Current-induced hole spin polarization in a quantum dot via a chiral quasi bound state

Abstract

We put forward a mechanism for the current-induced spin polarization in semiconductor heterostructures, which is based on the complex structure of the valence band. It takes place for a light hole in a quantum dot side-coupled to a quantum wire with heavy holes. In stark contrast with the traditional mechanisms based on the linear in momentum spin–orbit coupling, an exponentially small bias applied to this structure is enough to create the 100% spin polarization in the quantum dot. Microscopically, this effect is related to the formation of the chiral quasi bound states and the spin-dependent tunneling of holes from the quantum wire to the quantum dot. This new concept is equally valid for the GaAs-, Si- and Ge-based nanostructures.

---

New concepts

We put forward a new mechanism for the electrical spin orientation in semiconductor heterostructures, which exploits the complex structure of the valence band. This is an alternative to the old concepts of the current-induced spin polarization, which are based on the momentum-dependent spin–orbit splittings of the conduction and valence bands. The main advantages of the new concept are the possibility to completely polarize hole spin by an exponentially small electric current and its universal applicability to Si-, Ge- and GaAs-based heterostructures. These advantages originate in the formation of the robust chiral quasi bound states of light holes localized in the quantum dot in the vicinity of the quantum wire with heavy holes. This specific structural design is not realized to date, so it represents a promising challenge for the fabrication techniques of the nanoscale heterostructures.

1 Introduction

With the approach of the quantum computation era¹ the localized spins in quantum dots (QDs) remain the most prominent candidates for scalable quantum simulations and quantum information processing.^2–4 The electron spins can be already efficiently transferred in chains of QDs using the well-controlled spin–spin interactions, entangled spin states can be generated, and CNOT-gates can be realized with very high fidelity.^5–10

However, electrical polarization of individual spins in QDs still remains a vital problem. Basically, this can be performed due to the pronounced spin–orbit interaction in most of the semiconductors. Historically, the current-induced spin polarization was first proposed theoretically and realized experimentally for bulk Te, which is a gyrotropic material.^11,12 Later the current-induced spin polarization was demonstrated for quantum wells made of GaAs-like semiconductors,^13,14 strained bulk semiconductors,¹⁵ and epilayers.^16,17 However, the maximum degree of current-induced spin polarization is limited to a few percent because of the weakness of the momentum-dependent spin–orbit splitting compared to the Fermi energy.¹⁸ Streaming and hopping conductivity regimes can increase the polarization a few times,^19,20 but it still remains much smaller than unity.

In this paper we demonstrate that the strong spin–orbit splitting of the valence band can be exploited to create 100% spin polarization of holes localized in QDs in specifically designed structures. This splitting is large; for example, it is of the order of 300 meV in GaAs and Ge and is about 70 meV in Si. Thus, our proposal is equally relevant for high-quality optically-addressable GaAs-based structures,^21–23 most technologically advanced Si-based structures,^24–26 and emerging Ge-based structures.^27–30

Microscopically, the current-induced spin polarization takes place due to the spin-dependent hole tunneling, which leads to the formation of chiral quasi bound states in the continuum.^31,32 This represents a new concept that has some similarities to the spin filtering in magnetic fields³³ and chiral photonics.^34–36 A similar chirality-induced spin selectivity is also used to control spin by non-magnetic means in various molecular systems.^37–40

2 System under study

The system under study represents a quantum dot weakly side-coupled to a quantum wire,^41–43 see Fig. 1. We assume the structure to be formed electrostatically in a two-dimensional hole gas with a strong splitting between heavy and light hole subbands in the material with the top of the valence band described by Γ₈ representation of the T_d group or Γ₈⁺ representation of the O_h group. We consider the Fermi energy of heavy holes in the wire to be close to the energy of light hole state localized in the QD, as shown in Fig. 1(b). The heavy hole state in the QD is assumed to be deeply below the Fermi energy, so this state is always doubly occupied. Thus we will take into account only the tunneling between heavy holes in the quantum wire and light holes in the QD.

