(4.17) of chapter 4, while for conduction electrons, the DP mechanism governs the spin relaxation time, Eq. (5.3).

## 5.4 Analysis of the results

We have experimentally determined in our four samples, for the temperature range 10 - 80 K, the long-lasting spin dephasing time $T_2^*$ [see Eq. (5.2)] at a fixed magnetic field of $B_0 = 0.56$ T. The experimentally measured dephasing time is affected by inhomogeneities caused by the local variations of the electron $g_e^\perp$ factors. As we have seen in chapter 4, $T_2^*$ is related with the spin relaxation time $\tau_s = T_2(0)$ of a single electron by the expression (see eq. (4.5) of chapter 4):

$$ \frac{1}{T_2^*} = \frac{1}{\tau_s} + \frac{\Delta g_e^\perp \mu_B B_0}{\hbar}. \qquad (5.10) $$

The width $\Delta g_e^\perp$ of the $g_e^\perp$ distribution for each sample has been determined in chapter 4. We have fitted the measured $T_2^{*-1}$ for samples B to F with Eq. (5.10) using $\tau_s^{-1}$ given by the model proposed by Harmon et al. [13] [Eq. (5.7)]:

$$ \frac{1}{T_2^*} = \frac{n_l}{n_{imp}} \left( \frac{1}{\tau_l} + \frac{\Delta g_e^\perp \mu_B B_0}{\hbar} \right) + \frac{n_c}{n_{imp}} \left( \frac{1}{\tau_s^c} + \frac{\Delta g_e^\perp \mu_B B_0}{\hbar} \right), \qquad (5.11) $$

In the remainder we note $\frac{1}{\tau_l} + \frac{\Delta g_e^\perp \mu_B B_0}{\hbar} = \frac{1}{\tau_l^*}$ the spin dephasing rate of the ensemble of localized electrons. We have neglected the inhomogeneities of $g_e^\perp$ factors for conduction electrons, because in a previous work on a 8-nm CdTe QW containing a 2D electron gas [25], the spin relaxation time of conduction electrons showed to be negligibly sensitive to these inhomogeneities.

### 5.4.1 Insulating regime

For samples B, C and E in the insulating regime, the spin dephasing rate $T_2^{*-1}$ is shown in Figs. 5.7 (a)-(c). We have considered that the conduction electrons relax according to the DP mechanism ($\tau_s^c = \tau_s^{DP}$), Eq. (5.3). It will be shown a posteriori that the contribution of the EY mechanism is negligible. In order to identify which type of scattering is dominant, we have fitted the experimental data scanning the values $\nu = 0, 1$ and 2 in Eq. (5.3). In this regime, the Fermi energy varies with the temperature-dependent concentration $n_c(T)$, determined by Eqs. (5.8) and (5.9). For each sample, the parameters to be determined are: the binding energy $E_B$, the spin dephasing time $\tau_l^*$, and the coefficient $\Xi$ (assumed to be a constant independent on T).

In Fig. 5.7, the blue solid line shows the spin dephasing rate $T_2^{*-1}$ as a function of the temperature, according to Eqs. (5.8) and (5.11), and using the type-I scattering ($\nu = 0$) in $\tau_s^{DP}$, Eq. (5.3). Since it is impossible to fit the experimental data assuming type-II ($\nu = 1$) or type-III ($\nu = 2$) scatterings, we can firmly identify the type-I scattering as the effective one in the studied samples. Table 5.1, on page 113, gives the fitting parameters $E_B$, $\tau_l^*$ and $\Xi$ for all studied samples.
At low temperatures, it has been shown that the mechanism that governs the spin
relaxation in sample B (with a low-doping concentration) is the hyperfine interaction
(see chapter 4). In chapter 1, the binding energy for an isolated donor has been
calculated to be E<sub>B</sub> = 24.7 meV in sample A. Taking this theoretical value as a starting
point, we have fitted the experimental data for sample B, with only τ<sub>l</sub><sup>*</sup> and Ξ as free
parameters [see dashed line in Fig. 5.7(a)]. We have obtained τ<sub>l</sub><sup>*</sup> = 1.47 ± 0.05 ns and
Ξ = 146 fs. As shown in Fig. 5.7(a), a lower binding energy is needed to agree better with
the experimental results; the solid line represents the spin dephasing time considering
a lower binding energy, E<sub>B</sub> = 19 meV. The value extracted for the spin dephasing time
of the localized electrons, τ<sub>l</sub><sup>*</sup> = 1.40 ns, compares well with 1.97 ns calculated (from the
the linear dependence of T<sub>2</sub><sup>*</sup>(B)<sup>-1</sup> found in chapter 4, see table 4.2) in the same magnetic
field B<sub>0</sub> = 0.56 T, at 2 K [black diamond in Fig. 5.7(a)].

In Eq. (5.8), the concentrations n<sub>l</sub>(T) [and n<sub>c</sub>(T)] of localized and [conduction electrons] have a strong dependence on the binding energy. The smaller the binding energy, the lower the temperature at which a donor is ionized. In QWs, there is no theory that predicts a value for the binding energy at any doping concentration. Nonetheless, in bulk systems, it has been generally shown, theoretically and experimentally, that the binding energy of a donor-bound electron decreases when the doping concentration increases [26-29]
<figure><img src="image_55.png" /><figcaption>**Figure 5.7** Measurements of the spin dephasing rates at different temperatures (red full disks) in (a) sample B; (b) sample C; (c) sample E. The blue continuous lines are theoretical fits according to Eqs. (5.3), (5.8) and (5.11), for the type-I scattering ($v = 0$). The fitting parameters are given in Table 5.1. The dashed line in Fig. (a) is a theoretical fit forcing $E_B = 24.7$ meV. The black full diamonds are data points calculated using the dependence of $T_2^{*-1}$ on the magnetic field for each sample (see chapter 4, Table 4.1).</figcaption></figure>
As can be seen in Fig. 5.8 (a), right axis, the density of conduction electrons in sample B starts to increase at a temperature around 25 K, and at T = 100 K, almost one half of the donors has been ionized. In Fig. 5.8 (a), the conduction spin relaxation time is also plotted as a function of the temperature, with E<sub>B</sub> = 19 meV and Ξ = 97 fs, known from the theoretical fit of Fig. 5.7(a). As the density of conduction electrons increases with the temperature, the spin relaxation time decreases.

Sample C contains an intermediate doping concentration, and the mechanism that governs the spin relaxation time of the localized electrons is an interplay between the hyperfine and the anisotropic exchange interaction (see Fig. 4.12, chapter 4). Since there is no theory that predicts a value of the binding energy for this doping concentration, we took it as a free parameter to fit the data of Fig. 5.7(b). The values extracted from the theoretical fit are: E<sub>B</sub> = 20 meV, τ<sub>l</sub><sup>*</sup> = 2.65 ns, and Ξ = 36 fs. The value of the localized spin dephasing time τ<sub>l</sub><sup>*</sup> is of the same order as the one calculated from the data of chapter 4, at 2 K, in the magnetic field B<sub>0</sub> = 0.56 T [black diamond in Fig. 5.7 (b)]. The difference between the values may come from slightly different experimental conditions, in particular, the density of excitation.

Finally, sample E contains a doping concentration that is close to the metal-insulator transition while keeping in the insulating regime. The principal spin relaxation mechanism for localized electrons in this region is the anisotropic exchange interaction (Fig. 4.12, chapter 4). The values extracted from the theoretical fit in Fig. 5.7(c) are: E<sub>B</sub> = 8 meV, τ<sub>l</sub><sup>*</sup> = 0.95 ± 0.05 ns, Ξ = 47 fs. The value for the localized spin dephasing time agrees well with the calculated value of 0.9 ns found, at 2 K, in the magnetic field B<sub>0</sub> = 0.56 T, following the study of chapter 4.

For this sample E, due to the small binding energy E<sub>B</sub> = 8 meV, the localized electrons start to ionize at low temperature, see Fig. 5.8(c). At high temperature, T ~ 100 K, the conduction spin relaxation time is of the order of 100 ps. This value compares well to the one observed in 2D electron gas of n-doped CdTe QW with a similar concentration at low temperature [25].

We suspect that, due to the kind of the studied system (donors inside a QW) and the range of temperatures 10-80 K, scatterings with neutral impurities and with screened ionized impurities (at the highest temperature) are the more probable scattering mechanisms. But we cannot exclude other mechanisms, as for example the one related to the interface roughness. Because we worked at relatively low temperature, we exclude however the scattering due to acoustic phonons.

We underline that Fig. 5.8, right axis, shows the calculated density of conduction electrons as a function of the temperature, which is equal to the density of ionized impurities. Then for samples B, C and E at the middle of the explored temperature domain, the density of ionized impurities becomes only 10% or 20% of the total amount of donors in the sample. At 80 K, the density of ionized impurities becomes larger and, in particular for sample E, can reach a value of 35% of the total amount of donors in the sample. This shows that, in the explored domain of temperatures, the donors are mostly in their neutral state. Moreover, the ionized donors are likely to be screened by conduction electrons which are confined inside the QW.
<figure><img src="image_56.png" /><figcaption>**Figure 5.8** Concentrations of conduction electrons (dashed lines, right axis), and spin relaxation times of the conduction electrons (continuous lines, left axis) versus temperature, in (a) sample B; (b) sample C; (c) sample E.</figcaption></figure>
### 5.4.2 Metallic regime, near the MIT

Under the Mott criterion [14], and following the measurements carried out in chapter 4, sample F is assumed to contain a doping concentration slightly beyond the MIT; conductance measurements could have confirmed this point, but our samples are not patterned for transport experiments. We found that the experimental data on this sample could not be fitted by considering only a population of conduction electrons and a dephasing time related to DP or/and EY mechanisms. For doping concentrations slightly above the MIT, it has been shown that there is a coexistence between localized electrons forming an impurity band, and itinerant electrons, leading to a filamentary electronic transport [4].

