Phase randomization due to the spin-orbit interaction

  • The authors have seen that the phonon bath can induce transitions between different spin-orbit admixed spin states, and absorb the spin flip energy.
  • Remarkably, to leading order in the spin-orbit interaction, there is no pure phase randomization of the electron spin, such that in fact T2=2T1 Golovach et al., 2004 .
  • For a magnetic field perpendicular to the plane of the 2DEG, this can be understood from the form of the spin-orbit Hamiltonian.
  • With B along ẑ, these terms lead to spin flips but not to pure phase randomization.

2. Effect of the Overhauser field on the electron-spin time evolution

  • The electron spin will precess about the vector of the total magnetic field it experiences, here the vector sum of the externally applied magnetic field B0 and the nuclear field B N. Throughout this section, the authors call the longitudinal component BN z .
  • First, the hyperfine field or Overhauser field BN will change if the local nuclear polarization kIk changes.
  • The time scale T2 * can be measured as the decay time of the electron-spin signal during free evolution, averaged over the nuclear field distribution Fig. 26 a .
  • In the rotating frame, the spin will then rotate about the vector sum of B1 and BN z Fig. 26 b , which may be a rather different rotation than intended.
  • A , the hyperfine interaction also leads to admixing of spin and orbital states.

3. Mechanisms and time scales of nuclear field fluctuations

  • The authors have seen that the nuclear field only leads to a loss of spin coherence because it is random and unknown—if B N were fixed in time, they could simply determine its value and the uncertainty would be removed.
  • I± are the nuclear spin raising and lowering operators.
  • This virtual electron-nuclear flip-flop process continues to be effective up to much higher B0 than real electronnuclear flip-flops.
  • Altogether the dipole-dipole and hyperfine interactions are expected to lead to moderate time scales 10–100 s for BN x,y fluctuations.

4. Electron-spin decoherence in a fluctuating nuclear field

  • B.2, the authors saw that they lose their knowledge of the electron-spin phase after a time T2 *, in case the nuclear field orientation and strength are unknown.
  • The reason is that T2 depends not only on the time scale of the nuclear field fluctuations tnuc , but also on the amplitude and stochastics of the fluctuations.
  • Also, contrary to the usual case, the echo decay is not well described by a single exponential.
  • Similar echolike decay times were observed in optical measurements on an ensemble of quantum dots that each contain a single-electron spin Greilich, Yakovlev, et al., 2006 .
  • Finally, the authors point out that it may be possible to extend tnuc, i.e., to almost freeze the nuclear field fluctuations.

C. Summary of mechanisms and time scales

  • The authors present understanding of the mechanisms and time scales for energy relaxation and phase randomization of electron spins in few-electron quantum dots is summarized as follows as before, most numbers are specific to GaAs dots, but the underlying physics is similar in other dot systems .
  • Energy relaxation is dominated by direct electronnuclear flip-flops near zero field or whenever the relevant electron-spin states are degenerate .
  • Spin-phonon coupling is inefficient, and occurs mostly indirectly, mediated either by the hyperfine interaction or by spin-orbit interaction.
  • As B0 further increases, the phonon density of states increases and phonons couple more efficiently to the dot orbitals the phonon wavelength gets closer to the dot size , so at some point relaxation becomes faster again and T1 decreases with field.
  • Phase coherence is lost on much shorter time scales.

VIII. SPIN STATES IN DOUBLE QUANTUM DOTS

  • The authors discuss the spin physics of double quantum dots.
  • Then, the authors show how the spin selection rules can lead to a blockade in electron transport through the double dot.
  • Finally, the authors describe how this spin blockade is influenced by the hyperfine interaction with the nuclear spins, and discuss the resulting dynamics.

A. Electronic properties of electrons in double dots

  • The authors first ignore the spin of electrons and describe the basic electronic properties of double quantum dots.
  • The properties of spinless electrons in double dots have been treated in detail by Van der Wiel et al. 2003 .
  • Here the authors give all theory relevant for electron spins in double dots without going into the details of the derivations.