Fig. 1 (a) QD side-coupled to the quantum wire. The difference of the tunneling probabilities for spin-up (red balls with arrows) and spin-down (blue ones) holes from the quantum wire to the QD and of the spin flip rates leads to the current-induced spin polarization in the QD. (b) Energy diagram and the geometric parameters of the system. (c) Heavy hole distribution function in the quantum wire. The light hole resonance energy E₀ is between the Fermi energies of the heavy holes in the quantum wire propagating from the left, E^L_F, and from the right, E^R_F, leads.

The proposed device is described by the C_2v symmetry group. We choose the coordinate frame to have the x axis along the quantum wire and the z axis along the structure growth axis, as shown in Fig. 1(a). Clearly, the electric current j_x of heavy holes flowing along the wire can linearly couple to the light hole spin S_z in the QD along the z axis. The symmetry of this effect is the same as that of the Mott scattering or the spin Hall effect, so the spin polarization would change sign for the QD placed at the opposite side of the quantum wire. The current-induced spin polarization does not require any magnetic field or microscopic symmetry reduction, i.e. Rashba or Dresselhaus spin–orbit interactions, and it appears even in the centrosymmetric materials such as Si and Ge along with GaAs. Because of this, a very large degree of spin polarization can be achieved, which we demonstrate below.

The Hamiltonian of the system can be written as follows:

where E₀ is the single light hole energy level in the QD, are the occupancies of this state by holes with the spin ±1/2 along the z axis, respectively, with d_± being the corresponding annihilation operators, U is the Coulomb interaction energy between the two localized light holes, E_k denotes the energy of a heavy hole in the quantum wire with the wave vector k, are the occupancies of the states in the wire with the spin ±3/2 with c_k,± being the corresponding annihilation operators. We assume the wire to be ballistic and neglect the interaction between holes in it.

Most importantly, V_k,± in eqn (1) denote the tunneling matrix elements between the quantum wire and the QD. They are produced by the off-diagonal elements of the Luttinger Hamiltonian^44,45 and can be calculated as follows:

where Φ_± denotes the localized light hole wave function of an isolated QD with the corresponding spin, Ψ_k,± is the heavy hole wave function of the isolated quantum wire, γ₂ is the second Luttinger parameter (we use the spherical approximation), m₀ is the free electron mass, and k_α = − i∂/∂α with α = x, y are the components of the wave vector operator. It follows from the time reversal symmetry that . We note that the heavy hole mass along the wire is given by m = m₀/(γ₁ + γ₂), where γ₁ is the first Luttinger parameter, and the dispersion is given by E_k = ħ²k²/(2m). We chose the state with k = 0 to be the energy reference.

To be specific, let us consider the Gaussian wave functions:

where a is the localization length, which is assumed to be the same for the QD and the quantum wire and d is the distance between the center of the QD and the wire axis, see Fig. 1(b). Assumption of the same localization length for the QD and the quantum wire greatly simplifies the analytical calculations and does not change the general results.

To calculate the matrix elements V_k,± after eqn (2) we note that k_x can be replaced with k, while ik_y = ∂/∂y. Then we solve the integrals over x and y analytically and obtain

with a prefactor independent of k and d†. The matrix elements are shown in Fig. 2(a) as functions of the wave vector for d/a = 3. One can see that for the given k the tunneling matrix elements are generally strongly different for the spin-up and spin-down holes, which allows one to expect a high degree of current-induced spin polarization.

Fig. 2 (a) Tunneling matrix elements calculated after eqn (2) for d = 3a. (b) Chirality [capital script C] of the quasi bound state calculated after eqn (5). The negative values of d correspond to the location of the QD at the opposite side of the quantum wire.

Intuitively, one can imagine that a hole tunnels from the quantum wire to the quantum dot along a rounding trajectory, as shown in Fig. 1(a). The orbital rotation of the hole in the (xy) plane makes the system chiral and results in an effective magnetic field acting on the hole spin due to the spin–orbit interaction. This field favors tunneling of holes with a certain spin direction and leads to conversion from heavy to light holes.

