Spin-orbit coherent oscillations and phonon-induced spin relaxation in a quantum dot

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(1)Spin–orbit coherent oscillations and phonon–induced spin relaxation in a quantum dot. Luis Felipe Ramı́rez Cifuentes Submitted in partial fulfillment of the requirements for the degree of Physicist Advisor Ángela Stella Camacho Beltrán, Dr. rer. nat.. Universidad de los Andes Department of Physics Bogotá D.C., Colombia May 2009.

(2) Contents Introduction 1. 2. 3. 4. iii. Electrons in quantum dots with parabolic confinement. 1. 1.1. The constant interaction model . . . . . . . . . . . . . . . . . . . . . .. 4. 1.2. Theory of a single–electron in a two–dimensional electron gas . . . . .. 7. 1.2.1. Energy Spectrum . . . . . . . . . . . . . . . . . . . . . . . . .. 8. 1.2.2. Fock-Darwin States . . . . . . . . . . . . . . . . . . . . . . . . 10. Spin–orbit interaction in quantum dots. 12. 2.1. Bychkov–Rashba interaction . . . . . . . . . . . . . . . . . . . . . . . 12. 2.2. Numerical solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13. 2.3. InGaAs quantum dots . . . . . . . . . . . . . . . . . . . . . . . . . . . 17. 2.4. GaAs quantum dots . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20. Spin–orbit coherent oscillations. 22. 3.1. The Jaynes-Cummings model . . . . . . . . . . . . . . . . . . . . . . . 23. 3.2. Coherent oscillations . . . . . . . . . . . . . . . . . . . . . . . . . . . 24. Phonon–induced decoherence 4.1. 30. Primary processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31. Conclusions. 35. Bibliography. 39. i.

(3) List of Figures 1.1 1.2 1.3 1.4 1.5 1.6 1.7. Density of states for ideal one–, two–, and three–dimensional electron gas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . a) Vertical quantum dot geometry. Taken from [24]. b) Schematic potential diagram in the z–direction . . . . . . . . . . . . . . . . . . . . Schematic diagrams of low–, and high–bias regime. Taken from [16] . a) Dot current versus gate voltage. b) the addition to circular orbits is shown schematically. Taken from [1] . . . . . . . . . . . . . . . . . . Low–lying Fock–Darwin energy spectrum . . . . . . . . . . . . . . . Fock–Darwin energy spectrum . . . . . . . . . . . . . . . . . . . . . Square of the Fock–Darwin states, |ψn,l (r, φ)|2 , for different (n, l) . .. Mesh geometry used to calculate the eigenenergies and eigenstates of Equations 2.9 and 2.10 . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Fock–Darwin energy spectrum; obtained numerically . . . . . . . . . 2.3 Square of the Fock–Darwin states with ~ωo = 3 meV, |ψ|2 ; obtained numerically . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Energy spectra of InGaAs for α = 0.5, 1, 3, 5 meV nm . . . . . . . . 2.5 Energy spectra of InGaAs for α = 10, 15 meV nm . . . . . . . . . . . 2.6 Energy spectra of InGaAs for α = 20 meV nm . . . . . . . . . . . . 2.7 Square of the unnormalized wavefunctions of the total Hamiltonian, Equation 2.6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.8 Energy spectra of GaAs for α = 0.5, 1, 3, 5 meV nm . . . . . . . . . . 2.9 Energy spectra of GaAs for α = 10, 15 meV nm . . . . . . . . . . . . 2.10 Energy spectra of GaAs for α = 20 meV nm . . . . . . . . . . . . . .. .. 2. . .. 3 6. . 6 . 9 . 10 . 11. 2.1. . 15 . 15 . . . .. 16 18 18 19. . . . .. 19 20 21 21. 3.1 3.2 3.3 3.4 3.5 3.6. Energy spectrum obtained from Equation 3.9 . . . . . . . . . . . . . . Taken from [12] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Coherent oscillations at B = 97 mT . . . . . . . . . . . . . . . . . . . Coherent oscillations with electrostatic characteristic length lo = 25 nm Coherent oscillations with a) lo = 40 nm, and b) lo = 50 nm . . . . . . Coherent oscillations with a) lo = 150 nm, and b) lo = 200 nm . . . . .. 24 26 27 28 29 29. 4.1 4.2. Numerical evaluation of F (η) . . . . . . . . . . . . . . . . . . . . . . 34 Phonon–induced relaxation rate . . . . . . . . . . . . . . . . . . . . . 34 ii.

(4) Introduction The study of low–dimensional systems obey not only to theoretical interests, but also technological applications. The development of electronic devices tends towards the miniaturization of its components. Ideas from classical physics have been used in modeling these devices. Thus, when electronic processes are taking place in structures with dimensions greater or equal to microns, a continuous formulation can be applied to describe what is happening. Nevertheless, since leading edge technologies are becoming increasingly small, producing devices with dimensions below the microscale, quantum phenomena emerge. The ability to manipulate and control processes that involve transitions between spin states is, at the moment, of extreme importance due to the recent applications in polarized spin electronics and quantum computation. Recently, the Rashba effect has been considered to be a possible control of electron spin states via gate voltage [2, 27]. More recently Debald and Emary have proposed an experimental scheme to observe a spin–orbit driven Rabi oscillations in quantum dot systems [12]. When studying this systems, the influence of the environment on the coherent oscillations rises an obstacle. Spin dephasing is the most critical aspect that should be considered in the elaboration of proposals of quantum computation based in single spin states as qubits in quantum dots (QD’s) [17]. While for bulk and for 2D systems the spin relaxation processes have been studied in some detail, the problem for QD’s still require deeper and further discussions. Several processes that can induce spin relaxation in semiconductors have been identified and were studied. At the moment remains in discussion which, between these processes, is dominant in zero-dimensional systems. Some experimental results have shown good agreement with the theoretical predictions for 2D systems [22] but, in general, the identification of the processes through direct comparison with the experimental results may become a formidable task. This problem is more critical for QD’s, since few experimental results exist and the theoretical discussion of the spin relaxation mechanisms is still an open subject. Extensive theoretical works in QD systems have studied the main phonon mediated spin-flip mechanisms, including admixture processes due to iii.

(5) spin-orbit coupling [19], and phonon coupling due to interface motion (ripple mechanism) [31]. In this work we focus on the Rashba spin–orbit interaction discussing the coherent oscillations in a single quantum dot. Furthermore, we study the spin–phonon interaction as source of decoherence and present results on the relaxation process as function of external magnetic field. This work is organized as follows: We first briefly outline the Fock–Darwin theory for a single–electron in a two–dimensional electron gas in chapter 1. Then, implementing a numerical method, we further study the influence of the Zeeman and Rashba effects on InGaAs and GaAs quantum dots in chapter 2. Additionally, in chapter 3 we discuss a scheme for generating coherent oscillations in a single quantum dot with Rashba spin–orbit interaction, which can be manipulated by an external gate voltage. In chapter 4 we calculate the effect of spin–phonon interaction on the dot. We analyze how spin–phonon relaxation process and magnetic field influences the Rabi oscillation.. iv.

