Motional-Narrowing-Type Dephasing of Electron and Hole Spins of

PRL 96, 117203 (2006)

PHYSICAL REVIEW LETTERS

week ending 24 MARCH 2006

Motional-Narrowing-Type Dephasing of Electron and Hole Spins of Itinerant Excitons in Magnetically Doped II-VI Bulk Semiconductors Kai E. Ro¨nnburg, Ernst Mohler, and Hartmut G. Roskos* Physikalisches Institut der Johann Wolfgang Goethe-Universita¨t, Max-von-Laue-Strasse 1, D-60438 Frankfurt am Main, Germany

Kai Ortner, Charles R. Becker, and Laurens W. Molenkamp Experimentelle Physik III, Universita¨t Wu¨rzburg, D-97074 Wu¨rzburg, Germany (Received 27 June 2005; published 21 March 2006) Time-resolved optical spin-quantum-beat measurements performed on magnetically doped II-VI bulk semiconductors reveal an increase of the electron spin dephasing time with rising temperature typical for motional narrowing. With the dephasing being notably faster than in undoped II-VI semiconductors, the magnetic dopants must play a key role, modifying the known dephasing mechanisms and introducing new ones. Focusing on the latter, we theoretically explore the spin dephasing channel arising from magnetization fluctuations sampled by the itinerant excitons. This mechanism suffices to explain quantitatively the results of our time-resolved Faraday-rotation experiments on optically excited Cd1x Mnx Te which we present here as a function of magnetic field, temperature and manganese dopant density. In addition to electron spin-quantum beats, some of our experiments reveal hole spin beats as well. DOI: 10.1103/PhysRevLett.96.117203

PACS numbers: 75.50.Pp, 42.50.Md, 72.25.Rb, 78.47.+p

Semiconductors dilutely doped with magnetic impurities are an interesting class of materials because of the strong modifications of the band structure and the magnetic properties of the undoped species by the impurities [1]. We focus in this contribution on paramagnetic II-VI bulk semiconductors without charge doping. Although this material class has been investigated thoroughly in the past and most of the static properties are well understood, this does not hold for the dynamics of the electron and hole spins of optically excited itinerant excitons. If one measures the spin dephasing times of the electrons by optical pumpprobe techniques, one finds that the spins lose their imprinted coherence on a time scale of several to tens of picoseconds [2 – 4]. Hole spins dephase even more rapidly [2,3]. In contrast, spin dephasing in magnetically undoped semiconductors is found to occur on a much longer time scale of hundreds of picoseconds [5–7]. The origin of this large difference has remained unclear. While recent theoretical work has led to a detailed understanding of the spin dephasing mechanisms active in semiconductors without magnetic doping [8], this is not the case for the magnetically doped species. The understanding of spin dephasing in these materials is made complicated by several factors. First, the research on the undoped materials has identified three major spin dephasing mechanisms for electrons, the D’yakonov-Perel’ mechanism [9], the Margulis-Margulis or variable-g factor effect [8,10], and the Elliott-Yafet spin-relaxation channel [11], with additional effects such as the Bir-Aronov-Pikus mechanism to be considered for p-doped semiconductors [12]. All of these mechanisms are expected to be active in the magnetically doped bulk species, too, but modified in an unknown manner with respect to their strength and their relative significance. Second, the dopants introduce addi0031-9007=06=96(11)=117203(4)$23.00