In order to explain the spin dephasing time observed in sample F, we have considered that at very low temperature a fraction $f$ of the doping concentration $n_{imp}$ is in the conduction band: $n_c(0) = fn_{imp}$. Thus, at any temperature the density of localized electrons is given by a refined version of Eq. (5.8):

$$n_l(T) = n_l(0) \frac{\sqrt{1 + Q(T, n_l(0))} - 1}{\sqrt{1 + Q(T, n_l(0))} + 1}, \quad (5.12)$$

with $n_l(0) = n_{imp}(1 - f)$ the initial concentration of localized donors. The density of conduction electrons at any temperature is evaluated from the conservation rule $n_c(T) = n_{imp} - n_l(T)$. Figure 5.9(a) shows the theoretical fit of the measured spin dephasing time according to Eqs. (5.11) and (5.12), using Eq. (5.3) for the spin relaxation time of the conduction electrons. We have taken as free parameters the binding energy $E_B$ of the donor-bound electron, the transport coefficient $\Xi$, the spin dephasing time $\tau_l^*$ of the localized electrons, and the fraction $f = n_c(0)/n_{imp}$ of conduction electrons at vanishing temperatures.

Scanning the values $v = 0, 1$ and $2$ in Eq. (5.3), we identify that the type of scattering undergone by the conduction electrons in sample F is also the type-I of scattering ($v = 0$). The fitting parameters are found to be: $f = 0.1$, $E_B = 6$ meV, $\tau_l^* = 1.63$ ns and $\Xi = 47$ fs. The black diamond in Fig. 5.9(a) represents the spin dephasing time calculated from the $T_2^{*-1}$ formula obtained in chapter 4, at 2 K, in the magnetic field $B_0 = 0.56$ T; this value agrees well with our experimental data.

In Fig. 5.9(b), the conduction spin relaxation time (left axis) is plotted as a function of the temperature, with $E_B = 6$ meV and $\Xi = 47$ fs; the density of conduction electrons is also plotted (right axis). As the temperature increases, due to the small binding energy, the density of conduction electrons increases at a faster rate in comparison with the samples in the insulating regime.
<figure><img src="image_57.png" /><figcaption>**Figure 5.9** (a) Measurements of the spin dephasing rates at different temperatures (red full disks) in sample F. The blue continuous line is a theoretical fit according to Eqs. (5.3), (5.11) and (5.12), for the type-I scattering ($v = 0$). The fitting parameters are given in Table 5.1. The black full diamond is a data point extracted from chapter 4. (b) Concentration of conduction electrons (dashed line, right axis), and spin relaxation time of the conduction electrons (continuous line, left axis) versus temperature, in sample F.</figcaption></figure>
For conduction electrons in materials with a large spin-orbit constant or narrow energy band gap, the EY mechanism of spin relaxation becomes dominant [13]. The spin-orbit constant of CdTe is three times larger than the one of GaAs, material for which it has been shown that EY mechanism is not effective as compared to the DP one [13]. Using $\tau_{tr} = \Xi$ (table 5.1) obtained here and expression (15) of Ref. [22], the DP relaxation rate is found more than 10 times larger than the EY one, in the range of temperatures studied and for all the samples, see Fig. 5.10. This justifies, a posteriori, our use of the DP mechanism to describe the spin relaxation of conduction electrons in our samples.

<figure><img src="image_58.png" /><figcaption>**Figure 5.10** Spin relaxation rate according to the DP mechanism, Eq. (5.3) (black-solid line), and to the EY mechanism, Eq. (5.6) (red-dashed line), for each of the samples in the insulating regime: (a) sample B, (b) sample C, (c) sample E, and in the metallic regime: (d) sample F.</figcaption></figure>
<table><caption>**Table 5.1**</caption><thead><tr><th>Sample</th><th>Doping concentration n<sub>imp</sub> (cm<sup>-2</sup>)</th><th>Spin dephasing time τ<sub>l</sub><sup>*</sup> (ns)</th><th>Binding energy E<sub>B</sub> (meV)</th><th>Transport coefficient Ξ (fs)</th></tr></thead><tbody><tr><td>B</td><td>3.2 × 10<sup>10</sup></td><td>1.40 ± 0.05</td><td>19 ± 1</td><td>97</td></tr><tr><td>C</td><td>9.7 × 10<sup>10</sup></td><td>2.65 ± 0.05</td><td>20 ± 1</td><td>36</td></tr><tr><td>E</td><td>2.9 × 10<sup>11</sup></td><td>0.95 ± 0.05</td><td>8 ± 1</td><td>47</td></tr><tr><td>F</td><td>3.6 × 10<sup>11</sup></td><td>1.63 ± 0.05</td><td>6 ± 1</td><td>47</td></tr></tbody></table>

**TABLE 5.1:** Extracted parameters from the fits of the experimental data for the spin relaxation time as function of the temperature, see Figs. 5.7 and 5.9 (a). For sample F, the values are obtained considering a concentration of conduction electrons $n_c(0) = 0.1 n_{imp}$ at zero temperature.
## 5.5 Conclusion

In summary, in this chapter we have shown a quantitative agreement between theory and experiment for the temperature dependence of the spin relaxation time of electrons bound to donors immersed in the center of a 8-nm CdTe QW, in the 10-80K temperature range, by considering that two types of spin systems coexist and interact by spin exchange: the electrons localized on donors, and the free electrons promoted to the conduction band.

We evidenced that the spin relaxation time for the localized electrons is imposed by the same mechanisms known at low temperature (hyperfine and anisotropic exchange interactions), while the spin relaxation time of the conduction electrons is governed by the DP mechanism. For the samples in the insulating regime, we have identified that the scatterings undergone by the conduction electrons are of type I [19]. Under the same framework, we have succeeded in explaining the observed behavior *versus* temperature of the spin relaxation time for the sample with a doping concentration slightly above the Mott transition, by considering that an initial concentration of delocalized electrons exists at low temperature. The scatterings of the conduction electrons in this sample is also of type I.
## 5.6 Bibliography of chapter 5

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# Chapter 6

## Longitudinal spin relaxation time of donor-bound electrons immersed in a CdTe QW

### Content

6.1 Introduction

6.2 Development of an extended PFR technique

6.3 Experimental results

6.4 Discussion

6.4.1 Comparison with bulk semiconductor results

6.4.2 Magnetic field dependence of $T_1$

6.5 Conclusion

6.6 Bibliography of chapter 6
## 6.1 Introduction

This chapter is devoted to the study of the longitudinal spin relaxation time in samples A, C, D and E of chapter 4. Up to this work, the only studies carried out in bulk zinc-blende semiconductors are those reported by Refs. [1-6] in GaAs, CdTe and InP for different doping concentrations in the insulating regime and at low temperature. In bulk wurzite materials, a recent study has been performed by Ref. [7] in high-purity bulk ZnO.

In general, two principal challenges have been addressed in Refs. [1-7]:
(a) the development of an experimental technique that allows to measure long spin relaxation times

(b) the understanding of the principal mechanisms that dominate the longitudinal spin relaxation in presence of a longitudinal magnetic field.

Both points have been successfully overcame for high-purity direct-band-gap semiconductors by Linpeng *et al*. in Ref. [2]. In their work they measured, by means of a TRPL technique, the longitudinal spin relaxation time of donor-bound electrons as a function of the magnetic field in GaAs, InP and CdTe. The principle of the experimental technique is presented in Fig. 6.1 (a): A resonant pump pulse excites repetitively the $\pi$ transition to pump electrons from the $|↑\rangle$ state to the $|↓\rangle$ one. After this, the system freely evolves during a variable time $\tau_{wait}$ in which the electron population returns to thermal equilibrium and the $|↑\rangle$ state is repopulated. Then, a pulse resonant at the $\pi$ transition probes the $|↑\rangle$ state and the PL from the $\sigma$ transition is detected. The experiment is repeated with different time delays $\tau_{wait}$. An example of the PL recorded for InP is presented in Fig. 6.1 (b).

The experimental results obtained by Ref. [2] are presented in Fig. 6.1 (c). For the 3 samples, the authors observed at high magnetic fields a $B^{-x}$ dependence of $T_1(3 < x < 4)$. For GaAs at low magnetic fields, $T_1$ approaches a $B^2$ dependence. At low magnetic field, the behavior was explained as a result of the interaction of the donor-bound electron with the nuclear environment (hyperfine interaction) in a presence of a magnetic field, while at high magnetic fields, the spin-orbit interaction [2]. For higher doping concentrations, no formal theory has been developed by Refs. [3-6].