(6) Chapter 1 Electrons in quantum dots with parabolic confinement A system with electrons confined in regions comparable to their de Broglie wavelength shows quantum effects. When only one direction is subjected to this strong confinement (e.g., z-direction) with free motion in the others, a two–dimensional electron gas (2DEG) is created. Similarly, confinement in two directions (say, y and z) with free motion in the x–direction gives a one–dimensional electron gas (1DEG), and confinement in every direction (x, y, z) yields a zero–dimensional electron gas (0DEG). All of these gases are customary called ideal low–dimensional systems. In Figure 1.1 we show the density of states for an ideal one–, two–, and three–dimensional electron gas. These are obtained using the energy dispersion (see [3]) 1D : E =. ~2 kx2 , 2m ~2 k||2. ~2 2 (k + ky2 ), 2m 2m x ~2 2 ~2 k 2 = (k + ky2 + kz2 ). 3D : E = 2m 2m x. 2D : E =. =. We see that the density of electronic states has a strong dependence on the spatial dimension of the system. Nevertheless, in real low–dimensional systems, confinement is never perfect. Restricting the motion of an electron can be achieved only up to a finite, but small region. This imperfect confinement results in energy quantization where more than one such energy is allowed; hence, producing a difference of the density of states between a real low–dimensional system and an ideal one [3]. A quantum dot is a zero–dimensional system. Particularly, in this chapter we introduce the model of a vertical quantum dot. A detailed explanation of the fabrication 1.

(7) Figure 1.1: Density of states for ideal one–, two–, and three–dimensional electron gas. technique, and dot operation can be found in ref. [1]. We begin by presenting the geometry of the semiconductor heterostructure that contains the dot. Then, the constant interaction model and the single particle theory of a two–dimensional electron gas are outlined. A comprehensive review on these topics can be found in refs. [16, 28]. Figure 1.2a shows a diagram of a vertical pillar, semiconductor heterostructure. It has a width of about 0.5 µm. The quantum dot is located between the two AlGaAs tunnel barriers, and has the shape of a disc. The barrier layers have a thickness of about 10 nm. A voltage Vg applied to the side gate controls the magnitude of the lateral, electrostatic confinement potential; hence, it can reduce the effective diameter of the dot. When Vg = 0, the number of electrons inside the dot is about 80. Upon application of a negative voltage, the number of electrons N decreases, one by one, until a voltage around Vg = −1.5 V depletes the dot; that is, N = 0. This is why they are called few–electron quantum dots; one has great control on the number of electrons inside the dot. Besides the side gate, there are electrodes connected to the top and bottom of the vertical pillar. These allow one to apply a drain–source voltage Vsd , and additionally perform current measurements. Therefore, the AlGaAs barriers give rise to geometrical confinement of the electrons in one direction (say, z-direction), resulting in a 2DEG, and, in addition, the lateral electrostatic potential restricts the motion in the other two directions, thus generating a quasi–zero–dimensional system. In Figure 1.2b we show the potential profile in the z–direction, and the electron distribution (dashed line). With Vsd = 0, the height of the potential in both sides (source and drain) is the same. It stems from the different electron effective masses at the heterointerface (e.g., InGaAs/AlGaAs interface). However, when applied, the Vsd voltage produces a difference of heights; the potential at the drain interface is raised, whereas that at the source interface is lowered. This potential height variation produces what is known as the structural inversion asymmetry (SIA). As we shall see in section 2.1, the spin–orbit (Rashba) interaction has among its origins this lack of inversion symmetry.. 2.

(8) Figure 1.2: a) Vertical quantum dot geometry. Taken from [24]. b) Schematic potential diagram in the z–direction. The dashed line represents the electron distribution. As another consequence, the electrons are drifted to the source interface, increasing even more the z–direction confinement. This pillar structure operates like a three–terminal field–effect transistor. In fact, the current through the pillar can be switched on/off controlling the side gate voltage. Remarkably, in this type of transistor there is not just one threshold voltage but a somewhat periodic set of voltages where the current switches. This is a consequence of the Coulomb Blockade effect, which we shall address in the following section. Indeed, only a small fraction of a single–electron charge is sufficient to activate the switch. Because of that, these devices are named single–electron transistors, or SETs (cf. [18]). In atoms, the three–dimensional spherically symmetric potential yields degeneracies known as the shells: 1s, 2s, 2p, etc. The electronic configuration becomes particularly stable when these shells are filled with electrons. It gives rise to the sequence of magic numbers for this potential (i.e. 2, 10, 18, 36, ...). Those are just the atomic numbers of the most stable atoms. Similarly, in the circular pillar quantum dot (see Figure 1.2) the high degree of symmetry is responsible for degeneracies in the energy spectrum. As mentioned previously, the confinement potential can be controlled by applying a negative voltage to the side gate. This repulsive, confinement potential can be approximated by a harmonic potential [1]. This two–dimensional cylindrically confinement potential leads to a two– dimensional shell structure with magic numbers 2, 6, 12, 20 ...; then producing a lower magic number sequence than atoms. We further discuss the origin of this sequence in subsection 1.2.1. In what follows, we introduce the constant interaction model.. 3.

(9) 1.1. The constant interaction model. The constant interaction model is based on two assumptions [1]: • The Coulomb interactions between the electrons inside the dot, and between these electrons and those in the environment (source, drain, etc.) are parameterized by a constant capacitance C. • The discrete, single–particle energy spectrum is independent of those interactions, and thus of the number of electrons. The capacitance C is the sum of the capacitances between the dot and the source CS , the drain CD , and the gate CG . Therefore, C = CS + CD + CG . Under the previous assumptions, the total energy of a dot U (N ) with N electrons in the ground state, with voltages VS , VD , and VG applied to the source, drain, and gate, respectively, is given by (see [16]) N. [−|e|(N − N0 ) + CS VS + CD VD + CG VG ]2 X En (B) U (N ) = + 2C n=1. (1.1). where e is the electron charge, N0 |e| is the charge in the dot compensating the positive background charge originating from the donors in the heterostructure, and B is an applied magnetic field. The Ci Vi (i = S, D, G) terms represent an effective induced charge that changes the electrostatic potential on the dot. The last term, En , is the single–electron eigenenergies for the parabolic confinement (see subsection 1.2.1). Additionally, let us define the electrochemical potential µ(N ) of the quantum dot as µ(N ) ≡ U (N ) − U (N − 1), (1.2) and substituting Equation 1.1 into the previous equation we have 1 EC µ(N ) = N − N0 − EC − (CS VS + CD VD + CG VG ) + EN , 2 |e|. (1.3). where EC = e2 /C is the charging energy. The Coulomb repulsion between electrons in the dot results in an energy for adding an extra electron. Hence, extra energy is required and no current will flow through the dot until increasing the voltage provides this energy. This phenomenon is known as Coulomb Blockade. The electrochemical potentials of the transitions between successive ground states are spaced by the so–called addition energy Eadd (N ) = µ(N + 1) − µ(N ) = EC + ∆E 4. (1.4).