The single particle states of the Hamiltonian (1) are well known from the studies of Anderson and Fano.^46–48 The coupling between the QD and the quantum wire leads to the formation of the quasi bound states at the energy . Their chirality (also termed directionality) [capital script C] can be defined as the difference of the probabilities for a light hole with a given spin to tunnel from the QD to the quantum wire states propagating to the right and to the left:^32,34,49

where due to the time reversal symmetry, the chirality is opposite for the two spin states, so the definition of its sign is ambiguous.

The chirality of the quasi bound state is shown in Fig. 2(b) as a color map. The regions with k₀ < 0 and d < 0 show that the chirality is odd under reflection in the (yz) and (xz) planes, respectively. Generally, the absolute value of the chirality is of the order of unity.

Notably, it turns to unity exactly along the four lines given by the equation

k₀a = ±d/a ± 1, (6)

see yellow and light blue lines in Fig. 2(b). These relations are specific for the wave functions (3) and their exact form is not significant for the following†. At the corresponding QD energies, one of the tunneling matrix elements vanishes, which is shown by yellow and light blue circles in Fig. 2(a). This happens because of the interplay between the k_x and k_y-related terms in eqn (2). Here, the operator k_x can be simply replaced with k, so the relative role of the corresponding contribution increases with increase of the energy of the QD state. As a result the sign of the total matrix element V_k0,± changes two times as a function of k₀.

As a result, completely chiral bound states in the continuum are formed at certain energies, where the light hole state in the QD couples to the states in the wire propagating only in one direction. These completely chiral bound states in the continuum are robust and appear almost for any choice of the QD and the quantum wire wave functions, as we have checked.

The Gaussian form of the wave functions (3) corresponds to the parabolic localization potential, so at k₀a > 2 the coupling to the second size quantized subband of the quantum wire can play a role. Still in the most realistic region of k₀a < 2 and d/a ≳ 2 the chirality is very close to unity.

3 Formalism

To calculate the current-induced spin polarization in the nonequilibrium steady state, we use the Keldysh diagram technique.^50–52 We start from the bare Hubbard retarded Green's function of an isolated QD:^53,54where 〈n_σ〉 with σ = ± are the average occupancies of the corresponding light hole spin states, δ → 0⁺, and we measure frequencies in the units of energy for brevity. Henceforth we focus on the limit of strong Coulomb repulsion (as compared with the quasi bound state width), while the general case is described in the ESI.† Then using the standard self-energy we obtain from the Dyson equationwhere is the width of the quasi bound state with being the total density of states in the quantum wire. This expression includes the suppression of the tunneling rate Γ due to the Coulomb blockade. We assume Γ to be much smaller than the band width E_1/a (wide band approximation) and neglect the quasi bound state energy renormalization. Note that for Γ ≪ E_1/a the corresponding uncertainty of k is much smaller than 1/a, so the dependence of the tunneling matrix elements on k can be neglected†.

The occupancies of the QD states are given by the lesser Green's function:

which in the steady state is given by G^<_σ = G^R_σΣ^<_σG^A_σ. Here the lesser self-energy depends on the Fermi energies in the left and right leads attached to the quantum wire, E^L_F and E^R_F, respectively:where θ (ω) is the Heaviside step function. Hence eqn (9) yields:

This set of two equations allows us to find self-consistently the occupancies of the spin states 〈n_σ〉. Ultimately, the degree of current-induced spin polarization in the QD is given by P = (〈n₊〉 − 〈n₋〉)/(〈n₊〉 + 〈n₋〉).

We note that this approach is valid on one hand for the temperatures below Γ, when the heavy hole distribution functions in the leads can be approximated by the step functions. The role of finite temperature is discussed in the ESI.† On the other hand, the temperature is assumed to be larger than the Kondo temperature T_K, so that the high order correlations between holes can be neglected.⁵⁴ In fact, the retarded Green's function (8) without quasi bound state width renormalization can be obtained from the equations of motion in the Hartree–Fock approximation.⁵⁵ The Kondo effect at low temperatures can be taken into account, for example, using equations of motion truncated in an appropriate way at the temperatures of the order of T_K^56,57 or much smaller than it.⁵⁵ In the Kondo regime the very sharp peak in density of states⁵⁸ may lead to a strong increase of current-induced spin polarization at small biases. This question, however, deserves a deeper investigation.

