Relaxation and coherent oscillations in the spin dynamics of II-VI diluted magnetic quantum wells
Florian Ungar, Moritz Cygorek, Pablo I. Tamborenea, V. Martin Axt
RRelaxation and coherent oscillations in the spindynamics of II-VI diluted magnetic quantum wells
F. Ungar , M. Cygorek , P. I. Tamborenea , and V. M. Axt Theoretische Physik III, Universit¨at Bayreuth, 95440 Bayreuth, Germany Departamento de F´ısica and IFIBA, FCEN, Universidad de Buenos Aires, CiudadUniversitaria, Pab. I, C1428EHA Buenos Aires, Argentina
Abstract.
We study theoretically the ultrafast spin dynamics of II-VI diluted magneticquantum wells in the presence of spin-orbit interaction. We extend a recent study where itwas shown that the spin-orbit interaction and the exchange sd coupling in bulk and quantumwells can compete resulting in qualitatively new dynamics when they act simultaneously.We concentrate on Hg − x − y Mn x Cd y Te quantum wells, which have a highly tunable Rashbaspin-orbit coupling. Our calculations use a recently developed formalism which incorporateselectronic correlations originating from the exchange sd -coupling. We find that the dependenceof electronic spin oscillations on the excess energy changes qualitatively depending on whetheror not the spin-orbit interaction dominates or is of comparable strength with the sd interaction. Ultrafast spin dynamics in semiconductors is attracting nowadays a great deal of attention.In a recent article we explored the interplay between the exchange sd interaction (EXI) and thespin-orbit interaction (SOI) in II-VI diluted magnetic semiconductors (DMS) [1]. In that studywe found that the EXI and the SOI can be tuned to overrun each other or to compete on anequal level in realistic bulk and quantum well systems. Importantly, we found that the inclusionof the SOI introduces oscillations in the spin dynamics which are completely absent when onlythe EXI is relevant. In the present article, we characterize systematically the decay and theperiods of oscillations seen in the spin dynamics in quantum wells, as functions of the excessenergy of the electron population in the conduction band (mean energy of a Gaussian occupationof spin-polarized photoexcited electrons). We employ a microscopic density-matrix theory thatmodels on a quantum-kinetic level the spin dynamics, taking into account the exchange-inducedcorrelations and the localized character of the Mn spins [2, 3]. For the sake of brevity we shallonly sketch the model here and refer the reader to Refs. [1] and [3] for a complete description.We consider conduction band electrons in Mn-doped II-VI DMS coming from low-intensityoptical excitations. We work in the regime of electron densities n e much lower than the Mndensity n Mn , in which the Mn spin variables are nearly stationary [3, 4, 5, 6]. The spin dynamicsis described by (cid:104) s ⊥ k (cid:105) ( t ) and (cid:104) s (cid:107) k (cid:105) ( t ), the mean electronic spin components, perpendicular andparallel to the Mn magnetization, respectively, corresponding to the conduction-band state k .In this study we concentrate on Hg − x − y Mn x Cd y Te quantum wells, since this alloy offersgreat flexibility in the control of the Rashba SOI. This control is achieved thanks to thestrong dependence of the band gap on the doping fraction x + y [7]. With this material,Rashba coefficients of the order of α R ≈
10 ps − nm can be obtained for realistic quantumwell specifications [1]. Throughout the paper we shall assume the Mn magnetization to beperpendicular to the quantum well. Furthermore, we take a Gaussian electron occupation a r X i v : . [ c ond - m a t . m e s - h a ll ] S e p P a r a ll e l m ea n s p i n Time (ps)E c = 0 α R [ps -1 nm] = 4816-1-0.500.51 0 50 100 150 200 250 300 P a r a ll e l m ea n s p i n Time (ps)E c = 0 0 50 100 150 200 250 300Time (ps)E c = 8 meV0 50 100 150 200 250 300Time (ps)E c = 8 meV Figure 1.