(10) with ∆E = EN +1 − EN being the energy spacing between two discrete, successive quantum levels (see subsection 1.2.1). Notice that ∆E can be zero when two states are spin–degenerated. Electron tunneling through the dot depends on the alignment of the electrochemical potential in the dot with respect to those of the source µS , and the drain µD . Application of a bias voltage VSD = VS − VD opens up an energy window between µS and µD , ∆µSD = µS − µD = −|e|VSD .. (1.5). This energy window is named the bias window. Figure 1.3b shows a schematic diagram of the electrochemical potential levels. When the electrochemical potentials are properly aligned, that is, µS > µ(N ) > µD , electron tunneling occurs from states of the source, with a tunneling rate ΓS , through states of the dot toward empty states in the drain, with tunneling rate ΓD . Thus, the number of electrons inside the dot alternate between N − 1 and N , resulting in a single–electron tunneling current. In contrast, if µ(N ) lies outside the bias window, then no current will flow through the dot due to Coulomb Blockade, as in Figure 1.3a. This case, when only one dot level is inside the bias window, −|e|VSD < ∆E, Eadd , is called the low–bias regime. On the other hand, when multiple levels lie inside the bias window, −|e|VSD ≥ ∆E and/or −|e|VSD ≥ Eadd , it is called the high–bias regime. Figure 1.3e,f shows a schematic diagram of the latter regime, where several electrons flow through the dot at the same time. Moreover, by sweeping the gate voltage and measuring the current Idot through the dot a trace is obtained. Figure 1.3c shows this current as a function of the gate voltage. The spacing between the peaks corresponds to Eadd ; hence, this sweeping provides a tool for exploring the energy spectrum of the dot. Figure 1.4a shows, again, the dot current versus the gate voltage. In the inset, note that the addition energy is higher for some numbers. It turns out that these numbers constitute the sequence of magic number mentioned in the foregoing section. Referring to Figure 1.4b, where the addition to circular orbits is shown schematically, we observe that adding an electron requires the energy e2 /C; i.e., the charging energy EC . However, for the argument’s sake, if the first shell is filled, containing two electrons, extra energy is needed (that is, ∆E) to add the third electron, because it must go into the next energy level. Alike, when adding the seventh electron, it is needed the energy Eadd = EC + ∆E because the second shell is already filled, then forcing this electron into the next energy level –the third shell. In the next section we present the theory of a single–electron in a 2DEG, which leads to the Fock–Darwin states. 5.

(11) Figure 1.3: a) The electrochemical potentials are not properly aligned. Thus the dot is Coulomb blockaded. b) The electrochemical potentials are aligned, then generating a single–electron current flow through the dot in the low–bias regime. c) Dot current as a function of gate voltage. d) Schematic diagram of the electron path through the dot. e) Current flowing through multiple tunneling events in the high–bias regime. f) Wider bias windows spanning even more energy levels of the dot. Taken from [16].. Figure 1.4: a) Dot current versus gate voltage. b) the addition to circular orbits is shown schematically. Taken from [1].. 6.

(12) 1.2. Theory of a single–electron in a two–dimensional electron gas. Consider the Hamiltonian of a single electron in a 2DEG with parabolic confinement, and a perpendicular magnetic field Ho =. (p − eA)2 mωo2 2 + (x + y 2 ), 2m 2. (1.6). where m is the effective mass of the electron, ~ωo is the electrostatic confinement energy, p = (px , py , 0) is the pseudo–momentum of the electron, and A = (−y, x, 0)B/2 is the magnetic vector potential written in the symmetric gauge. This system can be solved analytically (cf. [28, 29, 30]), leading to the so–called Fock–Darwin states1 [10, 13]. Notice that the multiple–particle effects are not taken into account in this formulation; this is due to the regime in which the dots operate (see section 1.1). In order to gain some insight, we will rewrite Equation 1.6 in a suitable form. First, expanding the canonical momentum operator (p − eA)2 = (p − eA)(p − eA) = p2x + p2y − eB(px , py ) · (−y, x) + (eB/2)2 (x2 + y 2 ) = p2x + p2y − eB(xpy − ypx ) + (eB/2)2 (x2 + y 2 ) we can write Equation 1.6 as Ho =. p2x + p2y mωt2 2 ωc + (x + y 2 ) − (xpy − ypx ), 2m 2 2. (1.7). where ωc = eB/m is the cyclotron frequency, ωt = (ωo2 + ωc2 /4)1/2 , and defining the oscillator operators 1 x ilt 1 x ilt † ax = √ + px , a x = √ − px , (1.8) ~ ~ 2 lt 2 lt 1 y ilt 1 y ilt † + py , a y = √ − py , (1.9) ay = √ ~ ~ 2 lt 2 lt h i ai , a†j = δij , (1.10) 1. After V. Fock and C. Darwin, who first solved this problem, independently from each other, in the late 1920s and the early 1930s.. 7.

(13) with lt =. p ~/mωt , we obtain i~ωc ay a†x − ax a†y . Ho = ~ωt a†x ax + a†y ay + 1 − 2. (1.11). Using the following operators √ a± = (ax ∓ iay )/ 2. (1.12). the x and y oscillation modes decouple into eigenmodes ω± = ωt ± ωc /2 [11], 1 1 + ~ω− n− + , (1.13) Ho = ~ω+ n+ + 2 2 where n± = a†± a± . It is usual to rewrite the n± modes as radial and angular momentum quantum numbers n ≡ min(n+ , n− ), l ≡ n− − n+ . (1.14) Table 1.1 shows the different parameterization of the lowest shells. Notice that in the (n, l) form, the quantum numbers for this system differ from those of the hydrogen atom. (n+ , n− ) (n, l) (0,0) (0,0) (0,1) (0,1) Second (1,0) (0,-1) (0,2) (0,2) Third (1,1) (1,0) (2,0) (0,-2) Shell First. Table 1.1: Different parameterization (see Equation 1.14) of lowest modes. 1.2.1. Energy Spectrum. The energy spectrum of a one–dimensional harmonic oscillator, En = ~ω(n + 1/2), becomes En,l = ~ωo (2n + |l| + 1) in the two–dimensional case, where n = 0, 1, 2, 3, ... is the radial quantum number, l = 0, ±1, ±2, ... is the angular momentum quantum number, and ωo is the oscillator frequency. The eigenenergies En,l , in presence of a perpendicular magnetic field, are [1, 28] En,l = ~ωt (2n + |l| + 1) − l~ωc /2. 8. (1.15).