1. Charge stability diagram

  • Consider two quantum dots, labeled 1 and 2, whose electrochemical potentials are controlled independently by the gate voltages VG,1 and VG,2, respectively.
  • This allows the absolute number of electrons to be determined unambiguously in any region of gate voltage space, by simply counting the number of charge transition lines from the 0,0 region to the region of interest.
  • The bright lines in between the triple points in Fig. 28 b are due to an electron moving from one dot to the other.
  • When the tunnel coupling tc becomes significant, electrons are not fully localized anymore in single dots but rather occupy molecular orbitals that span both dots Van der Wiel et al., 2003 .
  • Rev. Mod. Phys., Vol. 79, No. 4, October–December 2007 A = 1 − 2. 34 When the single dot states are aligned, the energy of bonding orbital is lower by tc than the energy of the single dot orbitals, and the energy of the antibonding orbital is higher by the same amount.

2. High bias regime: Bias triangles

  • When the source-drain bias voltage is increased, two different types of tunneling can occur.
  • When there are no aligned levels elastic tunneling is suppressed and inelastic tunneling dominates the electron transport.
  • The energy window that is being probed is determined by the misalignment between the levels in the two dots.
  • When the source-drain bias voltage is increased, the triple points evolve into bias triangles, as depicted in Fig. 30 for weak tunnel coupling.
  • Moving upwards along the left leg of the triangle, 1 is fixed 1 1,0 is aligned with the source electrochemical potential and only 2 is changed.

B. Spin states in two-electron double dots

  • The physics of one- and two-electron spin states in single dots was discussed in Sec. IV.
  • The authors repeat the description of the single dot states, as discussed in Sec. IV.
  • The energy difference between the lowest-energy singlet and triplet states J depends on the tunnel coupling tc and the single dot charging energy EC.
  • In a finite magnetic field, the triplet states are split by the Zeeman energy.
  • In the strict sense of the word, exchange energy refers to the difference in Coulomb energy between states whose orbital wave functions differ only in their symmetry symmetric for a spin singlet and antisymmetric for a spin triplet Ashcroft and Mermin, 1974 .

D. Hyperfine interaction in a double dot: Singlet-triplet mixing

  • Early experiments in semiconducting heterostructures in the quantum Hall regime demonstrated that spinpolarized currents could be used to polarize the nuclei in the substrate Wald et al., 1994; Dixon et al., 1997 .
  • The importance of the hyperfine field becomes apparent when considering two spatially separated electron spins in a double dot structure.
  • The leakage current decreased suddenly for fields exceeding 0.9 T. Measurements of the leakage current for the opposite magnetic-field sweep direction showed hysteretic behavior.
  • The leakage current in the Pauli spin blockade region occurs due to spin relaxation from T− 1,1 to S 1,1 and the hysteretic behavior observed in Fig. 36 a can be explained in terms of triplet-to-singlet relaxation via hyperfine-induced flip-flops with the spins of the lattice nuclei in the dot.
  • A pulse then shifts the gate voltages to the 1,1 region of the charge stability diagram.

A. Single-spin manipulation: ESR

  • Alternatively, spin rotations could be realized by electrical or optical excitation.
  • Optical excitation can induce spin flips via Raman transitions Imamoglu et al., 1999 or the optical Stark effect Gupta et al., 2001 .
  • Furthermore, the alternating electric fields that are unavoidably also generated along with the alternating magnetic field can kick the electron out of the dot via photon-assisted tunneling PAT processes Platero and Aguado, 2004 .
  • Instead, ESR detection in quantum dots has been realized using two quantum dots in series, tuned to the spin blockade regime described in Sec. VIII.C.
  • The measured dot current oscillates periodically with the rf burst length Fig. 45 , demonstrating driven, coherent electron-spin rotations, or Rabi oscillations.

B. Manipulation of coupled electron spins

  • It has been shown that single spin rotations combined with two-qubit operations can be used to create basic quantum gates.
  • Loss and DiVincenzo have shown that a XOR gate is implemented by combining single-spin rotations with SWAP operations Loss and DiVincenzo, 1998 .
  • In the previous section experiments demonstrating single spin manipulation were reviewed.
  • In this section the authors review experiments by Petta et al. that have used fast control of Rev. Mod.
  • Phys., Vol. 79, No. 4, October–December 2007 the singlet-triplet energy splitting in a double dot system to demonstrate a SWAP operation and implement a singlet-triplet spin-echo pulse sequence, leading to microsecond dephasing times Petta, Johnson, Taylor, et al.,.