Refs. [3,5] have principally focused their work on point (a), by modifying the techniques described in chapter 2. In Ref. [3], by the inclusion of additional pump and probe pulses, an extended version of the TRPL technique was developed. This technique, that allows to measure times of the order of microsecond, uses 4 pulses (see Fig. 6.2): two pump pulses ($\sigma^+$ and $\sigma^-$ polarized) that create the spin polarization in the sample and two probe pulses that detect the PL coming from each pump pulse. The available delay between each pair of pulses is up to 20 microseconds. The PL is collected with a double-grating spectrometer and measured with a two-channel photon counter: $\sigma^+$ and $\sigma^-$ are separately recorded.

As already remarked, one of the main limitations imposed by the usual PFR technique, described in chapter 2, is the restricted time range that can be explored due to the finite length of mechanical delay lines. To overcome this restriction, in Ref. [5] they developed an extended PFR technique. The physical principle of this technique is the same as the
usual one, the novelty is that the pump-probe delay is controlled electronically. Figure 6.3 shows the experimental set-up developed by Ref. [5]: an electro-optic and an acousto-optic modulator are used to produce trains of pulses for the pump beam and single pulses for the probe beam; the delay between them is controlled electronically. With this technique, they measured the longitudinal spin relaxation time for donor-bound electrons in GaAs at different doping concentrations in the insulating regime. A scheme of this experimental technique is presented in Fig. 6.3

In this chapter, we have focused our efforts on point (a), developing an extended PFR technique according to the set-up proposed by Ref. [5]. With this technique, we measured the longitudinal spin relaxation time $T_1$ for donor-bound electrons immersed in the middle of a CdTe QW for different doping concentrations, in the microsecond regime with a 0.03 µs time resolution.

<figure><img src="image_59.png" /><figcaption>**Figure 6.1** Figure taken from [2]: (a) energy level diagram for the donor system, (b) PL recorded for bulk InP. The amplitude of the signal is proportional to the $|↑〉$ population. (c) Longitudinal spin relaxation time $T_1$ for electrons bound to donors, in high-purity bulk GaAs, InP and CdTe, versus Zeeman energy $gμ_B B$.</figcaption></figure>

<figure><img src="image_60.png" /><figcaption>**Figure 6.2** Figure taken from [3]: principle of a TRPL technique with pump-probe pulses.</figcaption></figure>
<figure><img src="image_61.png" /><figcaption>**Figure 6.3** *Figure taken from [5]: scheme of the extended PFR experiment. The delay between the pump train and the probe pulse is generated electronically.*</figcaption></figure>

**6.2 Development of an extended PFR technique**

In this section, we describe the extended PFR technique developed to measure spin relaxation times in the microsecond regime. A scheme of the experimental set-up is presented in Fig. 6.4.

The light source is a Ti: sapphire laser (*Coherent Mira*) optically pumped with a continuous laser *Verdi* (532 nm, 15 W), emitting 2-ps pulses with a repetition rate of 76 MHz (repetition period $T_R = 13.1$ ns). The divergence of the laser is corrected by focusing the principal beam with a 2-mm focal lens on a pinhole of 50 micrometers of diameter and then collimated by a 40-mm focal lens.

The laser output is split into pump and probe beams using a beam splitter (70/30). In the pump path, an electro-optic modulator (EOM) selects trains of $N$ pulses separated by $T_R$, with arbitrary long delay between the trains. The time width of each train $w_{pump} = NT_R$ can be chosen. The acousto-optic modulator (AOM) is used in the probe path for the same purpose. The time delay between a train produced by the EOM and one produced by the AOM is controlled electronically, allowing a large change of the delay between the pump trains and the probe ones.

In this experimental set-up, the EOM is controlled in such a way to produce linearly-polarized pump trains. The polarization direction is fixed by a polarizer set at the exit of the EOM; this direction is at 45° to the neutral axes of a photo-elastic modulator (PEM). The PEM is used to modulate the polarization of the pump beam between $\sigma_+$ and $\sigma_-$ at a frequency of 50 kHz, to avoid nuclear spin polarization in the samples (see section 2.3.2).

In order to perform synchronous detection (see section 2.3.4), the intensity of the probe beam is modulated with an optical chopper, at $f_{probe} = 500$ Hz. The signal given by the optical bridge is then modulated at two frequencies $f_{pump} = 50$ kHz and $f_{probe} = 500$ Hz.
The limitation in time range of this technique is imposed by the modulation frequency of the PEM. Since its period of modulation is 20 microseconds, the maximum temporal window is then 3 microseconds (in order to average at least on 3 trains of the pump-beam over half a period of the PEM).

The minimum temporal window is 30 ns (being limited by the time width of the trains). Figure 6.5 shows the optical trains obtained when the EOM and the AOM are driven by 100 ns electronic gates: around $N = 8$ picosecond pulses are present in the trains; the delay between the probe and pump trains can be swept at will (within the 3µs window ).

<figure><img src="image_62.png" /><figcaption>**Figure. 6.4** Scheme of the extended PFR technique. The time width of the trains, and the delay between the pump and probe trains are controlled electronically. The width for the trains shown in the figure is 50 ns (3 pulses). A magnetic field B is applied in Faraday geometry (not shown).</figcaption></figure>

<figure><img src="image_63.png" /><figcaption>**Figure 6.5** Pump and probe 100-ns optical trains delayed by 100 ns. Each train contains around 8 pulses separated by the repetition period of the laser, 13.1 ns. (The time width of each pulse is limited by the bandwidth of the used photodiode)</figcaption></figure>
## 6.3 Experimental results.

In figure 6.6, we present the longitudinal electron spin relaxation measured for the samples A, C, D and E. As usual, the laser wavelength was set to the $D^0X$ transition of each sample indicated by its PL spectrum performed at T = 10 K (see chapter 4, Fig. 4.2). Here we used a power of excitation of 3 mW for the pump beam and 60 microwatts for the probe beam. The widths of the trains, and the needed temporal window chosen for each sample, are indicated in the corresponding figure captions. One of the principal difficulties faced during the measuring process on the samples is the very low level of signal; especially for the heavily doped sample E, the signal was hardly detectable. In order to keep a detectable level of signal, the widths of the gates have been increased only when a longer temporal window was needed.

Figures 6.6 shows that the application of a longitudinal magnetic field slows down the spin dynamics in the studied samples. The signal in Fig. 6.6 can be described by a double exponential decay with a fast and a slow component. The fast component observed corresponds to the overlapping at t = 0 of the pump and probe trains. In Refs. [5,6] they identified a fast component, no related with this overlapping, due to the use of single probe pulses. For our samples, trains of probe pulses are needed to increase the level of the signal.

As it has been observed in bulk GaAs, CdTe and InP [2], residual doped samples show longer spin relaxation times: for sample A at a magnetic field of B = 0.08 T (Fig. 6.6 (a)) the measured time of $T_1 \sim 10\mu s$, reaches the maximum relaxation time than can be measured with our experimental set-up.

For the sample with an intermediate doping concentration, sample C, we observe a continuous increase of the spin relaxation time with the applied magnetic field. For samples D and E, there is first an increase and after a decrease of the relaxation time in the same range of magnetic fields. Figure 6.7 gathers the measured spin relaxation times, for the different samples, as function of the Zeeman splitting energy. As the concentration is increased, the maximum measured $T_1$ decreases.

Table 6.1 gathers the values of the magnetic field $B_{max}$ at which the measured longitudinal spin relaxation time is maximum, for samples A, C, D and E.
T = 2 K

normalized signal

<figure><img src="image_64.png" /></figure>

<figure><img src="image_65.png" /></figure>

<figure><img src="image_66.png" /></figure>

<figure><img src="image_67.png" /></figure>

delay time (ns)

**Figure 6.6** Longitudinal spin signal versus time of the donor-bound electrons, in samples A, C, D, E in different applied magnetic fields. The width of the trains for the pump and probe beams (*w*<sub>pump,probe</sub>) and the time windows are: (a) *w*<sub>pump</sub>= 100 ns, *w*<sub>probe</sub>= 50 ns, window = 1 µs; (b) *w*<sub>pump</sub>= *w*<sub>probe</sub>= 150 ns, window = 3µs; (c) *w*<sub>pump</sub>= *w*<sub>probe</sub>= 100 ns, window = 3µs; (d) *w*<sub>pump</sub>= *w*<sub>probe</sub>= 60 ns, window = 1µs.
<figure><img src="image_68.png" /><figcaption>**Figure 6.7** Longitudinal spin relaxation time $T_1$ as a function of the Zeeman splitting energy, for samples A, C, D and E.</figcaption></figure>

<table><caption>**Table 6.1** Magnetic field and maximum longitudinal spin relaxation time measured for samples A, C, D and E.</caption><thead><tr><th>Sample</th><th>Doping concentration (donors/cm²)</th><th>Electron Landé factor $g_e^\perp$</th><th>B<sub>max</sub> (T)</th><th>T<sub>1,max</sub> (µs)</th></tr></thead><tbody><tr><td>A</td><td>1 × 10<sup>9</sup></td><td>1.39</td><td>0.08</td><td>~9</td></tr><tr><td>C</td><td>9.7 × 10<sup>10</sup></td><td>1.44</td><td>4.0</td><td>5.0</td></tr><tr><td>D</td><td>1.6 × 10<sup>11</sup></td><td>1.45</td><td>0.64</td><td>3.0</td></tr><tr><td>E</td><td>2.9 × 10<sup>11</sup></td><td>1.40</td><td>0.8</td><td>0.7</td></tr></tbody></table>
## 6.4 Discussion of the results

In this section, we compare the results obtained for samples A, C, D and E with the available experimental data reported in bulk GaAs.