Electron spin dynamics in a Hg − x − y Mn x Cd y Te quantum well with Rashbainteraction and no exchange sd coupling for a Gaussian electron distribution centered at E c .Lines: simulations for different values of α R expressed in units of ps − nm. Red dots: simple fitsto the first oscillation of the curves with α R = 4 ps − nm (see text for details).centered at an energy E c above the conduction-band edge with a standard deviation of E s = 3 meV and initial spin-polarization rotated 30 ◦ with respect to the Mn magnetization.It is instructive to examine first the spin dynamics resulting only from the Rashba spin-orbitinteraction (i.e., no exchange sd coupling is accounted for). In this case the Mn magnetizationdoes not enter the dynamics. For later comparison we used, however, the above describedinitial condition where the direction of the initial electronic spin is related to direction of theMn magnetization. Figure 1 shows the time evolution of the summed parallel spin component, (cid:104) s (cid:107) (cid:105) ( t ) = (cid:80) k (cid:104) s (cid:107) k (cid:105) ( t ), for three different values of the Rashba coupling constant α R . The cases E c = 0 (Gaussian occupation centered at the band edge) and E c = 8 meV are shown. We seethat the Rashba interaction produces well-defined oscillations and decay. Note that without theEXI, the time evolution of the total spin is given by coherent precessions of individual electronspins around the k -dependent magnetic Rashba field, which are collectively responsible for thedecay. Red dots in both panels of Fig. 1 are fits to the initial oscillations of the α R = 4 ps − nmevolutions, done with a function f ( t ) ∝ exp [ − ( t/τ ) ] cos (2 πt/T ). For a Gaussian electrondistribution with spins pointing in the growth direction, we find from Eq. (17) of Ref. [1]: (cid:104) s (cid:107) ( t ) (cid:105) = C (cid:90) ∞ dk k exp (cid:34) − (cid:0) (cid:126) k − m ∗ E c (cid:1) (2 m ∗ E s ) (cid:35) cos(2 α R kt ) , (1)where m ∗ is the effective mass and C is a constant determined by the initial value of the totalspin. The integral in Eq. (1) is close to the (half-sided) Fourier transform of a function witha single central peak indicating in time regime a damped oscillation with roughly the peakfrequency. An initially exponential decay of (cid:104) s (cid:107) ( t ) (cid:105) would require a Lorentzian decay in theenergy domain. However, the function in Eq. (1) decays much faster for large k explainingwhy the initial behavior of (cid:104) s (cid:107) ( t ) (cid:105) is much better approximated by a Gaussian than by anexponential. Indeed, Fig. 1 reveals that the Gaussian fit is almost perfect at early times butworsens somewhat later. Applying the Gaussian fit to a number of different cases we findthat for given E c both τ ∝ α − R and T ∝ α − R hold to a very good approximation. A similarbehavior is observed for |(cid:104) s ⊥ (cid:105) ( t ) | = | (cid:80) k (cid:104) s ⊥ k (cid:105) ( t ) | , whose initial evolution can be well fitted with τ , T ( p s ) E c (meV) τ T Figure 2.
Gaussian electron spin relaxationtime, τ , and period of the oscillations, T ,in a Hg − x − y Mn x Cd y Te quantum well withRashba interaction α R = 4 ps − nm and noexchange sd coupling as a function of theexcess energy E c . Symbols: full calculation;lines: parabolic fit. P a r a ll e l m ea n s p i n Time (ps) 0.01 0.1 1 0 100 200 300Time (ps)
Figure 3. (cid:104) s (cid:107) ( t ) (cid:105) with exchange sd and noRashba interaction. x Mn = 0.3%, S = 0 . E c = 0 (red line), 4 meV (green circles),8 meV (blue squares). Inset: E c = 0, S = 0 . x Mn : 0.1% (red), 0.2% (green), 0.3%(blue), 0.4% (magenta), 0.5% (cyan). f ( t ) ∝ exp [ − ( t/τ ) ] cos (2 πt/T ) + 1, with the same values of τ and T as for (cid:104) s (cid:107) (cid:105) ( t ). We foundthat τ and T can be precisely fitted as functions of E c with parabolas, as seen in Fig. 2. Thedecrease of T and the increase of τ with rising E c reflect the fact that the effective Rashba fieldis k dependent becoming stronger for larger k .Let us now look at the spin dynamics under the influence of the EXI only (no Rashba SOI).For the EXI coupling constant we take the value J sd = 26 . [1]. The effects of theEXI on the carrier spins can be controlled via the Mn concentration, x Mn , and the initial netMn magnetization, S = |(cid:104) S (cid:105)| [1]. The dynamics of the electron spin component parallel to theMn magnetization is shown in Fig. 3. As found previously, it is approximately described by anexponential decay to an in general non-zero equilibrium value with a decay time [cf. Eq.