X. PERSPECTIVES

  • This review has described the spin physics of fewelectron quantum dots.
  • These are only the first experimental results and further improvements are expected.
  • These two- and three-electron qubit encodings eliminate the need for the technologically challenging single-spin rotations.
  • This may allow for new experiments exploring quantum coherence in the solid state, for instance involving entanglement and testing Bell’s inequalities.
  • The coherence time is currently limited by the randomness in the nuclear-spin system.

ACKNOWLEDGMENTS

  • The authors acknowledge the collaboration with many colleagues, in particular those from their institutes in Tokyo, Delft, and at Harvard.
  • The authors thank David Awschalom, Jeroen Elzerman, Joshua Folk, Toshimasa Fujisawa, Toshiaki Hayashi, Yoshiro Hirayama, Alex Johnson, Frank Koppens, Daniel Loss, Mikhail Lukin, Charlie Marcus, Tristan Meunier, Katja Nowack, Keiji Ono, Rogerio de Sousa, Mike Stopa, Jacob Taylor, Ivo Vink, Laurens Willems van Beveren, Wilfred van der Wiel, Stu Wolf, and Amir Yacoby.
  • J.R.P. acknowledges support from the ARO/ARDA/DTO STIC program.
  • S.T. acknowledges financial support from the Grant-in-Aid for Scientific Research A Grant No. 40302799 , the Special Coordination Funds for Promoting Science and Technology, MEXT, and CREST-JST.

1. Sign of the spin ground states

  • The different signs in the magnetic moment is due to the difference in the polarity of the electron and proton charge.
  • Since both free electrons and protons have a positive g factor, the spins in the ground states of a free electron spin down and a proton spin up are antiparallel to each other.
  • Hence both the nuclei and electrons in the ground state in GaAs have their spin aligned parallel to the external field, i.e., they are spin up.

2. Sign and magnitude of the thermal nuclear field

  • The authors can calculate the thermal average of the spin I of each isotope using the Maxwell-Boltzmann distribution.
  • Since I is always positive in thermal equilibrium, the authors derive from Eq. A3 that the thermal nuclear field acts against the applied field.

3. Sign of the dynamic nuclear field

  • Nuclear polarization can build up dynamically via flipflop processes, where an electron and a nucleus flip their spin simultaneously.
  • Because of the large energy mismatch between nuclear and electron Zeeman energy, a flip-flop process where an electron spin is excited is very unlikely, since the required energy is not available in the system EZ,nucl kBT EZ,el .
  • This brings the nucleus to a different spin state with Iz=−1.
  • Many of these processes can dynamically build up a considerable polarization, whose sign is opposite to that of the thermal nuclear field.
  • This has already been observed in the ESR experiments on 2DEGs see, e.g., Dobers et al. 1988 , where the excited electron spin relaxes via a flip-flop process.