(14) Figure 1.5 shows the low–lying energy spectrum of the Hamiltonian shown in Equation 1.6 as a function of the magnetic field, which is expressed as a dimensionless ratio of the cyclotron and confinement frequencies. There, each level En,l is two–fold spin– degenerate. Additionally, notice that at B = 0 the first shell, formed by the level labeled (0, 0), is two–fold degenarate (i.e., due to spin), the second shell, with labels (0, 1) and (0, −1), is four–fold degenerate (i.e., double–orbital and spin degeneracies), the third shell, formed by (0, 2), (1, 0) and (0, −2), is six–fold degenerate (i.e., triple–orbital and spin degeneracies), and so forth. However, as B is increased, a single particle state with a positive or negative angular momentum l shifts into a lower or higher energy, respectively. Therefore, the orbital degeneracies at B = 0 are lifted in a magnetic field. At this point, the relation of the shell structure to the sequence of magic numbers mentioned in section 1.1 becomes apparent. This sequence is: 2, 6, 12, 20 ...; hence, the magic number 6, for instance, means that the first shell, which can accommodate up to two electrons, and the second shell, holding up to four electrons, are completely filled, producing an artificial two–dimensional atom energetically stable. 5. 4. 3. 2. 1. 0. 0. 1. 2. 3. 4. 5. Figure 1.5: Low–lying Fock–Darwin energy spectrum as a function of dimensionless magnetic field. When the magnetic field is increased, the electron in the highest energy state is forced into different orbital states. For the argument’s sake, suppose there are seven non–interacting electrons in the quantum dot. Therefore, at low B, the highest occupied state is (0, 2) (two electrons in the first shell, four in the second, and the seventh 9.

(15) occupying the first state of the third shell). This state decreases in energy with B. After the first crossing, the highest energy state is (0, −1). Afterwards, there will be another crossing, between (0, −1) and (0, 3), leaving the latter as the highest occupied state. In Figure 1.6 we show several Fock–Darwin energy levels. Notice that, after the last crossing, the levels characterized by the quantum number (0, l), with l ≥ 0, gather together. Electrons occupying these states conform the so–called lowest orbital Landau level [21].. Figure 1.6: Fock–Darwin energy spectrum as a function of dimensionless magnetic field.. 1.2.2. Fock-Darwin States. The eigenstates of Equation 1.6, ψn,l (r, φ), are usually called Fock–Darwin states, as it was mentioned previously. These states can be written as (cf. [28]) s |l| 2 ilφ e n! r r 2 −r2 /4lB |l| √ e Ln (1.16) ψn,l (r, φ) = √ 2 2lB 2πlB (n + |l|)! 2lB where lB = (~/mωt )1/2 is the characteristic magnetic length, with ωt = (ωo2 +wc2 /4)1/2 , |l| and Ln are generalized Laguerre polynomials. The square of several eigenstates is plotted in Figure 1.7. Note that n sets the number of nodes in the radial direction, whereas l determines the size of the valley in the center and the radial extent of the wavefunction. 10.

(16) Furthermore, when l 6= 0, an additional node appears at r = 0. When increasing B, the characteristic magnetic length lB decreases, indicating that the confinement becomes stronger for larger B. At some point, when the external magnetic field is sufficiently strong, electrons occupying the same energy level will be pushed together, generating stronger Coulomb interactions not negligible; hence, the constant interaction model will fail at this large B (∼ 10 T, see [1]).. Figure 1.7: Square of the Fock–Darwin states, |ψn,l (r, φ)|2 , for different (n, l) quantum numbers. Obtained analytically.. 11.

(17) Chapter 2 Spin–orbit interaction in quantum dots In this chapter we outline the procedure to obtain the Rashba Hamiltonian. We then introduce a method to solve the total Hamiltonian, including the parabolic confinement, the Zeeman effect, and the Rashba effect. Using this method, we calculate the eigenenergies and eigenstates first for Ho , in order to compare them with our previous results, and then for the total Hamiltonian for InGaAs and GaAs quantum dots. The spin–orbit interaction in semiconductor heterostructures can stem from an electric field perpendicular to the two–dimensional electron gas (2DEG). From the point of view of an electron, this electric field is felt as an effective magnetic field lying in the plane of the 2DEG, perpendicular to the wave vector of the electron [26]. This effective magnetic field Beff couples to the electron spin –just as in the Zeeman effect– and this interaction lifts the spin degeneracy, even without an external magnetic field (Bext = 0). This is referred to as the Bychkov–Rashba interaction [7].. 2.1. Bychkov–Rashba interaction. The Bychkov–Rashba interaction is caused by the so–called structural inversion asymmetry (SIA), which arises from electric fields associated with asymmetric confinement potentials. Consider an electron in the 2DEG moving with a velocity v in the presence of an electric field E. Using the rest frame of the electron, this transforms into an effective magnetic field [26] 1 Beff = − 2 v × E (2.1) 2c. 12.

(18) where c is the speed of light. Because of the existence of this effective field, there is a coupling between the latter and the electron spin, producing a resulting spin–orbit interaction HSO = µ · Beff = −. 1 µ · (v × E) 2c2. e~2 = σ · (k × E), (2mc)2. (2.2). seeing that µ = −e~σ/(2m), and v = ~k/m, where σ is the vector of Pauli matrices, and k is the wave vector of the electron. Next, since the electric field is along the z–axis, perpendicular to the 2DEG, Equation 2.2 reduces to, e~2 σ · (k × E) = α0 hEz i (−i∇ × σ)z 2 (2mc) α0 = hEz i (σ × p)z ~ α = (py σx − px σy ), ~. HSO =. (2.3) (2.4). where α0 = e~2 /(2mc)2 , and α = α0 hEz i is the spin–orbit coupling constant, which is proportional to the interface electric field that confines the electrons in the xy plane. Furthermore, if an external magnetic field is applied along the z–axis, with magnetic vector potential A = (−y, x, 0)B/2, Equation 2.3 becomes HSO =. α [σ × (p − eA)]z . ~. (2.5). Therefore, the total Hamiltonian which describes the system (including Rashba and Zeeman effects, and the parabolic confinement potential) is H =. (p − eA)2 mωo2 2 α 1 + (x + y 2 ) + [σ × (p − eA)]z + gµB Bσz , 2m 2 ~ 2. (2.6). In what follows we present a method to solve Equation 2.6 using typical parameters.. 2.2. Numerical solution. In this section we solve the total Hamiltonian, Equation 2.6, utilizing the finite element method (cf. [9], [25]). First, we rewrite Equation 2.5 as follows. 13.