4 Results

It follows from eqn (11) that the chirality of the quasi bound state [capital script C] produces explicit spin-dependence of the occupancies provided a nonzero bias eV = E^L_F − E^R_F is applied to the quantum wire. The current-induced spin polarization is plotted in Fig. 3 as a function of bias for the different positions of the Fermi energy E_F = (E^L_F + E^R_F)/2 for the completely chiral quasi bound state, [capital script C] = 1. Generally, it is an odd function of the applied voltage and for the large voltages its absolute value reaches 100%. For the resonant case, E_F = E₀, the spin polarization saturates at eV ∼ Γ and for the detuned case it saturates at eV ∼ 2|E_F − E₀|, as clearly seen from the inset in Fig. 3. Generally, the current-induced spin polarization is the largest when the energy of the quasi bound state E₀ lies between the Fermi energies of the left and right leads, as it is shown in Fig. 1(c). In this case the holes flowing from the left can tunnel to the one spin state in the QD only due to its chirality, while the holes flowing from the right at the corresponding energy are absent due to the large bias. For [capital script C] < 1 the shape of this dependence is qualitatively the same.

Fig. 3 Current-induced spin polarization in the chiral bound state, [capital script C] = 1, as a function of the applied bias for the different Fermi levels E_F − E₀. The inset shows the spin polarization as a color map with the blue and red colors corresponding to P = −1 and +1, respectively.

Thus the large spin polarization can be induced by the voltages of the order of Γ. It is proportional to the squared tunneling matrix elements, which are given by eqn (4), and contain an exponentially small factor exp[−d²/(2a²)]. As a result, the tunneling rate Γ can be extremely small, which means that the spin polarization can saturate, in principle, at exponentially small bias. The ultimate limit for Γ is set by the spin relaxation time in an isolated QD. In the absence of a magnetic field, the nuclear spin fluctuations lead to the spin relaxation at the nanosecond time scale,^59,60 which corresponds to Γ ∼ 1 μeV. However, to achieve this giant spin sensitivity the temperature of the system should be as low as a few millikelvins. Otherwise the applied bias should exceed the thermal energy, see the corresponding analysis in the ESI.†

The spin polarization in the limit of large bias can be found from eqn (11) by setting the first arctangent to π/2 and the second one to −π/2. This gives:

which is determined solely by the chirality of the quasi bound state. In this limit one can consider the holes flowing from the left only, and the chirality determines the ratio of the tunneling rates to the two spin states in the QD. The maximum spin polarization degree is shown in Fig. 4 as a function of the quasi bound state energy for different distances between the QD and the quantum wire. Here the range of E₀ < 4[scr E, script letter E] with [scr E, script letter E] = ħ²/2ma² corresponds to the energies below the bottom of the second size quantized band of the quantum wire. Generally, the maximum spin polarization decreases with increase of the distance between the quantum wire and the QD because of the chirality decrease shown in Fig. 2(b). However, it is still close to unity almost in the whole range of the quasi bound state energies E₀ up to d ∼ 10a. We note that this corresponds to the suppression of the quasi bound state width by a giant factor of e^d2/a2. The inset in Fig. 4 explicitly shows the range of parameters where the current-induced spin polarization at large bias exceeds 90%. This range can be even larger for the different form of the wave functions†.

Fig. 4 Spin polarization in the limit of the large bias as a function of the quasi bound state energy for d/a = 3 (black solid curve), 5 (red dashed curve), and 10 (blue dotted curve). The inset shows the same as a color map. The yellow curves show the levels of P_max = 0.3, 0.6 and 0.9.