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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
Published 1 October 2007; publisher error corrected 4 October 2007
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 1 the rotation of an electron spin
into a superposition of up and down, 2 the measurement of the quantum state of an individual spin,
and 3 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 i.e., single spintronics.
DOI: 10.1103/RevModPhys.79.1217 PACS numbers: 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
0034-6861/2007/794/121749 ©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 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
Rugar et al., 2004. 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 Ashcroft and Mermin, 1974. It remains
striking that an abstract concept as antisymmetrization
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 Wolf
et al., 2001; Zutic et al., 2004. In addition to applications,
important scientific discoveries have been made in the
field of spintronics Awschalom and Flatte, 2007, in-
cluding magnetic semiconductors Ohno, 1998 and the
spin Hall effect Sih et al., 2005. 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 for an overview, see,
e.g., Gunther and Barbara 1994.
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 via field-
effect gates 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 Kouwenhoven et al., 2001. Electrons are
1
See URL: http://www.lorentz.leidenuniv.nl/history/spin/
goudsmit.html
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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 1998
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 DiVincenzo,
2000: 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 or holes. The dot can be
coupled via tunnel barriers to reservoirs, with which
electrons can be exchanged see Fig. 1. 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 2005.
3
See, e.g., Atature et al. 2006, Berezovsky et al. 2006,
Greilich, Oulton, et al. 2006, Krenner et al. 2006, 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 Park et al., 2002, normal metal Petta
and Ralph, 2001, superconducting Ralph et al., 1995;
von Delft and Ralph, 2001, or ferromagnetic nanopar-
ticles Guéron et al., 1999, self-assembled quantum dots
Klein et al., 1996, semiconductor lateral Kouwen-
hoven et al., 1997 or vertical dots Kouwenhoven et al.,
2001, and also semiconducting nanowires or carbon
nanotubes Dekker, 1999; McEuen, 2000; Björk et al.,
2004.
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 van Houten et al.,
1992. 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
Kouwenhoven et al., 2001.
The physics of dots containing more than two elec-
trons has been previously reviewed Kouwenhoven et
al., 1997; Reimann and Manninen, 2002. 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 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.
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 see Fig. 2. By doping the AlGaAs layer
with Si, free electrons are introduced. These accumulate
at the GaAs/AlGaAs interface, typically 50100 nm be-
low the surface, forming a two-dimensional electron gas
2DEG—a thin 共⬃10 nm sheet of electrons that can
only move along the interface. The 2DEG can have high
mobility and relatively low electron density typically
10
5
−10
7
cm
2
/V s and ⬃共1−5 10
15
m
−2
, respectively.
The low electron-density results in a large Fermi wave-
length 共⬃40 nm 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 see Fig. 2a.
Electron-beam lithography enables fabrication of gate
structures with dimensions down to a few tens of na-
nometers Fig. 2, 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 a a lateral
geometry and b in a vertical geometry. The quantum dot
represented by a disk 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. a Schematic view. Negative voltages applied to
metal gate electrodes dark gray lead to depleted regions
white in the 2DEG light gray. Ohmic contacts light gray
columns enable bonding wires not shown to make electrical
contact to the 2DEG reservoirs. b, c Scanning electron mi-
crographs of b a few-electron single-dot device and c a
double dot device, showing the gate electrodes light gray on
top of the surface dark gray. 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 dots to the other reservoir. For the device in
c, 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 b courtesy of A. Sachrajda.
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resistance Ohmic 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 Kouwenhoven et al., 2001. 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 Ciorga et al.,
2000. Their device, shown in Fig. 2b, 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 and thereby electron number 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 Elzerman et al.,
2003. Their design is shown in Fig. 2c for more details
on design considerations and related versions of this
gate design, see Hanson 2005. In addition to the
coupled dots, two quantum point contacts QPCs 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 Petta et al.,
2004; Pioro-Ladrière et al., 2005; 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 Hanson et al., 2005.
C. Measurement techniques
In this review, two all-electrical measurement tech-
niques are discussed: i measurement of the current due
to transport of electrons through the dot, and ii 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. 1992 and Field et al. 1993. But
whereas current measurements were already used exten-
sively in the first experiments on quantum dots Kou-
wenhoven et al., 1997, charge sensing has only recently
been fully developed as a spectroscopic tool Elzerman,
Hanson, Willems van Beveren, Vandersypen, et al., 2004;
Johnson, Marcus, et al., 2005. Several implementations
of electrometers coupled to a quantum dot have been
demonstrated: a single-electron transistor fabricated on
top of the heterostructure Ashoori et al., 1992; Lu et al.,
2003, a second electrostatically defined quantum dot
Hofmann et al., 1995; Fujisawa et al., 2004, and a quan-
tum point contact QPC兲共Field et al., 1993; Sprinzak et
al., 2002. 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 see, e.g.,
Vandersypen et al. 2004 for a discussion on noise
sources
. 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 most notably via cotunnel-
ing, charge detection is preferred for quantum informa-
tion purposes since it still functions for very small cou-
plings to a single 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. 1992 and Kouwenhoven et al. 1997,
2001. 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
. In general, capacitances to
multiple gates and other parts of the 2DEG will also
play a role; they can simply be added to C. 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 UN 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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  • ... centers in diamond (Doherty et al., 2012; Wrachtrup and Jelezko, 2006) possess good coherence properties, which allow long storage times. Furthermore, rapid progress has been made with quantum dots (Hanson et al., 2007; Loss and DiVincenzo, 1998; Zwanenburg et al., 2012), which can be fabricated on a chip and controlled relatively easily using electric signals. Remarkable progress has also been made on other system...

    [...]

  • ...Furthermore, rapid progress has been made with quantum dots (Hanson et al., 2007; Loss and DiVincenzo, 1998; Zwanenburg et al., 2013), which can be fabricated on a chip and controlled relatively easily using electric signals....