### 6.4.1 Comparison with bulk semiconductor results

In the insulating regime, at low temperature, the longest spin relaxation times have been measured in high-purity bulk semiconductors by Linpeng et al. [2]: $3 \times 10^{13} \text{ cm}^{-2}$ for GaAs, $2.3 \times 10^{14} \text{ cm}^{-3}$ for InP and $1 \times 10^{14} \text{ cm}^{-3}$ for CdTe (see Fig. 6.1 (c)).

Similarly we observe that sample A, with a residual doping concentration, exhibits the longest spin relaxation time $T_1 \sim 9 \text{ µs}$ at the energy $g_e\mu_B B = 0.006 \text{ meV}$. This value is larger than $T_1 = 3 \text{ µs}$ reported in Ref. [6] for bulk GaAs with a residual doping concentration of $n_d = 5.5 \times 10^{14} \text{ cm}^{-3}$, at the same Zeeman splitting energy (see Fig. 6.8 (a)), $B = 0.2 \text{ T}$. For bulk CdTe, there is no reported value in Ref. [2], at the same Zeeman energy.

For intermediate doping concentrations in bulk GaAs, the available experimental data are from Refs. [3,4,6]. The samples studied by Refs. [3,4] consisted in a 1µm-thick GaAs layer in an AlGaAs heterostructure, containing a doping concentration of $n_d = 1 \times 10^{15} \text{ cm}^{-3}$ and $n_d = 3 \times 10^{15} \text{ cm}^{-3}$. The results are shown in Fig. 6.8. The sample studied by Ref. [6] consisted in a 20µm thick GaAs layer, with a doping concentration of $n_d = 1 \times 10^{15} \text{ cm}^{-3}$. The results are presented in Fig. 6.9 (b).

<figure><img src="image_69.png" /><figcaption>**Figure 6.8** Figures taken from [3,4]: $T_1$ relaxation time of electrons bound to donors in bulk GaAs, with a doping concentration of $n_d = 1 \times 10^{15}$ (left) and $n_d = 3 \times 10^{15} \text{ cm}^{-3}$ (right), versus magnetic field [on the right, Fig. (b) is the relaxation rate].</figcaption></figure>
As can be seen in Fig. 6.8, the sample with a concentration of $n_d = 1 \times 10^{15} \text{cm}^{-3}$ (left column), shows a longitudinal spin relaxation time of $T_1 = 17 \text{ µs}$ at $B = 6 \text{ T}$ and $T = 1.5 \text{ K}$. In Ref. [6], for the same concentration of donors (see Fig. 6.9 (b)) the spin relaxation time shows a maximum relaxation time $T_{1,max} = 1 \text{ µs}$ at $B_{max} = 1 \text{ T}$, and after a decrease to hundreds of nanoseconds at $B = 6 \text{ T}$. Therefore, the spin relaxation times measured by V. V. Behlyk et al. [6] are shorter than the ones reported by Colton et al. [4]. The origin of this discrepancy is probably the different experimental conditions used in each case: in Ref. [4] the energy of excitation is fixed above the band gap, as has been already discussed in chapter 4, this produces faster-relaxing free carriers that shortens the spin relaxation time due to the interaction with localized electrons, while in Ref. [6] the excitation is done at the $D^0X$ transition energy.

For the sample with a concentration of $n_d = 3 \times 10^{15} \text{cm}^{-3}$ (right column of Fig. 6.8), the maximum relaxation time measured is $T_{1,max} = 1.4 \text{ µs}$ at $B_{max} = 5 \text{ T}$.

<figure><img src="image_70.png" /><figcaption>**Figure 6.9** Figures taken from [6]: $T_1$ spin relaxation time of electrons bound to donors in bulk GaAs with different doping concentrations (indicated on the figures), at different magnetic fields.</figcaption></figure>

The measured times reported by Refs. [2-6] are listed in Table 6.2. We remark that samples C and D, with an intermediate doping concentration, show a longer longitudinal spin relaxation time than samples studied in Ref. [4] and Ref. [6] (see tables 6.1 and 6.2).

For samples with a doping concentration next to the MIT (see Fig. 6.10), the available experimental data belongs to Ref. [5]. In this reference, the authors measured at low temperature, the longitudinal spin relaxation time $T_1$ in the range 0 -6T, for bulk GaAs. At low magnetic field they found a $T_1$ of the order of 260 ns, which increases up to $T_{1,max} = 450 \text{ ns}$ at $B_{max} = 6 \text{ T}$. Sample E with a doping concentration next to the MIT exhibits a longer $T_{1,max}$ (700 ns) than the one reported by Ref. [6].
<table><caption>**Table 6.2** Maximum longitudinal spin relaxation time reported for different doping concentrations in bulk GaAs in the insulating regime.</caption><thead><tr><th>Reference</th><th>n<sub>d</sub><br>(donors/cm<sup>3</sup>)</th><th>B<sub>max</sub> (T)</th><th>T<sub>1,max</sub> (μs)</th></tr></thead><tbody><tr><td>Linpeng et al. [2]</td><td>3 × 10<sup>13</sup></td><td>4</td><td>1400</td></tr><tr><td>V. V. Belykh et al. [6]</td><td>5.5 × 10<sup>14</sup></td><td>1</td><td>~10</td></tr><tr><td>Colton et al., [3]</td><td>1 × 10<sup>15</sup></td><td>3</td><td>19</td></tr><tr><td>Colton et al., [4]</td><td>3 × 10<sup>15</sup></td><td>6</td><td>1.4</td></tr><tr><td>V. V. Belykh et al. [6]</td><td>1 × 10<sup>15</sup></td><td>1</td><td>1</td></tr><tr><td>V. V. Belykh et al. [5]</td><td>1.4 × 10<sup>16</sup></td><td>6</td><td>0.45</td></tr></tbody></table>

<figure><img src="image_71.png" /><figcaption>**Figure 6.10** Figure taken from [5]: T<sub>1</sub> spin relaxation time of electrons bound to donors in bulk GaAs with a doping concentration close to the MIT n<sub>d</sub> = 1.4 × 10<sup>16</sup>cm<sup>-3</sup>, at different magnetic fields.</figcaption></figure>
### 6.4.2 Magnetic field dependence of $T_1$.

Up to know, there is no unified model that explains the magnetic field dependence of $T_1$ at a given doping concentration and for any magnetic field, either in bulk or QWs.

In high-purity bulk semiconductors, the increase of the spin relaxation time at moderate magnetic fields has been explained by the suppression of the hyperfine interaction [2,9], while at high magnetic fields, the decrease of the spin relaxation time observed by Linpeng et al. [2] (see Fig. 6.1 (c)) is induced by the spin-orbit interaction and phonons following a $B^{-x}$ law.

The role of the hyperfine interaction has particularly been discussed by Yao et al. [10], for the long time range. In this range, the spin relaxation time is induced by an inhomogeneous hyperfine coupling of the electron spin with the fluctuating nuclear spins. The fluctuation of the nuclear spin is induced by two different processes:

* the second-order hyperfine interaction which couples two nuclei via the same electron.

* the nuclear dipole interaction.

The first process is dominant at low magnetic field and it can be screened by a longitudinal magnetic field. The second process is less sensitive to B and could fix a maximum value of the spin relaxation time depending on the material and its intrinsic parameters.

In a semiconductor QW the magnetic field dependence of $T_1$ has been treated theoretically by Refs. [11,12] considering hopping diffusion. This model is based on the idea that the ensemble of localized electrons is grouped in small clusters, inside of which an electron is coupled with the rest of the donor-bound electrons by isotropic exchange. but it also may visit other clusters by hopping diffusion. In this references the spin relaxation time is calculated considering the hyperfine interaction and spin-orbit interaction (no dipole-dipole interaction is considered). We discard this model, since it is related to a hopping time which is still to be defined and determined in our system.

## 6.5 Conclusion

In this chapter we have measured the longitudinal spin relaxation measurements for donor-bound electrons in a CdTe semiconductor, by means of an extended PFR rotation technique. Our experiment brought the first results of this kind in a CdTe QW. For any doping concentration, we have observed an increase of the spin relaxation time when a magnetic field is applied. Particularly, the longest longitudinal relaxation time is for a residual doping concentration as it has been observed in previous references for other compounds. More theoretical studies are needed in order to understand the mechanisms that govern the longitudinal spin relaxation at any doping concentration.
## 6.6 Bibliography of chapter 6

[1] K. C. Fu, W. Yeo, S. Clark, C. Santori, C. Stanley, M. C. Holland, and Y. Yamamoto, Phys. Rev. B **74**, 121304(R) (2006).

[2] X. Linpeng, T. Karin, M. V. Durnev, R. Barbour, M. M. Glazov, E. Ya. Sherman, S. P. Watkins, S. Seto, and K. C. Fu, Phys. Rev. B **94**, 125401 (2016).

[3] J. S. Colton, T. A. Kennedy, A. S. Bracker, and D. Gammon, Phys. Rev. B **69**, 121307(R) (2004).

[4] J. S. Colton, M. E. Heeb, P. Schroeder, A. Stokes, L. R. Wienkes, and A. S. Bracker, Phys. Rev. B **75**, 205201 (2007).

