Ultrafast spin dynamics in magnetic wide‐bandgap semiconductors

Magnetic wide‐bandgap semiconductors based on ZnO and GaN are promising for spintronics applications and interesting for studying the interaction of charge carriers with magnetic ions. We use time‐resolved Faraday rotation spectroscopy to investigate the ultrafast spin dynamics in Zn1−xMnxO and Ga1−xMnxN. The mean field electron–magnetic ion exchange constant is determined by measuring the transient effective g‐factor (g*) for these materials. The relevant scattering processes are revealed by analyzing the ensemble spin dephasing time T2* .

1 Introduction Spintronic devices based on diluted magnetic semiconductors (DMS) have received a lot of attention recently [1]. Exploiting the exchange interaction between magnetic dopants and charge carriers is a promising approach to control carrier spins due to the giant Zeeman splitting effect [2]. The wide-bandgap semiconductors ZnO and GaN doped with magnetic ions are interesting candidates for fabricating such devices [3], because room temperature ferromagnetism has been reported in these materials [4,5]. However, the origin of this effect is still under debate [6]. Time-resolved optical experiments employing ultrafast laser pulses have shown to be a powerful tool for revealing physical processes in semiconductors. In particular, time-resolved Faraday rotation (TRFR) spectroscopy has recently been used to unveil the ultrafast spin dynamics in bulk semiconductors and semiconductor nanostructures [7][8][9][10].
2 Ultrafast spin dynamics in wide-bandgap semiconductors 2.1 Ultrafast spin dynamics in GaN, ZnO, and Zn 1Àx Co x O Soon after the surprising observation of ultralong electron spin dephasing times in bulk n-doped GaAs exceeding 100 ns [11], the ultrafast spin dynamics in the wide-bandgap semiconductor GaN was studied. Dephasing times reaching 20 ns at cryogenic temperatures have been measured [12]. Electron spin coherence at room temperature has been detected [12]. In 2005, time-resolved experiments on the ultrafast spin dynamics in ZnO yielded spin dephasing times of 190 ps at room temperature [13].
Only recently, the ultrafast spin dynamics in a magnetically doped wide-bandgap semiconductor has been investigated for the first time. Using TRFR spectroscopy Zn 1Àx Co x O sol-gel thin films have been studied [14]. The mean-field electron-Co 2þ exchange energy in Zn 1Àx Co x O N 0 a ¼ þ0.25 AE 0.02 eV has been determined. The ensemble spin dephasing time T Ã 2 increased with rising temperature, allowing spin precession to be observed even at room temperature. This anomalous temperature dependence of T Ã 2 has also been found in magnetically undoped ZnO with a record T Ã 2 in the range of nanoseconds. This effect has been attributed to hole trapping at grain surfaces in the sol-gel thin films. It showed the importance of charge-separated states to control the electron spin dynamics [14].
2.2 Ultrafast spin dynamics in Zn 1Àx Mn x O In the following, we present the ultrafast electron spin dynamics in chemically prepared Zn 1Àx Mn x O sol-gel films, measured via TRFR spectroscopy in the ultraviolet. ZnO films are prepared by modification of a sol-gel synthesis method reported previously [14]. To fabricate Zn 1Àx Mn x O films, a fraction of the Zn(OAc) 2 was replaced by a stoichiometric amount of Mn(OAc) 2 . The Zn 2þ ion in ZnO has a completely filled 3d shell and, correspondingly, carries no magnetic moment. If Zn is replaced by Mn, the valence charge 2þ is not changed, leading to Mn 2þ ions on Zn 2þ lattice sites [15]. The obtained sol was spin-coated onto 500 mm thick c-plane oriented sapphire substrates layer by layer and annealed for 1.5 h at 280 8C to form the final sol-gel film. Finally, it was baked at 550 8C for 3 h. We use TRFR spectroscopy to directly probe the electron spin dynamics in Zn 1Àx Mn x O. In this ultrafast pump-probe technique, ultraviolet laser pulses pump and probe spin-polarized carriers [14]. To ensure maximum excitation efficiency, we tune the laser wavelength in resonance with the D 0 X exciton transition at 368.4 nm.