    [...]

  • ...s in solids generally fall into two classes: quantum dots and atomic impurities. Quantum dots are small nanostructures where electrons are trapped in a potential well and have discrete energy levels (Hanson et al., 2007; Loss and DiVincenzo, 1998; Zwanenburg et al., 2012), see Fig. 1(c). These come in several forms. One is electrostatically-dened quantum dots, where the distribution of electrons is controlled by vo...

    [...]

  • ...…trapped ions and atoms (Blatt and Wineland, 2008; Bloch, 2008; Buluta et al., 2011), spins (Buluta et al., 2011; Hanson and Awschalom, 2008a; Hanson et al., 2007), and superconducting circuits (Buluta et al., 2011; Clarke and Wilhelm, 2008; Makhlin et al., 2001; Wendin and Shumeiko, 2007;…...

    [...]

  • ...Quantum dots are small nanostructures where electrons are trapped in a potential well and have discrete energy levels (Hanson et al., 2007; Loss and DiVincenzo, 1998; Zwanenburg et al., 2013); see Fig....

    [...]

References
More filters
Book
01 Jan 2000
TL;DR: In this article, the quantum Fourier transform and its application in quantum information theory is discussed, and distance measures for quantum information are defined. And quantum error-correction and entropy and information are discussed.
Abstract: Part I Fundamental Concepts: 1 Introduction and overview 2 Introduction to quantum mechanics 3 Introduction to computer science Part II Quantum Computation: 4 Quantum circuits 5 The quantum Fourier transform and its application 6 Quantum search algorithms 7 Quantum computers: physical realization Part III Quantum Information: 8 Quantum noise and quantum operations 9 Distance measures for quantum information 10 Quantum error-correction 11 Entropy and information 12 Quantum information theory Appendices References Index

25,929 citations

01 Dec 2010
TL;DR: This chapter discusses quantum information theory, public-key cryptography and the RSA cryptosystem, and the proof of Lieb's theorem.
Abstract: Part I. Fundamental Concepts: 1. Introduction and overview 2. Introduction to quantum mechanics 3. Introduction to computer science Part II. Quantum Computation: 4. Quantum circuits 5. The quantum Fourier transform and its application 6. Quantum search algorithms 7. Quantum computers: physical realization Part III. Quantum Information: 8. Quantum noise and quantum operations 9. Distance measures for quantum information 10. Quantum error-correction 11. Entropy and information 12. Quantum information theory Appendices References Index.

14,825 citations

Journal ArticleDOI
16 Nov 2001-Science
TL;DR: This review describes a new paradigm of electronics based on the spin degree of freedom of the electron, which has the potential advantages of nonvolatility, increased data processing speed, decreased electric power consumption, and increased integration densities compared with conventional semiconductor devices.
Abstract: This review describes a new paradigm of electronics based on the spin degree of freedom of the electron. Either adding the spin degree of freedom to conventional charge-based electronic devices or using the spin alone has the potential advantages of nonvolatility, increased data processing speed, decreased electric power consumption, and increased integration densities compared with conventional semiconductor devices. To successfully incorporate spins into existing semiconductor technology, one has to resolve technical issues such as efficient injection, transport, control and manipulation, and detection of spin polarization as well as spin-polarized currents. Recent advances in new materials engineering hold the promise of realizing spintronic devices in the near future. We review the current state of the spin-based devices, efforts in new materials fabrication, issues in spin transport, and optical spin manipulation.

9,917 citations


"Spins in few-electron quantum dots" refers methods in this paper

  • ...Logic gates have been realized based on magnetoresistance effects as well Wolf et al., 2001; Zutic et al., 2004 ....

    [...]

Journal ArticleDOI
TL;DR: Spintronics, or spin electronics, involves the study of active control and manipulation of spin degrees of freedom in solid-state systems as discussed by the authors, where the primary focus is on the basic physical principles underlying the generation of carrier spin polarization, spin dynamics, and spin-polarized transport.
Abstract: Spintronics, or spin electronics, involves the study of active control and manipulation of spin degrees of freedom in solid-state systems. This article reviews the current status of this subject, including both recent advances and well-established results. The primary focus is on the basic physical principles underlying the generation of carrier spin polarization, spin dynamics, and spin-polarized transport in semiconductors and metals. Spin transport differs from charge transport in that spin is a nonconserved quantity in solids due to spin-orbit and hyperfine coupling. The authors discuss in detail spin decoherence mechanisms in metals and semiconductors. Various theories of spin injection and spin-polarized transport are applied to hybrid structures relevant to spin-based devices and fundamental studies of materials properties. Experimental work is reviewed with the emphasis on projected applications, in which external electric and magnetic fields and illumination by light will be used to control spin and charge dynamics to create new functionalities not feasible or ineffective with conventional electronics.