[5] V. V. Belykh, E. Evers, D. R. Yakovlev, F. Fobbe, A. Greilich, and M. Bayer, Phys. Rev. B **94**, 241202(R) (2016).

[6] V. V. Belykh, K. V. Kavokin, D. R. Yakovlev, and M. Bayer, Phys. Rev. B **96**, 241201(R) (2017).

[7] X. Linpeng, M. L.K. Viitaniemi, A. Vishnuradhan, Y. Kozuka, C. Johnson, M. Kawasaki, and K.M. C. Fu, Phys. Rev. Applied **10**, 064061 (2018).

[8] Colton et al., Phys. Stat. Sol. (b) **233**, 445 (2002).

[9] M. M. Glazov, Phys. Rev. B **91**, 195301 (2015)

[10] W. Yao, R. B. Liu, and L. J. Sham, Phys. Rev. B **74**, 195301 (2006).

[11] I. S. Lyubinskiy, Pis'ma ZhETF **88**, 934 (2008).

[12] K.V. Kavokin, Semicond. Sci. Technol. **23**, 114009 (2008).
Conclusions and perspectives

This thesis has been devoted to study the spin relaxation time of donor-bound electrons immersed in a CdTe quantum well of thickness L = 8 nm, by means of a photo-induced Faraday and Kerr rotation techniques. We have experimentally studied three of the main variables that influences the spin relaxation time: doping concentration, temperature and longitudinal magnetic field.

One of the first principal results obtained in this work, is the theoretical determination of the exchange energy of two electrons-bound to donors placed in the middle of an infinite QW. The exchange energy determines the residence time of an electron-bound to a donor, which is fundamental to define the spin relaxation mechanisms in a doped semiconductor, in this sense its knowledge is of prime importance. The obtained expression is written in units of effective Hartree, and the distance between the donors is in units of effective Bohr radius, it can be then applied for any direct-band gap semiconductor QW of thickness L. In comparison with bulk systems, we have found that in the low doping regime, the electron exchange energy in a QW, $J_{QW}$, is smaller than in bulk: $J_{QW} < J_{3D}$, while for high doping concentrations the situation is reversed: $J_{QW} > J_{3D}$. Because we consider an infinite QW with a wave-function completely confined in the QW, our calculations do not describe systems for $L < a_B^{3D}$. Indeed, the electron wave-function of thin QWs overflows the barrier material which does not represent an infinite barrier for electrons.

By using a photo-induced Faraday rotation technique, we have measured the spin relaxation time of donor-bound electrons for different doping concentrations at low temperature in the insulating regime. A non-monotonous behavior is observed and it is successfully explained by the interplay of two mechanisms, as in GaAs: hyperfine interaction and anisotropic exchange interaction, being the hyperfine interaction the dominant mechanism at low doping and the anisotropic exchange interaction at high doping concentration. From the fit of the experimental data we determined the CdTe
spin-orbit constant: $α_{so} = 0.079$, which is in good agreement with previous theoretical calculations.

According with the spin relaxation mechanisms identified here, we have predicted that the spin relaxation time in bulk is always longer than in a QW, nevertheless the reported experimental values by Sprinzl et. al. [1] in bulk CdTe, are very small compared with those predicted by the theory and also with our experimental values. We concluded that the experimental conditions used by Ref. [1]: femtosecond pulses tuned at the band gap energy and a concentration of photo-excited carriers comparable with the nominal one, are at the origin of the observed shortening of spin relaxation time.

The results of the experimental study of the spin relaxation time in the range 10-80K, by means of photo-induced Kerr rotation technique, were successfully explained by invoking spin exchange between electron spins localized on donors and the spin of electrons promoted to conduction states. For the whole range of the studied doping concentrations, we demonstrated that, while the spin of localized electrons undergoes the effect of both hyperfine and anisotropic exchange interactions, the D'yakonov-Perel' mechanism governs the spin relaxation of the conduction electrons. Moreover, we have identified that the scatterings undergone by the conduction electrons are of type I [2]. We also demonstrated that the contribution of the Elliot Yafet mechanism to the spin relaxation time can be neglected. In particular, for the sample with a doping concentration slightly beyond the Mott transition, we found that the experimental data on this sample could not be fitted by considering only a population of conduction electrons and a dephasing time related to DP or/and EY mechanisms. In order to explain the spin dephasing time we considered that at very low temperature a fraction *f* of the doping concentration *n*<sub>imp</sub> it is already in the conduction band. A successful agreement between theory and experimental data was found.

Finally, we developed a new experimental set-up able to explore relaxation times in the microsecond regime. With this technique we have measured by first time the longitudinal spin relaxation time $T_1$ of donor-bound electrons immersed in the middle of a CdTe QW, for different doping concentrations as a function of a longitudinal magnetic field. From the observed experimental behavior, residual doped QW's appears as suitable candidates to obtain long spin relaxation times. The theory developed so far, shows that in high-purity-zinc-blende semiconductors the suppression of the hyperfine interaction is at the origin of the observed increased of $T_1$ at low magnetic fields, while at high magnetic fields the spin orbit interaction and phonons explain a decrease of the spin relaxation time with a $B^{-x}$ dependence. More studies must be carried out in order to explain the non-monotonous behavior of the longitudinal spin relaxation time for higher doping concentrations.

In summary, this work presents a fundamental study of the three main variables that influence the spin relaxation time in a semiconductor CdTe QW: doping concentration, temperature and longitudinal spin relaxation time. From this study we conclude that residual doped CdTe QWs are suitable candidates for obtaining long longitudinal spin relaxation times. In addition, we propose a zinc-blende II-VI material: bulk ZnSe and a
wurtzite material: bulk ZnO as promising candidates for obtaining long spin relaxation
times at low temperatures.

REFERENCES

[1] D. Sprinzl, P. Horodyská, N. Tesarová, E. Rozkotová, E. Belas, R. Grill, P. Malý, and P. Nemec, Phys. Rev. B **82**, 153201 (2010).

[2] J. Kainz, U. Rössler, and R. Winkler, Phys. Rev. B **70**, 195322 (2004).
# Appendix A

## Proof of $T_1 = T_2$ for a vanishing magnetic field

In this appendix, we argue that in absence of magnetic field, the longitudinal spin relaxation time $T_1$ is equal to the transversal relaxation time $T_2$.

Let's consider a two-level system of spin 1/2 under a vanishing magnetic field $\vec{B}_x \to 0$. The density matrix elements can be expressed in the x-basis as

$$ \rho_{\uparrow\uparrow}^{x}(t) = \frac{1}{2} + ae^{-t/T_1} \qquad (A.1) $$

$$ \rho_{\downarrow\downarrow}^{x}(t) = \frac{1}{2} - ae^{-t/T_1} \qquad (A.2) $$

$$ \rho_{\uparrow\downarrow}^{x}(t) = \rho_{\downarrow\uparrow}^{x}(0)e^{-t/T_2} \qquad (A.3) $$

with $|a| < 1/2$. A factor $e^{i\omega t}$ with $\omega = g\mu_B B/\hbar$ should appear in equation (A.3), but since $B \to 0$ then $e^{i\omega t} \to 1$: the dynamics of the coherence $\rho_{\uparrow\downarrow}^{x}(t)$ is dominated by the decay with the characteristic time $T_2$. Both populations relax towards the value 1/2, with the characteristic time $T_1$.

If we write now these elements in the z-basis, using the change of basis:

$$ |\uparrow x\rangle = \frac{1}{\sqrt{2}}(|\uparrow z\rangle + |\downarrow z\rangle) \qquad (A.4) $$

$$ |\downarrow x\rangle = \frac{1}{\sqrt{2}}(|\uparrow z\rangle - |\downarrow z\rangle), \qquad (A.5) $$

we get

$$ \rho_{\uparrow\uparrow}^{z}(t) = \frac{1}{2} + \mathcal{R}e\{\rho_{\uparrow\downarrow}^{x}(0)\}e^{-t/T_2} \qquad (A.6) $$
$$ \rho_{\downarrow\downarrow}^{z}(t) = \frac{1}{2} - \mathrm{Re}\{\rho_{\uparrow\downarrow}^{x}(0)\}e^{-t/T_2} \qquad (A.7) $$

$$ \rho_{\uparrow\downarrow}^{z}(t) = ae^{-t/T_1} + i \mathrm{Im}\{\rho_{\uparrow\downarrow}^{x}(0)^{*}\}e^{-t/T_2}. \qquad (A.8) $$

In the limit $\vec{B}_x \to 0$, the bases $(|\uparrow x\rangle, |\downarrow x\rangle)$ and $(|\uparrow z\rangle, |\downarrow z\rangle)$ play the same role, since there is no designated quantization axis. Then, the comparison of Eqs. (A.1) and (A.2) with Eqs. (A.6) and (A.7) shows that $T_2 = T_1$ for a vanishing magnetic field.
# Appendix B

## 1s and 2s hydrogenic states centred in an infinite quantum well

### 1. Method

A positive electric charge Ze (e: elementary charge) is at the origin of Cartesian coordinates xyz. The Hamiltonian $H$ of an electron (charge – e) in the Coulomb potential of Ze is

$$H = -\frac{\Delta}{2} - \frac{Z}{r}, \qquad (B.1)$$

where $\Delta$ is the Laplacian operator and $r$ is the distance of the electron to the origin. The distances are expressed in units of effective (bulk) Bohr radius $a_B^{3D}$, and energies in units of effective hartree $E_h^{3D}$ (for definitions of $a_B^{3D}$ and $E_h^{3D}$, see eq. (3.2) of chapter 3).