(19) HSO. α eBx eBy = σx py − − σy px + ~ 2 2 α ∂ eBx α ∂ eBy 0 1 0 −i = − i~ + + i~ − 1 0 i 0 ~ ∂y 2 ~ ∂x 2 eB α (iy − x) 0 0 ∇− 2 = α (2.7) + eB + 0 −∇ 0 ~ − 2 (iy + x). where ∇± = ∂/∂x ± i∂/∂y. We want to find the eigenenergies and eigenstates of the Schrödinger equation H Ψ = E Ψ, ψ↑ ψ↑ H = E . ψ↓ ψ↓. (2.8). Hence, using Equations 1.7, 2.6, and 2.7 we can write eB 1 − (iy − x) ψ↓ = E↑ ψ↑ , Ho + gµB B ψ↑ + α ∇ + 2 2~ 1 eB + Ho − gµB B ψ↓ − α ∇ + (iy + x) ψ↑ = E↓ ψ↓ , 2 2~ where. i~ωc ~2 ∇2 mωt2 2 + (x + y 2 ) + Ho = − 2m 2 2. (2.9) (2.10). ∂ ∂ −y x . ∂y ∂x. Then, we define a domain Ω ⊂ R2 and domain boundary ∂Ω on which we want to solve the coupled differential equations 2.9 and 2.10 as Ω = {(x, y) | x2 + y 2 < r2 }, 2. 2. 2. ∂Ω = {(x, y) | x + y = r },. (2.11) (2.12). where r is a constant. Additionally, we impose the following Dirichlet boundary condition ψ↑ (x, y) = ψ↓ (x, y) = 0 ∀x, y ∈ ∂Ω (2.13) Figure 2.1 shows Ω and ∂Ω. In order to solve the equations 2.9 and 2.10, the domain Ω is subdivided into 12 300 Lagrange, cubic, triangular elements. The central region is finer than the outer one because, given Figure 1.7, we expect rapid oscillations of the wave functions in there. For that reason, a greater precision is required in this area, in contrast with the slow changing outer one. 14.

(20) Figure 2.1: Mesh geometry used to calculate the eigenenergies and eigenstates of Equations 2.9 and 2.10. The mesh contains 12 300 Lagrange, cubic, triangular elements, giving rise to 24 700 degrees of freedom. The central region is finer than the outer one because the wave functions oscillate rapidly in there.. Figure 2.2: Fock–Darwin energy spectrum; obtained numerically. Compare with Figure 1.5.. 15.

(21) Figure 2.3: Square of the Fock–Darwin states, |ψ|2 ; obtained numerically. Compare with Figure 1.7.. 16.

(22) Before solving the total Hamiltonian (Equation 2.6), let us calculate the eigenenergies and eigenstates of Ho (i.e., an electron in the 2DEG with parabolic confinement, and without Rashba and Zeeman effects) for benchmarking purposes. In particular, we use m/mo = 0.05 (that is, InGaAs semiconductor), ~ωo = 3 meV, and r = 150 nm; being m/mo the effective to bare electron mass ratio, ~ωo the electrostatic confinement energy, and r the radius of the domain Ω. Figure 2.2 shows the low–lying Fock–Darwin energy spectrum, and Figure 2.3 shows several |ψ|2 of Fock–Darwin states. In the following sections we shall solve the total Hamiltonian. In section 2.3 and section 2.4 we solve Equations 2.9 and 2.10 for InGaAs, and GaAs quantum dots, respectively.. 2.3. InGaAs quantum dots. In this section we show the energy spectra of a InGaAs quantum dot, for different values of the spin–orbit coupling α. Thus, in this numerical study we use m/mo = 0.05, g = −4, and ~ωo = 3 meV (see [23]). This p electrostatic lateral confinement produces a characteristic electrostatic length lo = ~/mωo ≈ 22.5 nm. The magnetic field is in the range 0 ≤ B ≤ 2 T, producing pa characteristic magnetic length ranging from no confinement (at B = 0T) to lB = ~/mωc ≈ 18.2 nm; overcoming lo . Finally, the rashba constant is varied from α = 0.5 meV·nm to α = 20 meV·nm, producing spin– orbit interaction characteristic lengths of lSO = ~2 /2mα ≈ 1520 nm to lSO ≈ 38.1 nm. Figure 2.4 shows energy spectra for α = 0.5, 1, 3, 5 meV·nm. We notice that, as B is increased, the spin degeneracy is lifted for each shell. In InGaAs, this is mainly due to the Zeeman effect. In contrast, we will see in the next section that the spin– degeneracy splitting in GaAs is not primarily caused by the Zeeman effect, but by the Rashba effect; tacking into account the relatively small g factor, g = −0.44. An outstanding feature of the spin–orbit interaction in quantum dots is the lifting of degeneracy at zero magnetic field. Figure 2.5 and Figure 2.6 show this effect. In Figure 2.7 we present several eigenstates for this system. The first row corresponds to ψ↑ , and the second is for ψ↓ . The energy level increases from left to right. Comparing with Figure 2.3 qualitatively, we note that many low energy wavefunctions are hardly affected. However, for wavefunctions with higher energy, there are deviations from the Fock–Darwin states.. 17.

(23) Figure 2.4: Energy spectra of InGaAs for α = 0.5, 1, 3, 5 meV nm.. Figure 2.5: Energy spectra of InGaAs for α = 10, 15 meV nm.. 18.

(24) Figure 2.6: Energy spectra of InGaAs for α = 20 meV nm.. Figure 2.7: Square of the unnormalized wavefunctions of the total Hamiltonian, Equation 2.6. The first row corresponds to ψ↑ , and the second one is for ψ↓ . The energy level increases from left to right.. 19.

(25) 2.4. GaAs quantum dots. In this section we show the energy spectra as a function of the magnetic field for a GaAs quantum dot. We use the same variations in α and B as in the preceding section. In addition, we utilize m/mo = 0.063, g = −0.44, and ~ωo = 3 meV. p This electrostatic lateral confinement produces a characteristic electrostatic length lo = ~/mωo ≈ 22.5 nm. Similarly to the previous section, Figure 2.8 shows energy spectra for α = 0.5, 1, 3, 5 meV·nm. Due to the small g–factor, the Zeeman term in Equation 2.6 has little impact in the GaAs quantum dot. Figure 2.8a shows that the energy spectrum is almost the same as in Figure 2.2. Nevertheless, upon increasing α, a lifting of degeneracies appear evident. Indeed, Figure 2.9 and Figure 2.10 show lifting of orbital degeneracies at B = 0 and spin–splitting when B is increased, both of these effects due to the spin–orbit interaction alone.. Figure 2.8: Energy spectra of GaAs for α = 0.5, 1, 3, 5 meV nm.. 20.

(26) Figure 2.9: Energy spectra of GaAs for α = 10, 15 meV nm.. Figure 2.10: Energy spectra of GaAs for α = 20 meV nm.. 21.

(27) Chapter 3 Spin–orbit coherent oscillations Following a similar procedure to that which leads us to Equation 1.14, we can rewrite the Rashba Hamiltonian HSO =. α [σ × (p − eA)]z , ~. using the operators Equation 1.8 and Equation 1.12, as 2 α lt † † HSO = √ ax + ax − i ay − ay σ x 2 2lt 2lB 2 lt α † † ay + ay + i ax − ax σy +√ 2 2lt 2lB i αh † † γ+ a+ σ+ + a+ σ− − γ− a− σ− + a− σ+ = lt where γ± = 1 ±. lt2 1 , σ± = (σx ± iσy ). 2 2lB 2. (3.1). (3.2). (3.3). As we have seen, the g–factor in InGaAs or GaAs is negative. Tacking this into account, let us perform a unitary rotation of the spin such that σz → −σz and σ± → −σ∓ [11]. This rotation takes us back to the usual spin sequence of the Zeeman effect, where spin up states have a larger energy than spin down states. Thus, carrying out this rotation we obtain 1 1 1 H = ~ω+ n+ + + ~ω− n− + + |g|µB Bσz 2 2 2 i αh (3.4) + γ− a− σ+ + a†− σ− − γ+ a+ σ− + a†+ σ+ . lt. 22.