9,158 citations


"Spins in few-electron quantum dots" refers background or methods in this paper

  • ...In 2DEGs, spin-orbit coupling whether Rashba or Dresselhaus can lead to spin relaxation via several mechanisms Zutic et al., 2004 ....

    [...]

  • ...Logic gates have been realized based on magnetoresistance effects as well Wolf et al., 2001; Zutic et al., 2004 ....

    [...]

Book
01 Jan 1961

8,649 citations

Frequently Asked Questions (18)
Q1. What have the authors contributed in "Spins in few-electron quantum dots" ?

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. 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. Furthermore, experimental techniques are discussed that allow for 1 the rotation of an electron spin into a superposition of up and down, 2 the measurement of the quantum state of an individual spin, and 3 the control of the interaction between two neighboring spins by the Heisenberg exchange interaction. Electrical contacts can be made for charge transport measurements and electrostatic gates can be used for controlling the dot potential. 

Spin filling can also be deduced from excited-state spectroscopy without changing the magnetic field Cobden et al., 1998 , provided the Zeeman energy splitting EZ=2 EZ =g BB between spin-up and spin-down electrons can be resolved. 

due to the effect of the Zeeman splitting, the matrix element in Eq. 22 picks up another factor of EZ assuming only single-phonon processes are relevant . 

Since both nuclear spins and the localized electron spin are quantum objects, the hyperfine coupling could in principle create entanglement between them if both the electron spin and nuclear spins had a sufficiently pure initial state; see Braunstein et al., 1999 . 

Electric fields affect spins only indirectly, so generally spin states are only weakly influenced by their electric environment as well. 

In fact, the nuclear field has been the main limitation on the fidelity of spin rotations in recent electron-spin resonance experiments in a quantum dot see Sec. IX. 

by starting from zero electrons and thus zero spin and tracking the change in spin at subsequent electron transitions, the total spin of the ground state can be determined Willems van Beveren et al., 2005 . 

If the effect of the nuclear field on the electron-spin coherence could be suppressed, the spin-orbit interaction would limit T2, to a value of 2T1 to first order in the spin-orbit interaction , which, is as the authors have seen, a very long time. 

The phonon-induced transition rate between the renormalized states n , l , ↑ 1 and n , l , ↓ 1 is given by Fermi’s golden rule an analogous expression can be derived for relaxation from triplet to singlet states, or between other spin states := 2n,l1 nl↑ 

Sz of the ground state changes by more than 12 , which can occur due to many-body interactions in the dot, can lead to a spin blockade of the current Weinmann et al., 1995; Korkusiński et al., 2004 . 

The probability that a triplet state is formed is given by 3 T / S+3 T , where the factor of 3 is due to the degeneracy of the triplets. 

The signal-to-noise ratio is enhanced significantly by lock-in detection of GQPC at the pulse frequency, thus measuring the average change in GQPC when a voltage pulse is applied Sprinzak et al., 2002 . 

If the tunnel rates for spin up and spin down are not equal, the amplitude of the current can be used to determine the spin filling. 

The authors now show that the same result follows from the quantum-mechanical description, where the spin-orbit coupling can be treated as a small perturbation to the discrete orbital energy-level spectrum in the quantum dot. 

the authors note that the full quantum description is required to analyze correlations between microscopic nuclear spin states and the single electron spin state, as, e.g., in a study of the entanglement between electron and nuclear spins. 

In GaAs, estimates for vary from 103 to 3 103 m/s, and it follows that the spin-orbit length lSO = / m* is 1–10 m, in agreement with experimentally measured values Zumbühl et al., 2002 . 

The error probabilities are found to be =0.15 and =0.04, where is the probability that a measurement on the state S T yields the wrong outcome T S . 

The sensitivity of the charge sensor to changes in the dot charge can be optimized using an appropriate gate design Zhang et al., 2004 .