The 1s and 2s states in bulk ($r = \sqrt{x^2 + y^2 + z^2}$) or in 2D ($\rho = \sqrt{x^2 + y^2}$) are known [1]. They are presented in the following Table B.1.
<table><caption>**Table B.1** Wavefunctions and energies for the 1s and 2s hydrogenic states in 3D and 2D.</caption><thead><tr><th></th><th>Wavefunction φ</th><th>Energy E</th></tr></thead><tbody><tr><td>1s</td><td><sup>1</sup>&frasl;<sub>&radic;π</sub> Z<sup>3/2</sup> exp(&minus;Zr)</td><td>&minus;Z<sup>2</sup>/2</td></tr><tr><td><b>3D</b></td><td></td><td></td></tr><tr><td>2s</td><td><sup>1</sup>&frasl;<sub>2&radic;2π</sub> Z<sup>3/2</sup> exp(&minus;Zr/2) [1 &minus; Zr/2]</td><td>&minus;Z<sup>2</sup>/8</td></tr><tr><td>1s</td><td><sup>4</sup>&frasl;<sub>&radic;2π</sub> Z exp(&minus;2Zρ)</td><td>&minus;2Z<sup>2</sup></td></tr><tr><td><b>2D</b></td><td></td><td></td></tr><tr><td>2s</td><td><sup>4</sup>&frasl;<sub>3&radic;6π</sub> Z exp(&minus;2Zρ/3) [1 &minus; 4Zρ/3]</td><td>&minus;2Z<sup>2</sup>/9</td></tr></tbody></table>

We seek the 1s and 2s states when the charge Ze is placed in the middle of a QW of thickness L, simply modeled by two infinite barriers located at z = ±L/2; the confinement potential is supposed to be zero inside the QW (|z| < L/2). The 1s and 2s wavefunctions centred within this QW are taken in the following forms:

$$ \phi_1(\rho, z) = A_1 \exp(-\alpha_1 r) \cos\left(\pi \frac{z}{L}\right), \qquad (\text{B. 2a}) $$

$$ \phi_2(\rho, z) = A_2 \exp(-\alpha_2 r) [1 - \alpha_3 r] \cos\left(\pi \frac{z}{L}\right), \qquad (\text{B. 2b}) $$

where $r = \sqrt{\rho^2 + z^2}$. The postulated $\cos(\pi z/L)$ envelope function is a single sinusoid arch, and ensures that the wavefunctions $\phi_j(\rho, z)$, $j = 1$ or 2, vanish at the boundaries of the QW. The dependences on $r$ of both $\phi_j$ respect the ones which are met in the 2D ($L \to 0$) and 3D ($L \to \infty$) limits. In the following of this Appendix, the prefactors $A_j$ are fixed by normalization; $\phi_1(\rho, z)$ and its energy $E_1$ are first determined by means of the variational method; afterwards, using the 2s-1s orthogonality and the variational method, $\phi_2(\rho, z)$ and its energy $E_2$ are obtained.

## 2. 1s state

The normalization condition of the $\phi_1(\rho, z)$ wavefunction: $\int_{|z|<L/2} 2\pi\rho \, d\rho \, dz \phi_1(\rho, z)^2 = 1$, gives:

$$ A_1^2 = \frac{\alpha_1^3}{\pi N(\alpha_1 L)}, \qquad (\text{B. 3a}) $$

$$ \text{with } N(X) = 1 - \frac{\pi^2}{X^2 + \pi^2} + \frac{\pi^4/2}{[X^2 + \pi^2]^2} - \frac{\pi^2}{4} \left( \frac{X+4}{X^2 + \pi^2} - \frac{2\pi^2}{[X^2 + \pi^2]^2} \right) e^{-X}. \qquad (\text{B. 3b}) $$
Here, and in all the following calculations, we exploited the even parity of the integrand with respect to z. The integration over $\rho$ is first performed with the change of variable $r = \sqrt{\rho^2 + z^2}$, $z = \text{cte}$ ($\rho d\rho = \text{rdr}$), $r$ varying from $r = z$ ($> 0$) to $r = +\infty$; the integration over z is then performed, from $z = 0$ to $z = L/2$.

The forms of the normalization factors of the 3D and 2D 1s-wavefunctions can be retrieved from expressions (B.3): $A_1 \to \alpha_1^{3/2}/\sqrt{\pi}$ for $L \to \infty$, and $A_1\sqrt{L/2} \to \alpha_1\sqrt{2/\pi}$ for $L \to 0$.

The parameter $\alpha_1$ is obtained by minimization of the mean energy $\tilde{E}_1 = \langle \phi_1 | H | \phi_1 \rangle$, which is the sum of a kinetic term (possessing, after calculation, a remarkably simple form):

$$ \langle \phi_1 | -\frac{\Delta}{2} | \phi_1 \rangle = \frac{\alpha_1^2}{2} + \frac{\pi^2}{2L^2}, \qquad (B.4) $$

and a potential term due to the Coulomb interaction:

$$ \langle \phi_1 | -\frac{Z}{r} | \phi_1 \rangle = -Z \alpha_1 \frac{C(\alpha_1 L)}{N(\alpha_1 L)}, \qquad (B.5a) $$

$$ \text{with } C(X) = 1 - \frac{1}{2} \frac{\pi^2}{X^2 + \pi^2} (1 + e^{-X}). \qquad (B.5b) $$

Defining $g(X) = XC(X)/N(X)$, which is a smooth function close to the identity [$g(X) \approx X$], the minimum of $\tilde{E}_1$ is found to be reached when $\alpha_1$ is solution of the equation

$$ \frac{\alpha_1 L}{g'(\alpha_1 L)} = Z L. \qquad (B.6) $$

After a numerical determination of the derivative $g'(X)$ of the function $g(X)$, the left-hand side of the above equation can be calculated for a value of the parameter $\alpha_1 L$. One then obtains the thickness $L$ for which the starting parameter $\alpha_1 L$ corresponds to the minimum of $\tilde{E}_1$; finally, the associated $\alpha_1$ value is calculated through $\alpha_1 = (\alpha_1 L)/L$, and the energy $E_1$ of $\phi_1$ by substitution of this $\alpha_1$ in the expression for $\tilde{E}_1$. The procedure can be repeated for any starting parameter $\alpha_1 L$, and then allows to get $\alpha_1$ and $E_1$, and also $A_1$, as a function of the thickness $L$ of the QW, as shown in Fig. B1 for $Z = 1$ and $Z = 2$. One can remark that parameter $\alpha_1(Z = 1)$ for thickness $L$ coincides with $\alpha_1(Z)/Z$ for $ZL$. The prefactor $A_1$ goes to $\sqrt{Z^3/\pi}$ (3D limit) for $L \gg 1$, and behaves as $4Z/\sqrt{\pi L}$ for very small $L$ [see Figs B1(a) and (b)]. The parameter $\alpha_1$ goes to $Z$ (3D limit) for $L \gg 1$, and to $2Z$ (2D limit) for vanishing $L$ [see Figs B1(c) and (d)]. For decreasing values of $L$, the extension of the $\phi_1$ wavefunction first inflates slightly in the $xy$ plane ($\alpha_1$ becoming a little smaller than $Z$), and secondly globally shrinks for smaller values of $L$. The energy $E_1$, when considered by reference to the confinement energy $E_{conf.} = \pi^2/2L^2$, increases monotonically with $L$ from $-2Z^2$ (2D value) to $-Z^2/2$ (3D value) [see Figs B1(e) and (f)].
<figure><img src="image_72.png" /><figcaption>**Figure B1.** Parameters $A_1$ and $\alpha_1$ of the $\phi_1$ wavefunction, and its energy $E_1$, as a function of the thickness $L$ of the QW (thick continuous curves); the red thin dashed lines correspond to the 3D case. (a), (b): $A_1$ vs $L$ for $Z = 1, Z = 2$. (c), (d): $\alpha_1$ vs $L$ for $Z = 1, Z = 2$. (e), (f): $E_1 - E_{conf.}$ vs $L$ for $Z = 1, Z = 2$, where $E_{conf.} = \pi^2/2L^2$.</figcaption></figure>
### 3. 2s state

The orthogonality condition between the $\phi_1(\rho, z)$ and $\phi_2(\rho, z)$ wavefunctions:

$\int_{|z|<L/2} 2\pi \rho d\rho dz \Phi_1 \Phi_2 = 0$, implies the following relationship between $\alpha_1, \alpha_2$ and $\alpha_3$:

$$ \alpha_3 = \beta \frac{N(\beta L)}{N(\beta L) + M(\beta L)}, \qquad (B.7a) $$

with $\beta = \frac{\alpha_1 + \alpha_2}{2}$ and

$$ M(X) = \frac{1}{2} - \frac{3\pi^2/2}{X^2+\pi^2} + \frac{9\pi^4/4}{[X^2+\pi^2]^2} - \frac{\pi^6}{[X^2+\pi^2]^3} - \frac{\pi^2}{4} \left( \frac{1}{2} + \frac{3X+6-\pi^2/2}{X^2+\pi^2} - \pi^2 \frac{2X+9}{[X^2+\pi^2]^2} + \frac{4\pi^4}{[X^2+\pi^2]^3} \right) e^{-X}. \qquad (B.7b) $$

The relationships between $\alpha_1, \alpha_2$ and $\alpha_3$ in 3D and in 2D can be retrieved as limiting cases of the above expressions: $\alpha_3 \to (\alpha_1 + \alpha_2)/3$ when $L \to \infty$ and $\alpha_3 \to (\alpha_1 + \alpha_2)/2$ when $L \to 0$.