Recorded TRFR transients exhibit a delta-shaped artifact at zero pump-probe delay and a slowly varying background (data not shown), which are both subtracted before data analysis. Figure 1 shows TRFR signals of an undoped ZnO sol-gel film recorded at a temperature of T ¼ 10 K and an external transverse magnetic field of B ¼ 1.4 T, along with traces collected from Zn 1Àx Mn x O films at various Mn 2þ concentrations. Fitting the TRFR signal of undoped ZnO yields an effective g-factor of g Ã ¼ 1.98, which agrees well with g Ã of electrons for epitaxial and sol-gel ZnO thin films [13,14]. In contrast, effective g-factors of hole spins in ZnO have been determined to be 0.5 and 1.2 in epitaxial layers and commercial ZnO bulk substrates [15,16]. The observed Zn 1Àx Mn x O signals are biexponentially damped oscillations: where u F is the Faraday rotation angle, A 1 and A 2 are the amplitudes of the two precession components, v L1 are v L2 are the Larmor precession frequencies, t is the time delay between pump and probe pulses, T Ã 2 1 and T Ã 2 2 are the ensemble spin dephasing times, and w 1 and w 2 are the phase shifts of both components.
From v L , we can determine the effective g-factor (g Ã ) according to 2.3 Electron-Mn 2þ magnetic exchange coupling We first concentrate on the fast decaying component of the oscillation, which only lasts for the first tens of ps. This signal is attributed to spins of electrons that interact with magnetic dopant ions. Measurements of samples of different Mn 2þ concentrations allow us to determine the sign and magnitude of the mean field exchange energy N 0 a between electrons and Mn 2þ ions. A rising concentration of Mn 2þ ions (up to x ¼ 0.0014) increases the precession frequency by over 30%, which scales linearly with effective g-factor. Within the mean-field and virtual-crystal approximations, g Ã for a conduction-band electron in a DMS may be described using The mean field Mn 2þ -electron exchange parameter N 0 a can be extracted from the slope of the g Ã (x) dependence, as presented in Fig. 2. The first term in Eq. (3), g int , is the intrinsic electron g-value in the absence of magnetic dopants. From our undoped ZnO sol-gel film, g int ¼ þ1.98 was determined. The second term describes the contribution to g Ã induced by electron-Mn 2þ magnetic exchange coupling. hS x i is the expectation value of the Mn 2þ spin perpendicular to the c-axis of ZnO, i.e., along the external magnetic field (B x ). By convention, hS x i is defined as a negative number. The temperature dependence of hS x i is obtained from the Brillouin function.
A fit of the available data at a temperature of T ¼ 10 K and an external magnetic field of B ¼ 1.4 T, yields a value of N 0 a ¼ þ0.089 AE 0.019 eV. The sign of N 0 a is determined unambiguously from the observation that g Ã increases rather than decreases with Mn 2þ doping (see Eq. 3).
The obtained value for N 0 a is surprisingly low when compared to theoretical predictions for this material [17]. In principle, it is conceivable that in the Zn 1Àx Mn x O sol-gel film Mn 2þ ions are not entirely incorporated into the ZnO host lattice. As a consequence, the Mn 2þ concentration in Zn 1Àx Mn x O might be overestimated and our determined value for N 0 a might represent a lower bound. We therefore performed additional measurements on Zn 1Àx Mn x O nanocrystals, where the Mn 2þ concentration has been determined by inductively coupled plasma-atomic emission spectroscopy. The nanocrystals with diameters 4-5 nm were incorporated in a dodecylamine (DDA) matrix and spincoated on a sapphire substrate. As evident in Fig. 2, the TRFR measurements yielded N 0 a ¼ 0.082 AE 0.008 eV, which is in good agreement with the value obtained with the Zn 1Àx Mn x O sol-gel films.
N 0 a may also be determined with a temperature dependent measurement (Fig. 3). For the Zn 1Àx Mn x O solgel film with x ¼ 0.0014 N 0 a ¼ 0.090 AE 0.016 eV was determined, which is in good agreement with the values obtained by changing the magnetic ion concentration.