tional relaxation channels such as spin-flip scattering of electrons at the magnetic impurities [13–15], and (the focus of this Letter) precessional relaxation resulting from thermal fluctuations of the magnetization and from fluctuations of the spatial distribution of the dopants [16]. And finally, systematic experimental investigations of the evolution of the dephasing with temperature, magnetic field and dopant density are scarce. Because of the last point, we first present an experimental study of spin dephasing in bulk Cd1x Mnx Te, with x 0:001, 0.0073, 0.044, 0.148. Finding a pronounced motional-narrowing character, we then develop a spinrelaxation model which puts forth the itinerant nature of the spins and the ensuing sampling over environments with fluctuating magnetization. This approach turns out to quantitatively explain the measured data, which suggests that this relaxation channel is more significant in our experiments than the modifications of the spin dephasing effects also present. We performed time-resolved Faraday-rotation measurements on 600 nm thick Cd1x Mnx Te films grown without charge doping by molecular beam epitaxy on ZnTe (001) substrates. As illustrated in the inset of Fig. 1, spinquantum beats were induced and probed by 250 fs light pulses with their spectra always centered on the respective 1s exciton absorption lines. The bandwidth of the pulses was narrowed to 5 nm to minimize exciton-continuum excitation. The density of optically excited charge carriers (less than 1016 cm3 ) and the intensity of the probe beam (one percent of the pump intensity) were kept low in order to minimize sample heating and carrier-density-dependent effects [4]. Figure 1(a) displays measured transients for a Cd1x Mnx Te (x 0:0073) sample at a magnetic field of

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© 2006 The American Physical Society

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FIG. 1. Results of time-resolved pump-probe experiments on Cd1x Mnx Te (x 0:0073) at 8 T and for various temperatures: (a) Temperature dependence of the Faraday-rotation transients; (b) temperature and magnetic-field dependence of the electron spin precession frequency extracted from the transients; (c) fitted spin dephasing time as a function of temperature and magnetic field (fit procedure: see Refs. [2 – 4]). The lines in (b) and (c) are guides to the eye.

8 T. Numerical fits to the data and others taken at different magnetic fields yield the temperature and magnetic-field dependences of the oscillation frequency fT; B and the dephasing time e T; B of the electron spin-quantum beats, plotted in Figs. 1(b) and 1(c), respectively. From fT; B, after subtraction of the intrinsic Zeeman effect, we determine the effective Mn2 content x (the value given above) by fitting fT; B with a Brillouin function which includes the smoothly interpolated density-dependent correction factors of Ref. [17] for the residual antiferromagnetic interaction between the Mn2 ions. A salient feature of e is its increase with rising temperature, which can be interpreted as a strong indication for a motional-narrowing character of the dominant dephasing process. In addition, e decreases with rising magnetic field for temperatures at and above 8 K. Similar characteristics are observed for all of our four samples. Figure 2 displays a compilation of the data for e T for a fixed magnetic field of 5 T. The dephasing first becomes faster with increasing Mn2 content and then levels off. Coming to the theory, we first note that, in the case of CdMnTe quantum-well samples, spin dephasing times on the picosecond time scale were explained by spin-flip scattering of the electrons with the Mn ions [13,14], which should strongly increase upon transition from quantumwell to bulk conditions. The theory of Ref. [13] does not explain why we find values of e which are comparable to those measured in quantum wells instead of the much larger values expected for bulk materials. Furthermore, it is difficult to see how spin-flip scattering should reproduce the measured temperature dependence. We therefore explore the second of the two magneticdopant-induced relaxation channels addressed above, pre-


FIG. 2. Synopsis of the observed temperature dependence of the electron spin dephasing time e for four Cd1x Mnx Te samples; magnetic field fixed at 5 T. Inset: Measurement geometry. Excitation by circularly polarized Ti:sapphire-laser pulses (spot diameter, 250 m), probing with linearly polarized pulses of same wavelength and bandwidth.