The normalization condition of the $\phi_2(\rho, z)$ wavefunction gives

$$ A_2^2 = \frac{\alpha_2^3}{\pi} \left\{ N(\alpha_2 L) \left[ 1 - 2 \frac{\alpha_3}{\alpha_2} + \frac{3 \alpha_3^2}{2 \alpha_2^2} \right] + M(\alpha_2 L) \left[ -2 \frac{\alpha_3}{\alpha_2} + \frac{3 \alpha_3^2}{2 \alpha_2^2} \right] + \frac{1}{8} Q(\alpha_2 L) \frac{\alpha_3^2}{\alpha_2^2} \right\}^{-1}, \qquad (B.8a) $$

$$ \text{with } Q(X) = 3 + 3X^4 \frac{X^4 - 6\pi^2 X^2 + \pi^4}{[X^2+\pi^2]^4} - \frac{\pi^2}{2} \left( \frac{X^3}{X^2+\pi^2} + 3X^2 \frac{3X^2+\pi^2}{[X^2+\pi^2]^2} + 6X \frac{6X^4+3\pi^2 X^2 + \pi^4}{[X^2+\pi^2]^3} + 6 \frac{10X^6+5\pi^2 X^4 + 4\pi^4 X^2 + \pi^6}{[X^2+\pi^2]^4} \right) e^{-X}. \qquad (B.8b) $$

The energy $E_2$ is obtained as the minimum of the mean energy $\tilde{E}_2 = \langle \phi_2 | H | \phi_2 \rangle$, which only depends on $\alpha_2$ ($\alpha_1$ is known, $\alpha_3$ is a function of $\alpha_1$ and $\alpha_2$). $\tilde{E}_2$ is the sum of a kinetic term:

$$ \langle \phi_2 | -\frac{\Delta}{2} | \phi_2 \rangle = \frac{1}{4} \left( \frac{\pi A_2^2}{\alpha_2^3} \right) \left\{ \frac{\pi^2}{L^2} \left[ Q_1(\alpha_2 L) - \frac{\alpha_3}{\alpha_2} Q_2(\alpha_2 L) + \frac{\alpha_3^2}{4\alpha_2^2} Q_3(\alpha_2 L) \right] + 2\alpha_2^2 C(\alpha_2 L) \right. \\ \left. + \alpha_3^2 \left[ N(\alpha_2 L) + 2M(\alpha_2 L) \right] + 2\alpha_2\alpha_3 \left[ C(\alpha_2 L) - 2N(\alpha_2 L) \right] \right\}, \qquad (B.9a) $$

$$ \text{with} \qquad Q_1(X) = 1 - \left(1 + \frac{X}{2}\right) e^{-X} \qquad (B.9b) $$

$$ Q_2(X) = 3 - \left(3 + 2X + \frac{1}{2}X^2\right) e^{-X} \qquad (B.9c) $$

$$ Q_3(X) = 12 - \left(12 + 9X + 3X^2 + \frac{1}{2}X^3\right) e^{-X}, \qquad (B.9d) $$

and a potential term due to the Coulomb interaction:
$$ \langle \phi_2 | -\frac{Z}{r} | \phi_2 \rangle = -Z \frac{\pi A_2^2}{\alpha_2^2} \left\{ C(\alpha_2 L) + N(\alpha_2 L) \left[ -2 \frac{\alpha_3}{\alpha_2} + \frac{\alpha_3^2}{\alpha_2^2} \right] + M(\alpha_2 L) \frac{\alpha_3^2}{\alpha_2^2} \right\}. \quad (B.10) $$

Finding the parameter $\alpha_2$ [and consequently $\alpha_3$, with Eq. (B.7a)] which minimizes $\tilde{E}_2$ (at given $L$) requires a long numerical procedure, which is in contrast with the much easier work needed for the minimization of $\tilde{E}_1$. Figure B2 shows $\tilde{E}_2$ vs $\alpha_2$ for $L = 1.5$ and $Z = 2$; the minimum parameters indicated in the figure give one point in Figs. B3 (b), (d) and (f). Figure B3 shows $A_2$, $\alpha_2$, $\alpha_3$ and $E_2$ concerning the $\phi_2$ wavefunction, for different values of the thickness $L$, with $Z = 1$ and $Z = 2$: $A_2$ and $\alpha_2$ behave with $L$ just as $A_1$ and $\alpha_1$ do; $\alpha_3$ monotonically decreases with $L$; however, $E_2$, considered by reference to the confinement energy $E_{conf.} = \pi^2/2L^2$, goes with $L$ from $-2Z^2/9$ (2D value) to $-Z^2/8$ (3D value) but non-monotonically [see Figs B3(e) and (f)].

<figure><img src="image_73.png" /><figcaption>**Figure B2.** Mean energy $\tilde{E}_2$ (eqs. B.9 and B.10) as a function of the parameter $\alpha_2$, for $L = 1.5$ and $Z = 2$. The $\alpha_2$ step on the horizontal axis is $10^{-5}$. The energy $E_2$ and the parameter $\alpha_2$ concerning the wave-function $\phi_2$ are found as the minimum parameters of the curve.</figcaption></figure>
<figure><img src="image_74.png" /><figcaption>**Figure B3.** Parameters $A_2$, $\alpha_2$ and $\alpha_3$ of the $\phi_2$ wavefunction, and its energy $E_2$, as a function of the thickness $L$ of the QW (circular dots; continuous curves: guides for the eyes); the red thin dashed lines correspond to the 3D case. (a), (b): $A_2$ vs $L$ for $Z = 1, Z = 2$. (c), (d): $\alpha_2$ and $\alpha_3$ vs $L$ for $Z = 1, Z = 2$. (e), (f): $E_2 - E_{conf.}$ vs $L$ for $Z = 1, Z = 2$ ($E_{conf.} = \pi^2/2L^2$).</figcaption></figure>
# Appendix C

## Exchange energy of a “helium atom” centred in an infinite quantum well

### 1. Exact results in 3D

The non-relativistic energies of the (singlet) $(1s)^2$ ground state and of the first-excited triplet $1s2s$ state of the helium atom in 3D are known with an excellent precision: in Ref. [2] they are calculated with twenty significant digits. We only write here the first of them:

$$E_S^{\text{exact}} = -2.9037, \quad E_A^{\text{exact}} = -2.1752, \qquad (\text{C. 1a})$$

in units of effective hartree (see eq. (3.2), chapter 3). The subscript S, or A, recalls that the orbital part of the state is symmetric, or antisymmetric, under permutation of both electrons of the atom. The exchange energy $2J_0 = E_A - E_S$ is then exactly, in 3D:

$$2J_0^{\text{exact}} = 0.7285. \qquad (\text{C. 1b})$$

### 2. Exact results in 2D

In 2D, the non-relativistic energies of the $(1s)^2$ ground state and the first-excited triplet $1s2s$ state of the helium atom are also known with a very good accuracy: in Ref. [3] they are calculated with thirteen significant digits. We write here the first of them:

$$E_S^{\text{exact}} = -11.900, \quad E_A^{\text{exact}} = -8.296, \qquad (\text{C. 2a})$$

in units of effective hartree. The exchange energy is then exactly, in 2D:
$$2J_0^{\text{exact}} = 3.604. \qquad (\text{C. 2b})$$

### 3. Calculation of $2J_0$ in an infinite quantum well

The Hamiltonian describing both electrons in a helium atom writes

$$\hat{H} = -\frac{\Delta_1}{2} - \frac{\Delta_2}{2} - \frac{Z}{r_1} - \frac{Z}{r_2} + \frac{1}{r_{12}}, \qquad (\text{C. 3})$$

where $\Delta_j$ is the Laplacian operator acting on electron j ($j = 1$ or 2) and $r_j$ is the distance of electron j to the origin of the coordinates, where the Z = 2 nucleus is located; $r_{12} = |\vec{r}_2 - \vec{r}_1|$ is the distance between the electrons.