These relatively low values for N 0 a in Zn 1Àx Mn x O might be explained by localization of holes at the Mn 2þ ions [18,19,20], i.e., we measure an apparent value for N 0 a with TRFR.
2.4 Ultrafast spin dephasing in Zn 1Àx Mn x O As mentioned before, the first, fast decaying component with a time constant of a few tens of ps and effective g-value above 1.98 is attributed to electrons interacting with magnetic dopant ions. The second, slow component varies between 100 and 200 ps and exhibits an effective g-value around 2. It either originates from electrons not interacting with magnetic ions or from the Mn 2þ ions themselves. To resolve this question we have performed TRFR measurements as a function of temperature (Fig. 4).
The slow spin dephasing time stays constant in the temperature range T ¼ 10-110 K, which strongly suggests that this part of the TRFR signal is due to precessing Mn 2þ ions and not electrons. For electrons, a change of T Ã 2 with temperature would have been expected. Furthermore, the absolute value for the measured spin dephasing time in the range of 100-200 ps is in good agreement with previous measurements in (Zn, Cd, Mn) Se [21,22] and GaMnAs quantum wells [23].
2.5 Ultrafast spin dynamics in wurtzite GaN and Ga 1Àx Mn x N We use TRFR spectroscopy to directly probe the transient electron spin dynamics in wurtzite GaN and Ga 1Àx Mn x N. Wurtzite n-type GaN of 1 mm thickness is grown by plasma assisted molecular beam epitaxy (PAMBE) on a 500 mm thick c-cut sapphire substrate [24]. The electron density, as well as the density of Si and O impurities, are determined by elastic recoil detection (ERD) and room temperature Hall measurements and amount to n e ¼ n D % 1 Â 10 17 cm À3 . At this carrier concentration the longest electronic spin dephasing times have been reported in GaN [12]. Two Ga 1Àx Mn x N layers of 1.2-1.3 mm thickness are also prepared by PAMBE. Their manganese concentration is determined by ERD to be x Mn ¼ 0.0134 (sample 1) and 0.005 (sample 2) [25]. 98% of all dopant ions are in the Mn 3þ or "Mn 2þ þ hole" state. This fraction was determined by comparing the Mn ion concentration measured by ERD and electron spin resonance, which detects only Mn 2þ ions [25]. The high impedance prohibits unambiguous Hall measurements on this type of samples.
To characterize the magnetically undoped n-type GaN sample, we first carry out photoluminescence (PL) and transmission measurements at a temperature of T ¼ 10 K. Figure 5a depicts the PL spectrum, excited near-resonance at a photon energy of 3.550 eV. We observe a broad asymmetric emission centered at 3.490 eV at the well-known position of the free excitons A, B, and C in GaN. The three lines are not resolved, possibly due to lattice strain [26]. The bound neutral exciton (D 0 X) occurs at a slightly smaller energy around 3.47 eV compared to the free excitons FX A , FX B , and FX C [27,28]. TRFR measurements (i)-(ix) are performed at photon energies indicated by the vertical dotted lines in the PL and transmission spectra ( Fig. 5a and b). The narrow spectral width of the laser pulses of 0.15 nm (full width at half maximum) ensures a high energy resolution in this experiment. In the TRFR data the delta-shaped artifact at zero pump-probe delay and a slowly varying background (data not shown) are both subtracted before data analysis. Damped oscillatory signals ranging for nanoseconds are clearly visible (see inset of Fig. 5b).