(19)of Ref. [3]] τ (cid:107) = ( J sd n Mn m ∗ / (cid:126) d ) (cid:104) S − S (cid:107) (cid:105) , where d is the width of the quantum well and (cid:104) S − S (cid:107) (cid:105) is a second moment of the spin- Mn system. In particular, τ (cid:107) is linear in x Mn andindependent of the excess energy E c . Note that no oscillations appear in the time evolution ofthe parallel spin component when only the EXI is present. The perpendicular component decaysto zero with a slightly different rate while it precesses around the Mn magnetization [3].We now come to the combined effects of the Rashba SOI and EXI. Figure 4 shows the spindynamics for x Mn = 0 . S = 0 . α R ≈ − nm, and three different values of E c . Note thesemilog scale chosen to better visualize the long time behavior. This set of parameters definesa “strong sd” and “weak Rashba” situation. Accordingly, the initial decay is exponential (notGaussian like in Fig. 1), with an E c -independent decay rate, like in Fig. 3. However, at later timesfairly regular oscillations appear, with essentially constant, E c -dependent amplitude, due to thepresence of the Rashba interaction. We note that the frequency of the oscillations depends onlyslightly on E c , becoming higher for higher E c , which indicates a slight dependence on the Rashbamechanism. The frequency of the oscillations (period of about 35 ps) is close to the precessionfrequency about the net Mn magnetic field, which is independent of E c . The oscillations do notdecay because all electrons precess with nearly the same frequency governed mainly by the Mnmagnetization and thus do not dephase. The fact that the amplitude of the oscillations does not P a r a ll e l m ea n s p i n Time (ps) E c = 0 meV4 meV8 meV Figure 4.
Evolution of the mean parallelelectron spin under exchange sd and Rashbainteractions. Parameters: x Mn = 0 . S =0 . α R ≈ − nm, and three values of E c . P e r p e nd i c u l a r m ea n s p i n Time (ps) x Mn = 0 %0.1 %0.3 % Figure 5.
Evolution of the mean per-pendicular electron spin component under ex-change sd and Rashba interactions. Parame-ters: α R = 16 ps − nm, E c = 8 meV, S = 0 . x Mn .decay with time and the near independence of the period on E c distinguish qualitatively theseoscillations from the ones observed with Rashba coupling alone. Also note that again there is asaturation value different from zero as seen above in the sd-only case. However, a new featureproduced by the presence of the Rashba interactions is that the perpendicular spin componentdoes not go to zero as in the sd-only evolution (not shown for brevity).Finally, Fig. 5 shows the time evolution of the mean perpendicular spin component underEXI and SOI for fixed Rashba constant α R = 16 ps − nm, E c = 8 meV, and three different valuesof x Mn . This figure shows the effect of an increasing EXI coupling in the presence of a strongRashba coupling on the perpendicular spin component. We see that as the Mn concentrationincreases, starting from a Rashba-only situation in which the equilibrium value (cid:104) s ⊥ k (cid:105) is half itsinitial value [1], a decay to zero sets in. At the same time, the frequency of the oscillationsincreases and their amplitude goes down.In conclusion, we have studied theoretically the effects of the Rashba spin-orbit interaction inII-VI diluted-magnetic-semiconductor quantum wells. We characterized the dependence of thespin dynamics on the excess energy of a Gaussian population of electrons in the conduction band.Our findings provide qualitative signatures that could aid experimentalists in distinguishing therelative importance of spin-orbit and exchange interactions in DMS quantum wells.We gratefully acknowledge the financial support of the Deutsche Forschungsgemeinschaft(grant No. AX17/9-1), the Universidad de Buenos Aires (UBACyT 2011-2014 No.20020100100741), and CONICET (PIP 11220110100091). References [1] Ungar F, Cygorek M, Tamborenea P I and Axt V M 2015