#### a. Ground energy

The ground energy $E_S$ of the “helium atom” centred in a QW with infinite barriers is determined here by the variational method. The ground-state wavefunction $\Psi_S(\vec{r}_1, \vec{r}_2)$ of the pair of electrons 1 and 2 is taken as the product of two one-electron 1s-wavefunctions: $\Psi_S(\vec{r}_1, \vec{r}_2) = \phi_1(\vec{r}_1)\phi_1(\vec{r}_2)$; the mean energy $\tilde{E}_S = \langle \Psi_S | \hat{H} | \Psi_S \rangle$ is studied as a function of the parameter $\alpha_1$, which now can vary (see section 1 of Appendix B for the definitions of $\phi_1$ and $\alpha_1$), and its minimum is taken for the value of $E_S$. $\tilde{E}_S$ is the sum of a kinetic term:

$$\langle \Psi_S | -\frac{\Delta_1}{2} - \frac{\Delta_2}{2} | \Psi_S \rangle = \alpha_1^2 + \frac{\pi^2}{L^2}, \qquad (\text{C. 4a})$$

a potential term due to the Coulomb interaction of the electrons with the Z = 2 nucleus:

$$\langle \Psi_S | -\frac{2}{r_1} - \frac{2}{r_2} | \Psi_S \rangle = -4 \alpha_1 \frac{C(\alpha_1 L)}{N(\alpha_1 L)} \qquad (\text{C. 4b})$$

$[N(X)$ and $C(X)$: see Appendix B], and a Coulomb electron-electron term:

$$\langle \Psi_S | \frac{1}{r_{12}} | \Psi_S \rangle = 32\pi \int_0^\infty \int_0^\infty \frac{\rho_q d\rho_q dz_q}{\rho_q^2 + z_q^2} \left[ \int_0^L dz \cos(zqz) \int_0^\infty d\rho \rho J_0(\rho_q\rho) \phi_1(\rho, z)^2 \right]^2, \qquad (\text{C. 4c})$$

expressed using the fact that $1/r$ is the inverse Fourier transform of $4\pi/q^2$ [$J_0(x)$ is the Bessel function of first kind and of zeroth order]. This integral is calculated numerically. Figure C1 shows an example of calculation of $E_S$, for a QW of thickness L = 1.5. The mean energy $\tilde{E}_S = \langle \Psi_S | \hat{H} | \Psi_S \rangle$ is plotted as a function of the parameter $\alpha_1$. For this particular value of L, the ground energy is found to be $E_S = \min \tilde{E}_S \approx -0.198$, corresponding to the value $\alpha_1^{\min} \approx 1.701$. As expected, $\alpha_1^{\min}$ is slightly smaller than the $\alpha_1 \approx 2.0$ parameter of
the $\phi_1$ wavefunction for a single electron (see Fig. B1(d)), as a result of the screening of the $+2e$ nucleus charge by the other electron.

<figure><img src="image_75.png" /><figcaption>**Figure C1:** $\tilde{E}_s$ vs $\alpha_1$ as a function of the parameter $\alpha_1$ for $L = 1.5$ (dots: calculated points). The energy $E_s$ and the parameter $\alpha_1$ are found as the minimum of the curve. This procedure has been done for each $E_s$ value plotted in Fig. C2.</figcaption></figure>
b. First-excited state

The first-excited energy $E_A$ of the “helium atom” centred in a QW with infinite barriers is determined perturbatively here: the electron-electron interaction $1/r_{12}$ is treated as a small correction to the state $\Psi_A(\vec{r}_1, \vec{r}_2) = [\phi_1(\vec{r}_1)\phi_2(\vec{r}_2) - \phi_2(\vec{r}_1)\phi_1(\vec{r}_2)]/\sqrt{2}$ of energy $E_1 + E_2$ ($\phi_1, \phi_2, E_1$ and $E_2$ are given in Appendix B). The energy $E_A$ is of the form $E_A = E_1 + E_2 + \Delta E$, the corrective term being $\Delta E = \langle \Psi_A | 1/r_{12} | \Psi_A \rangle$; $\Delta E$ is the sum of a direct term:

$$ \int \int d^3 r_1 d^3 r_2 \frac{\phi_1(\vec{r}_1)^2 \phi_2(\vec{r}_2)^2}{r_{12}} \\ = 32\pi \int_0^\infty \int_0^\infty \frac{\rho_q d\rho_q dz_q}{\rho_q^2 + z_q^2} \left[ \int_0^{\frac{L}{2}} dz \cos(z_q z) \int_0^\infty d\rho \rho J_0(\rho_q \rho) \phi_1(\vec{r})^2 \right] \times \dots \\ \dots \left[ \int_0^{\frac{L}{2}} dz \cos(z_q z) \int_0^\infty d\rho \rho J_0(\rho_q \rho) \phi_2(\vec{r})^2 \right], \qquad (C. 5a) $$

and an exchange term:

$$ \int \int d^3 r_1 d^3 r_2 \frac{\phi_1(\vec{r}_1)\phi_2(\vec{r}_2)\phi_2(\vec{r}_1)\phi_1(\vec{r}_2)}{r_{12}} \\ = 32\pi \int_0^\infty \int_0^\infty \frac{\rho_q d\rho_q dz_q}{\rho_q^2 + z_q^2} \left[ \int_0^{\frac{L}{2}} dz \cos(z_q z) \int_0^\infty d\rho \rho J_0(\rho_q \rho) \phi_1(\vec{r}) \phi_2(\vec{r}) \right]^2. \qquad (C. 5b) $$

In 3D and in 2D, calculating $E_S$ by the variational method and $E_A$ perturbatively gives satisfactory values (for our purpose) of the exchange energy $2J_0 = E_A - E_S$: we so obtain $2J_0 = -2.124 + 2.848 = 0.724$ in 3D, and $-7.948 + 11.635 = 3.687$ in 2D, which are 0.6% smaller, and 2% larger, than the exact values, respectively. We are then confident that our calculated exchange energies $2J_0$ of a “helium atom” centred in infinite QWs of different thicknesses, possess an uncertainty of the 1-% order.

Figure C2 shows our calculated values of the ground energy $E_S$ and first-excited energy $E_A$, as a function of the thickness $L$ of the QW; the exact values in 2D ($L \to 0$) and in 3D ($L \to \infty$) are also indicated. $E_S$ and $E_A$ monotonically increase with the thickness $L$, and their difference, which is $2J_0$, monotonically decreases (see Fig. 3.3 of chapter 3).
<figure><img src="image_76.png" /><figcaption>**Figure C2.** Ground energy $E_S$ (full disks) and first-excited energy $E_A$ (full squares) of a "helium atom", as a function of the thickness $L$ of the infinite QW; the continuous curves are guides for the eyes. The empty circle and empty square, at $L = 0$, are the exact values in 2D. The exact values in 3D are indicated by two horizontal red segments, on the right of the graph.</figcaption></figure>
# Appendix D

## New interpolated formulas for the 3D and 2D exchange energies

Section 3.4 of chapter 3 described the procedure of Ref. [4] to build an interpolated expression of the exchange energy $2J(R)$ in 3D and 2D. In Eq. (3.27), in which $A$ is the only remaining free parameter, it is possible, alternatively, to impose that, for $R \gg 1$, the interpolated $2J(R)$ be in coincidence with the standard asymptotic form $2J(R\gg 1) = CR^\beta e^{-\omega R}$. We then obtain the following expression for $A$:

$$A = (C/2J_0)^{1/\beta}/e. \qquad (D.1)$$

So we can propose new interpolated formulas for the exchange energy $2J(R)$ in 3D and 2D, using values of $2J_0$ which are, nowadays, known with very good accuracy (see Appendix C, eqs. (C.1 b), (C.2 b)):

$$2J_{3D}(R) = 0.729 (1 + 0.259R^2)^{\frac{5}{4}} \times \dots \\ \exp[-0.003R - 1.271R \arctan(0.509R)]; \qquad (D.2a)$$

$$2J_{2D}(R) = 3.604 (1 + 1.549R^2)^{\frac{7}{8}} \times \dots \\ \exp[-0.579R - 2.178R \arctan(1.245R)]. \qquad (D.2b)$$

These expressions give $2J(R)$ values close to the corresponding ones of Ref. [4] for $R$ of the order of unity, and are more satisfactory, by construction, for large values of $R$.

Figure D1 shows the exchange energies $2J_{3D}(R)$ given by Ref. [4] (dashed line) and by Eq. (D.2a) (continuous line); for $R$ of the order of unity, the curves only differ by several percent, this difference approaching 10% for $R \ge 4$ (not shown). Figure D2 illustrates
the 2D case: $2J_{2D}(R)$ given by Ref. [4] is larger than the one by Eq. (D.2b), the difference being 10% for $R = 1$ and approaching 14% when $R \ge 2$ (not shown). The Eqs. (D.2) are in agreement with the asymptotic forms (3.22) and (3.23) of chapter 3.

<figure><img src="image_77.png" /><figcaption>**Figure D1.** Dashed line: exchange energy $2J_{3D}(R)$ according to Ref. [4]; continuous line: $2J_{3D}(R)$ according to Eq. (D.2a). Empty disks: numerical values cited in Ref. [4]; black square: exact value of the exchange energy $2J_{3D}(0)$ of the helium atom.</figcaption></figure>
<figure><img src="image_78.png" /><figcaption>**Figure D2.** Dashed line: exchange energy $2J_{2D}(R)$ according to Ref. [4]; continuous line: $2J_{2D}(R)$ according to Eq. (D.2b). Empty disks: numerical values calculated in Ref. [4]; black square: exact value of the exchange energy $2J_{2D}(0)$ of the 2D helium atom.</figcaption></figure>
References for appendices A-D

[1] X. L. Yang et al., Phys. Rev. A **43**, 1186 (1991).

[2] D. T. Aznabayev, A. K. Bekbaev, I. S. Ishmukhamedov, and V. I. Korobov, Phys. Part. Nucl. Lett. **12**, 689 (2015).

[3] L. Hilico, B. Grémaud, T. Jonckheere, N. Billy, and D. Delande, Phys. Rev. A **66**, 022101 (2002).

[4] I. V. Ponomarev, V. V. Flambaum, and A. L. Efros, Phys. Rev. B **60**, 5485 (1999).