The decay of the TRFR signal (i) recorded at a photon energy of E ¼ 3.488 eV is bi-exponential, i.e., it exhibits a fast and a slow dephasing component. v L1 and v L2 are extracted from the TRFR signals via the Fast Fourier Transform (FFT). For n-type GaN we obtain g Ã ¼ 1.953 AE 0.005, where v L1 ¼ v L2 . This value is in good agreement with previous works and a clear signature of electrons [12,29,31]. In contrast, the effective g-factor g Ã of holes in GaN was found to be g ¼ 2.17 and 2.27 in orthogonal directions [30]. Oscillatory TRFR signals at photon energies of 3.488 eV (i) and 3.483 eV (ii) are attributed to spin precession of electrons at the energy of the free exciton (FX). The decay of spin polarization is bi-exponential. The first component decays with T Ã 2;fast ¼ 30-50 ps. It is due to the fast initial exciton recombination after pulsed optical excitation and associated with the decrease of the number of spin-polarized electrons [31,32]. Indeed, carrier and spin dynamics in GaN are intimately linked in the first tens of picoseconds, as has already been shown for ZnO quantum dots [33]. The dephasing time of the second, long-lived component varies between T Ã 2;slow ¼ 2:0-2:5 ns. This TRFR signal is attributed to free doping-related electrons. Their spin relaxation is governed by the Dyakonov-Perel mechanism (DP) and in good agreement with earlier measurements on n-doped GaN (2.8 ns at B ¼ 2 T in Ref. [12]).  At a photon energy of 3.478 eV (curve (iii)), a prominent change in spin precession occurs. A striking phase shift of 1808 is observed in the TRFR oscillation at a time delay t D ¼ 50 ps. Trace (iv) still exhibits the short-lived precession component, but with a very low amplitude. At even lower photon energies 3.468-3.439 eV (v)-(ix), only a single exponential decay is present. The amplitude of the first oscillation in (i), (ii), and (iii) has a positive sign, indicated by vertical black arrows in Fig. 5c. For Faraday signals (iv)-(ix) it has a negative sign. We measure the Faraday rotation signal in curves (iv)-(ix) at the energies of the donor bound exciton D 0 X, as indicated by dashed vertical lines in the photoluminescence spectrum (Fig. 5a). Binding energies of the neutral donor bound exciton of 6-7 meV [26] and the positive donor bound exciton of 11.2 meV [34] in GaN suggest their presence at a low temperature of T ¼ 10 K. Interestingly, the observation of delocalized and localized electrons at high and low photon energies is also directly evident when analyzing the spin dephasing times T Ã 2 in Fig. 6.
There are clearly two regions with almost constant T Ã 2 over a wide energy range, if photon energies are resonant either with free excitons or donor bound excitons. At the FX energy, they amount to 2 ns, while at the D 0 X energy, they become twice as large (up to 4 ns). To gain insight into the spin dephasing mechanism, we investigate the temperature dependence of the spin dephasing time T Ã 2 (Fig. 7). In this measurement, photon energies of the laser were tuned between 3.483 eV at T ¼ 10 K and 3.415 eV at T ¼ 300 K to account for the gradual shift of the optical bandgap at elevated temperatures. Two regimes of spin dephasing exist with different functional dependencies around T ¼ 50 K. This value coincides with the Fermi temperature T F ¼ ( h 2 (3p 2 n e ) 2/3 )/(2m eff k B ) of 45 K for this system, with m eff being the effective electron mass in GaN of 0.2m 0 and n e ¼ 1 Â 10 17 cm À3 being the electron density of the sample. At temperatures T < T F , electron spin dynamics is governed by the Dyakonov-Perel mechanism due to ionized impurity scattering in the degenerate regime [35,36].
Our measured data (Fig. 7) are in good agreement with the expected T Ã 2 $ T 0 dependence [35]. Below T F ¼ 45 K, our data indicate a T Ã 2 $ T 0:29 temperature dependence at photon energies of the bound exciton D 0 X and an almost temperature independent behavior T Ã 2 $ T 0:07 at energies of the free exciton FX. For temperatures T > T F , the spin dephasing time T Ã 2 decreases with a T À2 proportionality (Fig. 7). Other works have reported a T Ã 2 $ T À1:3 proportionality for T > 25 K in GaN [12] and T Ã 2 $ T À2:5 for T > 30 K in GaAs [11] under similar experimental conditions, which is in fair agreement with our data. At these higher temperatures T > T F , the temperature dependence changes for ionized impurity scattering and additional scattering mechanisms, such as longitudinal optical photon scattering gain importance [35], leading in total to the observed T À2 proportionality.