(28) At this point we note that, analogously to quantum optics, this Hamiltonian can be regarded as a qubit coupled to two bosonic modes with energies ~ω± . In this case, the coupling is mediated by the spin–orbit interaction. We have already studied the spectral properties of these bosonic modes ω± (see subsection 1.2.1). In Equation 3.4, the coupling between bosonic modes and spin produces anticrossings in the spectrum, as we shall see in the next section. Additionally, we observe that the spin–orbit interaction leads to a coupling between adjacent ω± modes with opposite spin due to the operators a± and a†± in the last term of the Hamiltonian in Equation 3.4. Hence, there are no direct transitions between ω+ and ω− modes. Furthermore, making an analogy with quantum optics (cf. [14]), note that the terms preceded by γ+ in Equation 3.4 are counterrotating, and thus negligible under the rotating wave approximation when the spin–orbit coupling is small compared to the confinement.. 3.1. The Jaynes-Cummings model. Following the preceding line of reasoning, we have for the approximate Hamiltonian. where. H = ω+ n+ + HJC ,. (3.5). 1 HJC = ω− a†− a− + Ez σz + λ(a− σ+ + a†− σ− ), 2. (3.6). with m Ez = |g| 2mo 2 lo γ− λ = . 2lt lSO. 2 lo , lb. (3.7) (3.8). HJC is the well–known Jaynes–Cummings model of quantum optics. It is completely integrable, and has ground state |ψg i = |0, ↓i with energy EG = −Ez /2. The rest of the Hilbert space decomposes in two–dimensional subspaces {|n, ↑i, |n + 1, ↓i : n = 0, 1, ...}. Diagonalization in each subspace leads to the eigenenergies 1 ∆n (n,±) Eα = n+ ω− ± (3.9) 2 2. 23.

(29) where ∆n ≡. p. δ 2 + 4λ2 (n + 1),. δ = ω− − Ez ,. (3.10) (3.11). and the eigenstates ψα(n,±) = cos θα(n,±) |n, ↑i + sin θα(n,±) |n + 1, ↓i. (3.12). where |ψg i is the ground state, and tan θα± =. δ ± ∆n √ . 2λ n + 1. (3.13). In Figure 3.1 we show the low–lying energy spectrum of the Jaynes Cummings model normalized to ~ωo . We note that at Br = 87 mT there are anticrossings; that is, resonance occur at that point.. Figure 3.1: Energy spectrum obtained from Equation 3.9.. 3.2. Coherent oscillations. In this section we discuss the single–quantum dot based qubit proposed by Debald and Emary [12]. We shall see that it is possible to induce coherent oscillations between ψα+ 24.

(30) and ψα− by changing the spin–orbit coupling, α, via a voltage pulse applied to the side gate [12]. Consider the quantum dot operating near resonance. Therefore, we can address it as a single two–level system. In addition, let the electrochemical potentials of the source µS and the drain µD be tuned close to the nth anticrossing. The spin–orbit coupling is set to α1 , and the states participating in the oscillations are ψα±1 (we omit superscript “n”); i.e., the eigenstates of HJC (α1 ). These states are situated symmetrically around the drain electrochemical potential, see Figure 3.2a (µD and µS are the drain and source electrochemical potentials, respectively). Temperature is taken smaller than detuning, kb T δ, to avoid thermal broadening effects. Assuming Coulomb blockade regime, electrons can tunnel either from the source to ψα+1 and then leave the dot through the drain, or tunnel to ψα−1 , Figure 3.2b. Taking the source to dot tunneling rate, ΓS , grater than the dot to drain, ΓD , we assure that the dot is filled from the source, then maximizing the current. Hence, the dot is initialized in ψα−1 . With a electron trapped in this state, we apply a voltage pulse to the side gate. Two effects result from this pulse: a) the spin–orbit coupling is changed nonadiabatically to a new value α2 , and b) the system is drifted below both the source and drain electrochemical potentials. Since the change α1 → α2 is performed nonadiabatically, the electron remains in ψα−1 , until Rabi oscillations begin between this state and ψα+1 since these states are no longer eigenstates of the new Hamiltonian, HJC (α2 ), Figure 3.2c. Additionally, tunneling outside the dot does not occur because both of these levels are below the external electrochemical potentials. After a time tp , the voltage on the side gate is returned to its initial value, and the system comes back to its initial setup, Figure 3.2d. Tunneling out of the dot can occur provided that the electron is found in the upper state. Therefore, in order to estimate the current through the dot during the final setup, we first need to know the probability of finding the system in ψα+1 after a period of time tp . To do this, we shall calculate P (tp ) = =. ψα+1 Ψ(tp ). 2. ,. ψα+1 e−iH (α2 )tp /~ ψα−1. (3.14) 2. .. (3.15). Notice that ψα−1 can be written as a linear combination of the eigenstates of H (α2 ), ψα−1. = cos θα−1 |n, ↑i + sin θα−1 |n + 1, ↓i , = c+ ψα+2 + c− ψα−2 ,. 25. (3.16).

(31) Figure 3.2: Taken from [12] where c± = = = =. ψα±2 ψα−1 ,. (3.17). cos θα±2 hn, ↑ | + sin θα±2 hn + 1, ↓ | × cos θα−1 |n, ↑i + sin θα−1 |n + 1, ↓i , cos θα±2 cos θα−1 + sin θα±2 sin θα−1 , cos(θα±2 − θα−1 ) = cos ∆θ± . . Therefore, Equation 3.14 becomes P (tp ) =. hψα+1 |e−iH (α2 )tp /~ (c+ |ψα+2 i + c− |ψα−2 i) −. +. 2. =. hψα+1 |(c+ e−iEα2 tp |ψα+2 i + c− e−iEα2 tp |ψα−2 i). =. c2+ e−iEα2 tp + c2− e−iEα2 tp. =. e−iEα2 tp [c2+ e−i(Eα2 −Eα2 )tp + c2− ]. =. c2+ e−i(Eα2 −Eα2 )tp + c2− .. −. +. −. +. +. 2. 2. −. 2. 2. −. 26. (3.18).