5 Discussion

The current-induced spin polarization can be directly detected optically using the spin-induced Faraday rotation at the related resonances, for example, with the trion optical transitions in the QD. Alternatively, the spin polarization can be detected electrically using ferromagnetic contacts including magnetic microscope tips or using an additional QD weakly coupled to the main one due to the spin blockade effect and exchange interaction between the two QDs.⁶¹ There is also an inverse effect: the spin generation in the QD by external means would lead to the spin galvanic effect, e.g. to the current in the wire, which can be measured electrically.

We note also that if the two QDs are placed at the opposite sides of the quantum wire, the electric current produces the opposite spin polarizations in them similar to the spin Hall effect, as can be seen from the chirality shown in Fig. 2(b). However, for a single QD the hole spin flips are more efficient in one direction than in the opposite, so the total spin polarization of holes in the system is nonzero in contrast with the spin Hall effect.

We stress that the proposed mechanism of the current-induced spin polarization via the chiral quasi bound states does not require magnetic field or linear in momentum spin–orbit coupling in contrast to the previous approaches. The spin–orbit interaction required here describes the splitting of the valence band exactly at the center of the Brillouin zone. It results in the off-diagonal matrix elements of the Luttinger Hamiltonain [see eqn (2)], which are related to the diagonal ones as γ₂/γ₁. We stress that this ratio is not parametrically small and amounts, for example, to 0.28, 0.08, and 0.32 in GaAs, Si and Ge, respectively. The close to unity spin polarization is equally possible in all these materials.

To give a specific example of the system parameters to observe the effect, we consider a two-dimensional hole gas in the GaAs/AlGaAs heterostructure.^{41,43,62–70} The typical parameters of the system are a = 100 nm, Fermi energy E_F = 0.2 meV, coupling strength Γ = 10 μeV, and temperature T = 100 mK. The temperature should be low enough to consider a degenerate hole gas. Then at bias larger than 10 μV we predict the possibility of complete current-induced hole spin polarization.

6 Conclusions

We demonstrated that the complex valence band structure leads to chirality of the quasi bound states of light holes in the QD side-coupled to the quantum wire with heavy holes. The current flowing along the quantum wire produces nonequilibrium spin polarization of holes in the QD. This new mechanism of current-induced spin polarization has the following advantages: (i) it takes place for exponentially small applied voltages. (ii) It does not require magnetic fields and can be applied to centrosymmetric materials. (iii) In a broad range of the geometric parameters, the current-induced spin polarization reaches 100%.

Author contributions

V. N. M. and D. S. S.: conceptualization, data curation, formal analysis, funding acquisition, investigation, methodology, project administration, resources, software, supervision, validation, visualization, writing – original draft preparation, and writing – review and editing (equal).

Conflicts of interest

There are no conflicts to declare.

Acknowledgements

We thank L. E. Golub, M. M. Glazov, I. V. Krainov, and V. S. Khrapai for fruitful discussions. We acknowledge RF President Grant No. MK-5158.2021.1.2 and the Foundation for the Advancement of Theoretical Physics and Mathematics “BASIS”. Analytical calculations by D.S.S were supported by the Russian Science Foundation grant No. 21-72-10035. V. N. M. acknowledges support by a President grant for young scientists MD-781.2021.1 and support from the Interdisciplinary Scientific and Educational School of Moscow University “Photonic and Quantum technologies. Digital medicine”.