We now turn to magnetically doped GaN.Ga 1Àx Mn x N sample 1 with a manganese concentration of x Mn ¼ 0.0134 shows weak photoluminescence at photon energies of the bound exciton D 0 X, while sample 2 with x Mn ¼ 0.005 does not exhibit any detectable photoluminescence. The onset of transmission is virtually unchanged compared to magnetically undoped GaN, revealing only a slight redshift of 3 meV (data not shown). Both spectra are dominated by donorbound exciton related absorption.
Damped oscillatory Faraday rotation transients are observed in both Ga 1Àx Mn x N samples 1 and 2 (Fig. 8a). Interestingly, the oscillation frequency strongly changes with time delay. To extract the instant frequency and hence the effective g-factor, a wavelet analysis is used [37]. For sample 1 we find a steep drop of the effective g factor in the first tens of picoseconds before it stays constant at g Ã % 1.97 (Fig. 8b). Due to the random distribution of magnetic ions in the crystal, electrons exist in different magnetic surroundings and possess different effective g-factors g Ã [14]. Directly after zero pump probe delay the oscillation period of the TRFR signal is the smallest, indicative of an effective gfactor g Ã ¼ 2.8 at T ¼ 8 K. These electrons reside in volumes of highest Mn 3þ (or "Mn 2þ þ hole") concentration and dephase the fastest (T Ã 2;fast ). Electron spins surrounded by a lower Mn 3þ concentration are identified by their lower g Ã and slower dephasing. After 50 ps, all electron spins in the vicinity of Mn 3þ ions have dephased. What remains is a signal with a constant effective g-factor g Ã % 1.97. To clarify its origin, we investigate the spin dephasing time T Ã 2;slow depending on photon energy (at T ¼ 10 K) and on temperature. In Ga 1Àx Mn x N TRFR oscillations are only present in a narrow spectral range between E ¼ 3.460 and 3.485 eV, which is in contrast to magnetically undoped GaN. The absence of the FX-related TRFR signals in Ga 1Àx Mn x N is most probably attributed to a manganese-related blueshift of the free exciton energies (!40 meV) to a spectral region, not accessible by our laser system [38]. For Ga 1Àx Mn x N the photon energies, where oscillations are observed, lie in the vicinity of the neutral bound exciton D 0 X ( Fig. 9 (a)). T Ã 2;slow remains almost constant at 100 ps, only rising to 500 ps as the excitation is tuned to lower energies, where the TRFR signal eventually vanishes. The smaller value of T Ã 2;slow of several hundreds of picoseconds in Ga 1Àx Mn x N, compared to nanoseconds in magnetically undoped GaN, is most probably the consequence of increased impurity related scattering in the magnetically doped system. The temperature dependence of T Ã 2 depicted in Fig. 9b is investigated both at low (E ¼ 3.468 eV) and high (E ¼ 3.478 eV) energies of the D 0 X region. The general dependence on temperature is similar to magnetically undoped GaN (Fig. 7). T Ã 2 increases with temperature up to a maximum before it decreases again. For Ga 1Àx Mn x N the maximum occurs at T ¼ 75 K, which is slightly higher than for magnetically undoped GaN. The origin of this behavior is presently unclear, but might be explained by an increased number of impurity centers, i.e., the Mn 3þ ions, in Ga 1Àx Mn x N [39]. The measured relations of T Ã 2 $ T 0:35 (E ¼ 3.468 eV) and T Ã 2 $ T 0:38 (E ¼ 3.478 eV) in Ga 1Àx Mn x N at low temperature underline the assignment to the neutral bound exciton D 0 X, due to their resemblance of the T Ã 2 $ T 0:29 dependence observed in magnetically undoped GaN.
At higher temperatures we also find roughly a T À2 dependence, which is the same as for magnetically undoped GaN and indicative of the Dyakonov-Perel spin relaxation mechanism. At this temperature range, bound excitons cease to exist and dissociate to free excitons.
In contrast, Fig. 9c shows the temperature dependence of the fast component T Ã 2;fast , which is associated with electrons interacting with magnetic ions. T Ã 2;fast is observed at photon energies E ¼ 3.468 eV À E ¼ 3.435 eV and is monotonically increasing with rising temperature.