(32) Writing Ω ≡ (Eα+2 − Eα−2 )/2 yields 2. P (tp ) = cos2 ∆θ+ e−2iΩtp + cos2 ∆θ− .. (3.19). The sequence of states schematically depicted in Figure 3.2 is operated as a cycle, and the current through the dot is measured. We can estimate this current by seeing that −1 Idot ≈ eΓD P (tp ), using the simplification that Γ−1 D > tp , ΓS . Then, by sweeping tp one is able to resolve the time evolution of the Rabi oscillations. The ideal case of a nonadiabatic change α1 ↔ α2 from a finite value to zero produces oscillations with the maximum possible amplitude; that is, P = 1. Nevertheless, in a real situation, it is only possible changes between finite values of α, leading to a reduction in the oscillations amplitude. In fact, in experiments with 2DEG, it was achieved a change in α from 0.3 to 1.5 meV·nm in a InGaAs sample [20]. In Figure 3.3 we plot P (tp ) (Equation 3.19) calculated for the first anticrossing. We used α1 = 1.5 meV·nm and α2 = 0.3 meV·nm, with an external magnetic field B = 97 mT, close to the resonance magnetic field Br = 87 mT for the electrostatic characteristic length lo = 150 nm.. Figure 3.3: Coherent oscillations at B = 97 mT. Directed toward a better understanding of the influence of B on P (tp ), we show time traces of this transition probability, with the above parameters, as a function of the magnetic field in Figures 3.4, 3.5, and 3.6. We first note that each different setup (lo = 25, 40, 50, 150, 200 nm) produces a distinct resonance magnetic field magnitude: 27.

(33) Br ≈ 3200, 1250, 790, 87, 49 mT, respectively. In addition, at these Br ’s (δ = 0), there is a node. This is because, for δ = 0, the eigenstates of the Jaynes–Cummings √ model are (|n, ↑i ± |n + 1, ↓i)/ 2 ∀ α 6= 0. Moreover, using this setup, we see that α1 /α2 = 1.5/0.3 = 5 generates a maximum amplitude Pmax ≈ 0.45, and Rabi frequency of Ω. ≡ (3.9). =. =. Eα+2 − Eα−2 , 2 s √ 2 λ2 n + 1 δ √ + 4, 2 λ2 n + 1 2 GHz.. (3.20). Figure 3.4: Coherent oscillations with electrostatic characteristic length lo = 25 nm.. 28.

(34) Figure 3.5: Coherent oscillations with a) lo = 40 nm, and b) lo = 50 nm.. Figure 3.6: Coherent oscillations with a) lo = 150 nm, and b) lo = 200 nm.. 29.

(35) Chapter 4 Phonon–induced decoherence In the foregoing chapter, we discussed how to adapt the Jaynes–Cummings model of quantum optics to the current system, operating close to the first anticrossing. In this analogy, the roles of the atomic pseudospin and light field are replaced by the spin and orbital momentum of the electron, respectively [12]. It was shown that, using voltage pulses applied to the side gate, it is possible to induce coherent oscillations in the quantum dot. Observation of these oscillations requires the system to remain in an excited state for a rather long time; that is, the lifetime of state ψα+ has to be long enough. Any relaxation from the excited to the ground state produces dephasing. Thus, the phase coherence will be lost and the coherent oscillation signal will be destroyed. In this chapter we study the decoherence effects that influence the coherent oscillations. Decoherence is produced due to the coupling to the environment. The magnetic moment of a single–electron spin is µB = 9.27 × 10−24 J/T (cf. [16]). It is a very small quantity and, consequently, electron spin states are weakly perturbed by their magnetic environment. On the other hand, electric fields affect only indirectly the spins so, in general, spin states are weakly influenced by their electric environment, as well. For electron spins in semiconductor heterostructures, it turns out that one of the most important interactions with the environment is the spin–orbit coupling. However, as we saw in section 2.3, in our quantum dot the spin–orbit characteristic length lSO is much smaller than other relevant lengths; i.e., lo and lB . As a consequence, even without attempting to control it, the spin–orbit interaction is not per se a source of decoherence. Nonetheless, at the setpoint of our quantum dot system, hybridization in the qubit couples the spin to the orbitals, which in turn are sensitive to the electron–phonon scattering. Therefore, decoherence stems mostly from the coupling to phonons [15, 23]. In what follows, we shall study the spin–phonon process in which spin–flip is induced due to electron–phonon interaction, via spin–orbit coupling. 30.

(36) 4.1. Primary processes. The potential induced by a bulk acoustic phonon with wave vector q is [8] Vep (q) = λq eiq·r (bq + b†q ),. (4.1). with phonon creation (annihilation) operator b†q (bq ), and coupling parameter λq . At low temperatures, the electron–phonon interaction is dominated by the piezoelectric coupling for bulk acoustic phonons in the long–wavelength regime. The piezoelectric coupling is found from the piezoelectric equations [4]. For zincblende structures, the only non–vanishing independent piezoelectric constant is h14 , producing a coupling function Mq, = 2ieh14 (q̂x q̂y ξˆz, + q̂y q̂z ξˆx, + q̂z q̂x ξˆy, ), (4.2) where q̂i = (q/q)i , and ξˆi, = (ξ̂ )i , with the unit vector ξ̂ describing the phonon polarization . To obtain a more tractable form of the piezoelectric coupling, angular averages for the longitudinal and the two transverse modes are performed, and then added. Therefore, taking into account the different sound velocities clong , and ctrans associated with the longitudinal and transverse modes, respectively, we obtain the following angular average of the magnitude of the piezoelectric coupling function M [6] 12 16 clong 2 2 |M | = (eh14 ) + ≡P (4.3) 35 35 ctrans Hence, we get for the coupling parameter 1 |λq | = ωq V 2. . ~P 2ρm. 2 ,. (4.4). where ωq = cq is the phonon dispersion, V is the normalization volume, and ρm is the mass density. In order to evaluate the effect of Equation 4.1 on the phonon–induced transitions from the upper to the lower state in a Jaynes–Cummings subspace, Ψ+ → Ψ− , we have to find the associated matrix element. Since the electron–phonon coupling does not depend on the spin, only the matrix elements which are diagonal in spin subspace contribute. Then, using the Jaynes–Cummings eigenstates (Equation 3.12), we obtain hΨ+ |Vep (q)|Ψ− i = cos θ+ cos θ− hn+ , n− , ↑ |Vep (q)|n+ , n− , ↑i + sin θ+ sin θ− hn+ , n− + 1, ↓ |Vep (q)|n+ , n− + 1, ↓i (4.5). 31.