cessional dephasing of itinerant spins. We adopt the motional-narrowing concept of magnetic resonance spectroscopy quite in its original form because we deal essentially with the same situation for which the theory was developed, which is the dephasing of an ensemble of spins which do not couple to each other but experience fluctuating on-site fields during diffusive propagation. In the magnetically doped semiconductor, the local field is determined by the magnetization of the Mn2 ions. We assume that they are distributed statistically and that they do not interact. The motional character enters the picture by the fact that the optically generated mobile singlet excitons diffuse through the semiconductor, and that the electron and hole spins thereby experience a varying local magnetization (see sketch in the inset of Fig. 3). The coupling of the electron and hole spins by the electronhole exchange interaction is weak and will be neglected. Each spin species hence dephases independently from the other although the charge carriers remain bound to excitons by the Coulomb interaction. For a description of dephasing, we follow the theory of magnetic resonance spectroscopy where the phase loss of diffusing spins depends on the magnetic-field fluctuations B [18,19]. With the applied magnetic field B oriented along the z axis and the field fluctuations arising from the fluctuations of the local magnetization, we obtain for the dephasing rate: hMx 2 i hMy 2 i=2 1 2 2 : 0 hMz i T2 1 g0 B B=@ hMz i2 20 (1) Here, g0 is the intrinsic g factor of the itinerant electrons

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(holes), and B Bohr’s magneton. Furthermore, Jspd =n0 gMn B @. Jspd denotes the coupling constant of the exchange interaction between Mn2 ions and itinerant electrons (holes), n0 is the density of the unit cells of the host lattice, and gMn 2 the Mn2 g factor. The excitons experience an average magnetization hMz i, but locally the magnetization fluctuates by M, with Mi , i x; y; z being the components of the fluctuation. An exciton is subjected to any given magnetization for a time 0 , the correlation time. The first term of Eq. (1) describes secular dephasing due to amplitude (longitudinal) fluctuations of the local Mz . The second term, nonsecular broadening, is a result of the transverse (x; y) components of the magnetization fluctuations which tend to flip the aligned spins. The fluctuations have two contributions. The first is thermal and quantum-statistical in origin, while the second arises from the statistical spatial distribution of the Mn2 ions. For both, one has to take into account that each exciton samples over a limited number of Mn2 ions. With either the partition function or the fluctuationdissipation theorem, one determines the thermal contribution to the amplitude fluctuation of Mz from the susceptibility B; T 0 @hMz B; Tith =@B, which yields hMz 2 ith n0 xkB TB; T. The transverse magnetization fluctuations are obtained from the relation hJx2 ith hJy2 ith hJz2 ith JJ 1, yielding hMx2 ith hMy2 ith n0 xkB T30; T B; T hMz i2th . In addition, one also has to take into account the fluctuations due to the statistical spatial distribution of the magnetic ions. Assuming a Poisson distribution, we obtain a local density-fluctuational variance hMz2 idf hMz i2 of the Mz component of the magnetization. Since hMx i hMy i 0, Mx and My are only affected in higher order and their fluctuations can be neglected. Finally performing a spatial averaging over the volume of the exciton which is assumed to cover a mean number Nex of Mn2 ions, all variances are reduced by a factor of 1=Nex . Consolidating the preceding relations, we obtain for the dephasing rate: 1 2 0 n0 xkB TB; T hMz i2 T2 Nex n xk T30; T B; T hMz i2 0 B : (2) 2 2g0 B B=@ hMz i2 20 The task remains to determine Nex and the correlation time 0 of the fluctuations. Both are related to the spatial extension of the exciton which, above 3.5 K, is dominated by the exciton Bohr radius because the thermal localization p radius 0:37@= mkB T of the center-of-mass motion (m me mh being the exciton mass), is then smaller than the Bohr radius. The excitons, in the magnetic field, are mod2 eled as ellipsoids with a volume Vex of 4 3 aBz BaBx B. Here, aBz B and aBx B aBy B are the magnetic-field equivalents of the Bohr radius aB 0 and can be calculated


following Ref. [20] [with an exciton Bohr radius of 6.50 nm, determined with the parameters in the caption of Fig. 3, one obtains aBz 6:08 nm (6.35 nm) and aBx 5:59 nm (6.17 nm) at 8 T (4 T)]. The mean number Nex of Mn2 ions sampled by an exciton is then Nex B n0 xVex B. The correlation time 0 B; T is given by 1=0 Mn 2 1=prop 1=Mn sf . sf is the spin-flip time of the Mn ions which amounts to at least several hundred picoseconds [3] and will be neglected in the following. prop denotes the time of propagation after which an exciton sees a new magnetic environment assuming quasistatic local Mn2 moments. An exciton finds a changed magnetic environment after it has moved by a distance equal to its spatial hvth i extension. prop can then be calculated by prop 23 2a Bx 1 hvth i 1 3 2aBz , with 1=2 hvth i 8kT m