We now turn to the effective g-factor g Ã and the strength of the magnetic ion-electron exchange coupling, which we determine by a temperature dependent measurement of T Ã 2;fast ( Fig. 10).
Only the first oscillation period of the TRFR transient is used to determine g Ã , because of the already mentioned steep g-factor decrease with time delay (Fig. 8b). From the extracted frequency v L,1 we obtain the effective g-factor g Ã via Eq. (2). Figure 10 shows the temperature dependence of g Ã for the two different Mn concentrations (sample 1 with x ¼ 0.0134 and sample 2 with x ¼ 0.005) and magnetically undoped n-GaN for comparison. For sample 1 at T ¼ 8 K and E ¼ 3.473 eV, we obtain g Ã ¼ 2.80 AE 0.05. With increasing temperature g Ã monotonically decreases and approaches the electron g factor in undoped GaN of %1.96 at T ¼ 100 K. Sample 2 shows the same behavior, but smaller values of g Ã at lower temperatures due to the lower Mn concentration. This behavior is due to exchange coupling between the Mn 3þ ions and electrons. It is quantified by the mean field electron-Mn 3þ exchange energy N 0 a. For a conduction band electron in a DMS, g Ã may be described using Equation (4) differs from Eq. (3) only by x eff , which is the effective Mn 3þ concentration, reflecting a possible antiferromagnetic coupling among Mn 3þ ions at doping levels above 1% [40,41]. In sample 1 we assume an effective concentration of x eff ¼ 0.0114 due to partial antiferromagnetic coupling of the Mn 3þ ions (this correction is necessary at x > 0.01).
The extracted effective g-factors g Ã have slightly different values when measured at various photon energies for both Ga 1Àx Mn x N samples (Fig. 10). As a result, N 0 a changes, too. This fact might indicate a slightly increased magnetic coupling at photon energies of the bound excitons. However, the deviation of N 0 a is small compared to the error induced by the uncertainty of Mn concentration, temperature, and determination of the Larmor frequency. A global fit of the data for both Ga 1Àx Mn x N samples yields N 0 a ¼ 14 AE 5 meV.
Our result agrees well with values of N 0 a determined by electron paramagnetic resonance for GaN:Mn 3þ (N 0 a ¼ 0 AE 100 meV) [38] and GaN:Mn 2þ (N 0 a ¼ AE14 meV) for Mn 2þ ions [42]. The determined value for N 0 a in  Ga 1Àx Mn x N is an order of magnitude smaller than in wurtzite II-VI DMSs. Interestingly, a similarly small value of N 0 a ¼ À20 AE 6 meV was found for the III-V DMS (Ga,Mn)As at low Mn concentrations x 0.0013 [43]. In GaMnAs the small value of N 0 a was explained by a reduction of the s-d coupling, being a system with strongly bound holes [19]. However, we cannot distinguish between the Mn 3þ and "Mn 2þ þ h" configuration in our measurements in Ga 1Àx Mn x N. Figure 10b shows the effective g-factor g Ã of the slow dephasing component depending on temperature, measured at the same photon energies as in Fig. 10a. With changing temperature, it remains virtually unchanged with an effective g-factor ranging around g Ã ¼ 1.96. This value is identical to the g-factor of electrons in undoped GaN and corroborates our assignment of the long-lived spin precession to electrons not interacting with Mn ions.
3 Concluding remarks and outlook In conclusion, we have presented the ultrafast spin dynamics in widebandgap magnetic semiconductors. Time-resolved Faraday rotation spectroscopy was used to measure the transient effective g-factor g Ã and ensemble spin dephasing time T Ã 2 in Zn 1Àx Mn x O sol-gel thin films and Ga 1Àx Mn x N grown with molecular beam epitaxy. In that way, the mean field electron-ion exchange energies N 0 a could be determined and the types of observed carriers identified. This information is important for the design of spintronic devices based on these materials. Nanostructured systems fabricated via cost-effective techniques suited for mass production, such as sputtering [44,45] or sol-gel synthesis, will be in the focus of future research.