Notes and references

1. M. I. Dyakonov, Will We Ever Have a Quantum Computer?, Springer Nature, 2020.
2. T. F. Watson, S. G.-J. Philips, E. Kawakami, D. R. Ward, P. Scarlino, M. Veldhorst, D. E. Savage, M. G. Lagally, M. Friesen, S. N. Coppersmith, M. A. Eriksson and L. M.-K. Vandersypen, Nature, 2018, 555, 633.
3. J. Yoneda, K. Takeda, T. Otsuka, T. Nakajima, M. R. Delbecq, G. Allison, T. Honda, T. Kodera, S. Oda, Y. Hoshi, N. Usami, K. M. Itoh and S. Tarucha, Nat. Nanotechnol., 2018, 13, 102.
4. C. H. Yang, R. C.-C. Leon, J. C.-C. Hwang, A. Saraiva, T. Tanttu, W. Huang, J. Camirand Lemyre, K. W. Chan, K. Y. Tan, F. E. Hudson, K. M. Itoh, A. Morello, M. Pioro-Ladriére, A. Laucht and A. S. Dzurak, Nature, 2020, 580, 350.
5. K. C. Nowack, F. H.-L. Koppens, Y. V. Nazarov and L. M.-K. Vandersypen, Science, 2007, 318, 1430–1433.
6. D. M. Zajac, A. J. Sigillito, M. Russ, F. Borjans, J. M. Taylor, G. Burkard and J. R. Petta, Science, 2018, 359, 439.
7. F. Basso Basset, M. B. Rota, C. Schimpf, D. Tedeschi, K. D. Zeuner, S. F. Covre da Silva, M. Reindl, V. Zwiller, K. D. Jöns, A. Rastelli and R. Trotta, Phys. Rev. Lett., 2019, 123, 160501.
8. A. R. Mills, D. M. Zajac, M. J. Gullans, F. J. Schupp, T. M. Hazard and J. R. Petta, Nat. Commun., 2019, 10, 1063.
9. H. Qiao, Y. P. Kandel, S. Fallahi, G. C. Gardner, M. J. Manfra, X. Hu and J. M. Nichol, Phys. Rev. Lett., 2021, 126, 017701.
10. S. G. Carter, S. C. Badescu, A. S. Bracker, M. K. Yakes, K. X. Tran, J. Q. Grim and D. Gammon, Phys. Rev. Lett., 2021, 126, 107401.
11. E. L. Ivchenko and G. E. Pikus, JETP Lett., 1978, 27, 604.
12. L. E. Vorob’ev, E. L. Ivchenko, G. E. Pikus, I. I. Farbshtein, V. A. Shalygin and A. V. Shturbin, JETP Lett., 1979, 29, 441.
13. S. D. Ganichev, S. N. Danilov, P. Schneider, V. V. Bel’kov, L. E. Golub, W. Wegscheider, D. Weiss and W. Prettl, J. Magn. Magn. Mater., 2006, 300, 127.
14. A. Y. Silov, P. A. Blajnov, J. H. Wolter, R. Hey, K. H. Ploog and N. S. Averkiev, Appl. Phys. Lett., 2004, 85, 5929.
15. Y. K. Kato, R. C. Myers, A. C. Gossard and D. D. Awschalom, Phys. Rev. Lett., 2004, 93, 176601.
16. B. M. Norman, C. J. Trowbridge, D. D. Awschalom and V. Sih, Phys. Rev. Lett., 2014, 112, 056601.
17. I. Stepanov, S. Kuhlen, M. Ersfeld, M. Lepsa and B. Beschoten, Appl. Phys. Lett., 2014, 104, 062406.
18. S. D. Ganichev, M. Trushin and J. Schliemann, Spin polarization by current in Handbook of Spin Transport and Magnetism, ed. E. Y. Tsymbal and I. Zutic, Chapman & Hall, Boca Raton, 2012, p. 487.
19. L. E. Golub and E. L. Ivchenko, New J. Phys., 2013, 15, 125003.
20. D. S. Smirnov and L. E. Golub, Phys. Rev. Lett., 2017, 118, 116801.
21. D. Huber, M. Reindl, Y. Huo, H. Huang, J. S. Wildmann, O. G. Schmidt, A. Rastelli and R. Trotta, Nat. Commun., 2017, 8, 15506.
22. E. A. Chekhovich, S. F.-C. da Silva and A. Rastelli, Nat. Nanotechnol., 2020, 15, 999.