(37) The phonon–induced transition rate can be calculated using the following Fermi’s golden rule 2π X |hΨ+ |Vep (q)|Ψ− i|2 δ(E+ − E− − ~ωq ) ~ q Z V = d3 q |hΨ+ |Vep (q)|Ψ− i|2 δ(∆E − ~ωq ) 4π 2 ~. Γep =. (4.6) (4.7). with ∆E = E+ − E− , the normalization volume V, and the phonon frequency ωq . For representing the electron–phonon interaction (4.1) in the Fock–Darwin basis, we write the coordinate operators in terms of the creation and annihilation operator (1.12), leading to x = lt (a− + a†− + a+ + a†+ )/2. (4.8). y = lt (a− − a†− + a†+ − a+ )/2i.. (4.9). Furthermore, introducing the complex phonon wave vectors αq = lt (qy + iqx )/2,. (4.10). αq∗. (4.11). = lt (qy − iqx )/2,. and the displacement operators †. D± (α) ≡ eαa± −α. ∗a ±. ,. (4.12). we are able to rewrite eiq·r = ei(qx x+qy y+qz z) , = eiqz z D+ (αq )D− (αq∗ ), using Equation 4.9, and noting that qx = −i. αq + αq∗ αq − αq∗ , qy = . lt lt. Consequently, we can write Equation 4.1 as Vep (q) = λq eiqz z D+ (αq )D− (αq∗ )(bq + b†q ).. (4.13). Furthermore, to calculate the matrix elements in Equation 4.5 using Equation 4.13,. 32.

(38) we note that hn0+ , n0− , s0 |Vep |n+ , n− , si = δs0 ,s λq hn0+ |D+ (αq )|n+ ihn0− |D− (αq∗ )|n− i,. (4.14). where the factor eiqz z becomes of order unity due to the strong z–direction confinement. The matrix elements of the operators in Equation 4.12 are (see [5]) r 1 n! 0 2 0 0 hn |D(α)|ni = (α)n −n e− 2 |α| Lnn −n (|α|2 ), (4.15) 0 n! with n0 ≥ n, and the generalized Laguerre polynomial Lm n . Accordingly, using (4.5), (4.14), and (4.15) we have hΨ+ |Vep (q)|Ψ− i = λq Ln+ (|αq |2 )e−(|αq |. 2 +|α∗ |2 )/2 q. × [cos θ+ cos θ− Ln− (|αq∗ |2 ) + sin θ+ sin θ− Ln− +1 (|αq∗ |2 )],. (4.16). Finally, from (4.4), (4.7), and (4.16), for the lowest qubit (i.e., n+ = n− = 0; cf. [11]), we obtain for the phonon–induced relaxation rate √ 2mP lo2 Γep = sin2 θ+ sin2 θ− η 5 F (η), (4.17) 2 2 8πρm lt ~ c √ where η = lt ∆E/( 2~c) and F (η) =. Z. 1. dt √. 0. t5 2 e−(ηt) 1 − t2. (4.18). Figure 4.1 shows F (η) numerically evaluated. Using ρm = 5.7 × 103 m/s, ctrans = 3.8 × 103 m/s, clong = 2.6 × 103 m/s, eh14 = 3.5 × 106 eV/cm [12], and lo = 150 nm (that is, ωo ≈ 1011 ), we plot in Figure 4.2 the phonon–induced relaxation rate Γep as a function of the external magnetic field. Notice that it is minimum close to the resonance magnetic field, Br = 87 mT. Hence, in the vicinity of Br the rate Γep is quite suppressed, Γep ≈ ωo × 10−7 = 104 s−1 . Comparison with the Rabi frequency Ω = 2GHz, Equation 3.20, suggests that the spin qubit robustness is not significantly lessened by the hybridization of the spin and orbitals. In many situations other sources of decoherence may arise. There can be noise in the gate voltage, local electric field fluctuations, ripple mechanisms, hyperfine interaction with the host ions, higher order, virtual spin–phonon interactions, or those mentioned in the introduction of this work, to name a few [2, 17, 19, 22, 27, 31]. However, as it was mentioned above, the SO interaction is one of the most important, and, 33.

(39) since lSO lo , lB , the main source of decoherence appears close to resonance, when hybridization between orbital and spin degrees of freedom occurs. In this section we have shown that the spin–phonon process in which spin–flip is induced due to electron–phonon interaction, via spin–orbit coupling, does not affect significantly the robustness of the coherent oscillations.. Figure 4.1: Numerical evaluation of F (η).. Figure 4.2: Phonon–induced relaxation rate as a function of the magnetic field.. 34.

(40) Conclusions This work can be divided in two main parts. First, we studied the energy spectrum of a single–electron in a two–dimensional electron gas (2DEG) in the presence of a perpendicular magnetic field. We presented analytical results for the Fock–Darwin states. Then, we perform the same calculation numerically and compare the results to check the used program. Next, we introduce the Rashba and the Zeeman effects, and solve the Hamiltonian numerically. We have applied our calculation to InGaAs and GaAs for a large range of values of spin–orbit coupling, from 0, 5 meV·nm to 20 meV·nm, finding between 0, 5 and 3 meV·nm almost the same features in the whole spectrum as function of magnetic field up to 2 T. However, for higher values of spin– orbit coupling a remarkable feature of this interaction becomes apparent. The spin– orbit interaction in quantum dots lifts the orbital degeneracy at zero magnetic field. In addition, we saw in the GaAs case that orbital degeneracies at B = 0 are lifted and spin–splitting occurs upon increasing B, both of these effects due to the spin–orbit interaction alone, since the Zeeman term is negligible as a consequence of the small g–factor in this material. In contrast, the impact of these two interactions is mixed in the InGaAs case when B is increased, since Zeeman effect becomes significant. The approaches when dealing with this system, including Zeeman and Rashba effects, are mostly based on perturbation theory. Therefore, almost all calculations concern exclusively the energy spectra. Instead, we calculated numerically the eigenstates for the InGaAs quantum dot. Accordingly, we present several eigenstates for this system, and compare them qualitatively to the Fock–Darwin states. We noted that many low energy wavefunctions are hardly affected, but, for wavefunctions with higher energy, there are deviations from the Fock–Darwin states. On the other hand, the second part of this work consists of the study of coherent oscillations in the quantum dot, making an analogy with the Jaynes–Cummings model of quantum optics, and the sources of decoherence, particularly spin–phonon relaxation processes. It was shown that by sweeping the gate pulse time tp , one is able to resolve the time evolution of the Rabi oscillations. Moreover, we have seen that these oscillations 35.

(41) have a non–trivial dependence on the external magnetic field, and the electrostatic, lateral confinement potential. We presented various charts of probability of finding an electron in the excited state of the qubit as a function of the pulse time, and magnetic field, for different values lateral potential. We noticed that, the greater the electrostatic, lateral potential, the greater the magnetic field which causes resonance. Finally, it was shown that the strong confinement suppresses important channels of relaxation. In fact, we noted that since the spin–orbit characteristic length is much smaller than both the electrostatic and magnetic characteristic lengths, the SO interaction in this setup is not per se a relaxation channel, despite in semiconductor heterostructures it is among the most important ones. Furthermore, we studied the primary spin–phonon process in which spin–flip is induced due to electron–phonon interaction, via spin–orbit coupling. The effect of the spin–phonon relaxation process was estimated using a Fermi’s golden rule. Thus, the phonon–induced relaxation rate was calculated, and its dependence on the external magnetic field was plotted. We observed that, operating close to resonance, the relaxation rate is much smaller than the Rabi oscillation frequency. As a consequence, the robustness of the coherent oscillation signal is not significantly lessened.. 36.

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