hvth i being the average exciton velocity

assuming a Maxwell-Boltzmann distribution. For all temperatures of our experiments, prop is deep in the sub-picosecond range. The predictions of our model for the spin dephasing time T2 e at B 5 T are presented in Fig. 3. Using only parameters from the literature (see figure caption) and no fit parameter, the calculated data reproduce the experimental data of Fig. 2 remarkably well with respect to both the absolute values and the temperature and magnetic-field dependencies. Starting at low concentrations, the dephasing first becomes faster and then either levels off or becomes slower again.

FIG. 3. Dopant-density and temperature dependence of electron spin dephasing as predicted by our model for the following literature parameters: From Refs. [1,22], n0 1:47 1022 cm3 , gMn 2, JMn 5=2, g0 1:7, me 0:094 m0 , mh 0:72 m0 , Jspd 0:22 eV, exciton screened with S 10:2. Effective-temperature and Mn2 -spin corrections according to Ref. [17]. Inset: Sketch of exciton moving through the local moments of magnetic dopants. For x 0:01, an exciton with 6.50 nm radius samples about 180 local spins.

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hole spin beats [21]. Among the samples addressed before, this was the case for x 0:044 (data not shown). Figure 4 displays transients with more pronounced high-frequency oscillations for two other Cd1x Mnx Te samples with x 0:04 and 0.008, respectively. For each set of data, we extracted the ratio of the two oscillation frequencies and found them to correspond to the expected ratio of 4. The data set is too limited, however, to test spin-relaxation models. We acknowledge discussions with F. Bronold and P. Kopietz, and help by S. Glutsch (exciton volume).

FIG. 4. Measured Faraday-rotation transients exhibiting both electron and hole spin beats. Main panel: 500 nm thick Cd0:96 Mn0:04 Te film on a ZnTe substrate at 2 T and 2.5 K. Inset: 200 nm thick Cd0:992 Mn0:008 Te film on Cd0:96 Zn0:04 Te substrate at 8 T and for 2.5 and 4 K.

The initial decrease of the dephasing time results from the competition of the quadratic dependence of the variances on the Mn2 concentration x and the linear increase of Nex with x. The resulting 1=x dependence of e is well reproduced in our measurements (see Fig. 2). With increasing Mn2 concentration, the corrections due to the antiferromagnetic interaction of the Mn2 ions [17] become important leading to the observed leveling off, respectively, increase of the dephasing times. The dashed lines in Fig. 3 represent the contribution by the secular (longitudinal) magnetization fluctuations to the dephasing for the lowest (2.4 K) and the highest (40 K) temperatures of the graph. Comparison with the total dephasing time indicates that nonsecular dephasing is insignificant at low temperatures when the transverse fluctuations are frozen out. So far, we have assumed that the observed spin-quantum beats are due to the spin precession of the electrons alone. This is justified because hole spins, with their 4 times stronger exchange interaction with the Mn2 ions in Cd1x Mnx Te [1], precess by roughly a factor of 4 more rapidly, but also dephase at least 16 times faster than 2 electrons [see dominant term T2 / 1=2 / 1=Jspd in Eqs. (1) and (2)]. The small extension of the holes compared to the exciton radius will lead to an even faster dephasing. This makes it difficult to observe hole spin beats and has only allowed the detection of an overdamped signature until now [2,3]. In Faraday-rotation measurements on several samples held at or below 4 K, we observed rapidly damped highfrequency signals in addition to the beat signals discussed so far, and we attribute the high-frequency transients to

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