Spins in few-electron quantum dots
Summary (11 min read)
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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Cites background from "Spins in few-electron quantum dots"
...The resonance condition is obtained by shifting the energy levels with V g and Coulomb-blockade diamonds appear in differential conductivity maps as a function of source–drain voltage and V g (ref...
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...Here we are not treating the highly successful area of spins in semiconductors for solid-state quantum computation — for a review see ref...
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Cites background from "Spins in few-electron quantum dots"
... 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...
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...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....
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...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...
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...…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;…...
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...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....
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References
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14,825 citations
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"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 ....
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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 ....
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...Logic gates have been realized based on magnetoresistance effects as well Wolf et al., 2001; Zutic et al., 2004 ....
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Related Papers (5)
Frequently Asked Questions (18)
Q2. How can the authors deduce spin filling 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.
Q3. Why does the matrix element in Eq. 22 pick up another factor of EZ?
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 .
Q4. What is the simplest explanation for the hyperfine coupling?
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 .
Q5. What is the effect of electric fields on spin states?
Electric fields affect spins only indirectly, so generally spin states are only weakly influenced by their electric environment as well.
Q6. What is the main limitation on the fidelity of spin rotations in a quantum dot?
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.
Q7. How can the authors determine the total spin of the ground state?
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 .
Q8. how long would the effect of the nuclear field on the electron-spin interaction be suppresse?
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.
Q9. What is the phonon-induced transition rate between the renormalized states?
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↑
Q10. How many changes can lead to a spin blockade?
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 .
Q11. What is the probability that a triplet state is formed?
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.
Q12. What is the effect of lock-in detection of GQPC on the pulse frequency?
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 .
Q13. What is the amplitude of the current used to determine the spin filling?
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.
Q14. How can the spin-orbit coupling be treated?
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.
Q15. What is the full quantum description of nuclear spins?
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.
Q16. How long does the spin-orbit length lSO mean?
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 .
Q17. What is the error probability of a measurement on the state T?
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 .
Q18. How can the sensitivity of the charge sensor be optimized?
The sensitivity of the charge sensor to changes in the dot charge can be optimized using an appropriate gate design Zhang et al., 2004 .