23. D. Najer, I. Söllner, P. Sekatski, V. Dolique, M. C. Löbl, D. Riedel, R. Schott, S. Starosielec, S. R. Valentin, A. D. Wieck, N. Sangouard, A. Ludwig and R. J. Warburton, Nature, 2019, 575, 622.
24. M. Veldhorst, C. H. Yang, J. C.-C. Hwang, W. Huang, J. P. Dehollain, J. T. Muhonen, S. Simmons, A. Laucht, F. E. Hudson, K. M. Itoh, A. Morello and A. S. Dzurak, Nature, 2015, 526, 410–414.
25. X. Mi, J. V. Cady, D. M. Zajac, P. W. Deelman and J. R. Petta, Science, 2017, 355, 156–158.
26. F. Borjans, X. Croot, X. Mi, M. Gullans and J. Petta, Nature, 2020, 577, 195–198.
27. C. Kloeffel, M. Trif and D. Loss, Phys. Rev. B: Condens. Matter Mater. Phys., 2011, 84, 195314.
28. X.-J. Hao, T. Tu, G. Cao, C. Zhou, H.-O. Li, G.-C. Guo, W. Y. Fung, Z. Ji, G.-P. Guo and W. Lu, Nano Lett., 2010, 10, 2956–2960.
29. N. Hendrickx, D. Franke, A. Sammak, G. Scappucci and M. Veldhorst, Nature, 2020, 577, 487–491.
30. F. N.-M. Froning, L. C. Camenzind, O. A.-H. van der Molen, A. Li, E. P.-A. M. Bakkers, D. M. Zumbühl and F. R. Braakman, Nat. Nanotechnol., 2021, 16, 308–312.
31. C. W. Hsu, B. Zhen, A. D. Stone, J. D. Joannopoulos and M. Soljačić, Nat. Rev. Mater., 2016, 1, 16048.
32. A. Overvig, N. Yu and A. Alù, Phys. Rev. Lett., 2021, 126, 073001.
33. M. Vallejo, M. Ladrón de Guevara and P. Orellana, Phys. Lett. A, 2010, 374, 4928–4932.
34. F. Spitzer, A. N. Poddubny, I. A. Akimov, V. F. Sapega, L. Klompmaker, L. E. Kreilkamp, L. V. Litvin, R. Jede, G. Karczewski, M. Wiater, T. Wojtowicz, D. R. Yakovlev and M. Bayer, Nat. Phys., 2018, 14, 1043.
35. X. Yin, J. Jin, M. Soljačić, C. Peng and B. Zhen, Nature, 2020, 580, 467.
36. J. Lin, J. P.-B. Mueller, Q. Wang, G. Yuan, N. Antoniou, X.-C. Yuan and F. Capasso, Science, 2013, 340, 331.
37. R. Naaman, Y. Paltiel and D. H. Waldeck, Nat. Rev. Chem., 2019, 3, 250.
38. B. P. Bloom, B. M. Graff, S. Ghosh, D. N. Beratan and D. H. Waldeck, J. Am. Chem. Soc., 2017, 139, 9038.
39. S. P. Mathew, P. C. Mondal, H. Moshe, Y. Mastai and R. Naaman, Appl. Phys. Lett., 2014, 105, 242408.
40. O. Ben Dor, N. Morali, S. Yochelis, L. T. Baczewski and Y. Paltiel, Nano Lett., 2014, 14, 6042.
41. M. Sato, H. Aikawa, K. Kobayashi, S. Katsumoto and Y. Iye, Phys. Rev. Lett., 2005, 95, 066801.
42. H. Edlbauer, S. Takada, G. Roussely, M. Yamamoto, S. Tarucha, A. Ludwig, A. D. Wieck, T. Meunier and C. Bäuerle, Nat. Commun., 2017, 8, 1710.
43. I. V. Borzenets, J. Shim, J. C.-H. Chen, A. Ludwig, A. D. Wieck, S. Tarucha, H.-S. Sim and M. Yamamoto, Nature, 2020, 579, 210.
44. J. M. Luttinger, Phys. Rev., 1956, 102, 1030–1041.
45. E. L. Ivchenko, Optical spectroscopy of semiconductor nanostructures, Alpha Science, Harrow UK, 2005.
46. P. W. Anderson, Phys. Rev., 1961, 124, 41.
47. U. Fano, Phys. Rev., 1961, 124, 1866.
48. M. F. Limonov, M. V. Rybin, A. N. Poddubny and Y. S. Kivshar, Nat. Photonics, 2017, 11, 543.
49. P. Lodahl, S. Mahmoodian, S. Stobbe, A. Rauschenbeutel, P. Schneeweiss, J. Volz, H. Pichler and P. Zoller, Nature, 2017, 541, 473–480.
50. J. Rammer and H. Smith, Rev. Mod. Phys., 1986, 58, 323.
51. G. Stefanucci and R. van Leeuwen, Nonequilibrium Many-Body Theory of Quantum Systems: A Modern Introduction, Cambridge University Press, 2013.
52. P. I. Arseev, Phys.-Usp., 2015, 58, 1159.
53. J. Hubbard and B. H. Flowers, Proc. R. Soc. London, Ser. A, 1963, 276, 238.
54. H. Haug and A.-P. Jauho, Quantum kinetics in transport and optics of semiconductors, Springer, 2008, vol. 2.
55. C. Lacroix, J. Phys. F: Met. Phys., 1981, 11, 2389.
56. Y. Meir, N. S. Wingreen and P. A. Lee, Phys. Rev. Lett., 1993, 70, 2601.
57. R. S. Świrkowicz, J. Barnaś and M. Wilczyński, Phys. Rev. B: Condens. Matter Mater. Phys., 2003, 68, 195318.
58. A. C. Hewson, The Kondo problem to heavy fermions, Cambridge university press, 1997.
59. M. M. Glazov, Electron and Nuclear Spin Dynamics in Semiconductor Nanostructures, Oxford University Press, Oxford, 2018.
60. D. S. Smirnov, E. A. Zhukov, D. R. Yakovlev, E. Kirstein, M. Bayer and A. Greilich, Phys. Rev. B, 2020, 102, 235413.
61. R. Hanson, L. P. Kouwenhoven, J. R. Petta, S. Tarucha and L. M.-K. Vandersypen, Rev. Mod. Phys., 2007, 79, 1217.
62. I. Zailer, J. E.-F. Frost, C. J.-B. Ford, M. Pepper, M. Y. Simmons, D. A. Ritchie, J. T. Nicholls and G. A.-C. Jones, Phys. Rev. B: Condens. Matter Mater. Phys., 1994, 49, 5101–5104.
63. W. R. Clarke, A. P. Micolich, A. R. Hamilton, M. Y. Simmons, K. Muraki and Y. Hirayama, J. Appl. Phys., 2006, 99, 023707.
64. S. P. Koduvayur, L. P. Rokhinson, D. C. Tsui, L. N. Pfeiffer and K. W. West, Phys. Rev. Lett., 2008, 100, 126401.
65. J. Shabani, J. R. Petta and M. Shayegan, Appl. Phys. Lett., 2008, 93, 212101.
66. S. Sasaki, H. Tamura, T. Akazaki and T. Fujisawa, Phys. Rev. Lett., 2009, 103, 266806.
67. T. Otsuka, E. Abe, Y. Iye and S. Katsumoto, Phys. Rev. B: Condens. Matter Mater. Phys., 2009, 79, 195313.
68. T. Otsuka, E. Abe, Y. Iye and S. Katsumoto, Phys. Rev. B: Condens. Matter Mater. Phys., 2010, 81, 245302.
69. J. C.-H. Chen, D. Q. Wang, O. Klochan, A. P. Micolich, K. Das Gupta, F. Sfigakis, D. A. Ritchie, D. Reuter, A. D. Wieck and A. R. Hamilton, Appl. Phys. Lett., 2012, 100, 052101.
70. A. Srinivasan, D. S. Miserev, K. L. Hudson, O. Klochan, K. Muraki, Y. Hirayama, D. Reuter, A. D. Wieck, O. P. Sushkov and A. R. Hamilton, Phys. Rev. Lett., 2017, 118, 146801.

---

Footnote

† Electronic supplementary information (ESI) available: Detailed calculation of the tunneling matrix elements and the Keldysh formalism for arbitrary interaction strength and finite temperatures, as well as the detailed analysis of the limits of weak and strong interaction. See DOI: https://doi.org/10.1039/d1nh00685a

---