Time Resolved Faraday Rotation Measurements of Spin Polarized Currents in Quantum Wells

Transcription

1 Time Resolved Faraday Rotation Measurements of Spin Polarized Currents in Quantum Wells M. R. Beversluis 17 December Introduction For over thirty years, silicon based electronics have continued to progress as advances in lithography have allowed circuit elements to grow geometrically smaller and correspondingly faster. However, the increasing difficulty at which this performance is achieved has motivated the development of technologies which exploit quantum behavior. One such avenue is an electron spin-based technology, or Spintronics. Spin is environmentally robust and offers long-lived (tens of nanoseconds) macroscopic quantum behavior compared with many other solid-state quantum variables. In fact, magnetic bits in hard drives are now based on the giant magnetoresistive effect, which is a spin-polarized resistivity. This field hopes to further exploit spin for computational elements. Wolf s recent Science article [1] on spintronics points out that development of magnetoelectonics requires measurements of tiny magnetic moments of very small devices on very short time-scales, and Faraday rotation of a linearly polarized light pulse by a coherent collection of electron spins has begun to fulfil this need. Faraday rotation is a phenomenon by which the linear polarization of a electric field directed along k through a magnetized medium will be rotated by an amount proportional to the component of magnetization along k. This rotation is caused by the magnetic field s generation of different indices of refraction for left and right circularly polarized light, as shown in Figure 1. The two components of the circular decomposition of a linear polarization travel with different phase velocities, and after leaving the medium, have acquired a phase difference between them, causing the linear polarization to rotate. Furthermore, the presence of unequal populations of photoexcited carriers in the two spin states (σ + and σ ) affects the indices of refraction unequally and so leads to Faraday rotation effects. 2 Faraday Rotation Stone [2] has written a classical derivation of the Faraday effect, which is presented here. While the obvious differences between a semiconductor and gas of Lorentz atoms will make this a qualitative description, this derivation will give insight into the optical effect in other materials as well. To begin with, consider a low density gas of Lorentz atoms driven by an external electric field in the presence of a static magnetic field pointing in 1

2 n - 1 L n - 1 R n - n L R ω ω (a) ω 0 ω L ω 0 ω 0 + ω L ω 0 ω L ω 0 ω 0 + ω L (b) Figure 1: (a) Indices of refraction vs. frequency for left and right circular polarized light in a Faraday rotator, where ω 0 is the resonance in the absence of B, and ω L = eb/2mc is the Lamour frequency. (b) The differential index vs. frequency, which determines the polarization rotation. the z direction.. The driven oscillations of the electron are given by s + ω0 2 s + e mc ṡ B = e2 m E 0 e iωt. The damping term due to the radiation reaction is omitted for brevity, and will be added later. A trial solution is s = p 0 e e iωt, where p 0 is the complex amplitude of the dipole moment of the atom. Upon substitution into the equations of motion, the components of our solution for p 0 are: where the nonzero components of α are α xx = α yy = e2 m α xy = α yx = e2 m α zz = e2 m p 0 = α E, ω 2 ω0 2 (ω 2 ω0) 2 2 (2ω L ω) 2 i2ω L ω (ω 2 ω0) 2 2 (2ω L ω) 2 1. ω 2 ω0 2 The Larmor frequency ω L = eb/2mc indicates the energy of the interaction with the magnetic field. The asymmetry of the magnetic interaction with respect to left or right circular rotating waves causes an energy splitting between these polarizations. When the wave propagates through a gas chamber with N atoms, the amplitude of the displacement vector is D 0 = E 0 +4πNp 0.

3 For each displacement component, we write: with D 0x = n 2 1E 0x iβe 0y D 0x = n 2 1E 0y + iβe 0x D 0z = n 2 2E 0z n 2 1 = 1+4πNα xx = 1 4πNe2 (ω 2 ω 2 0)/m (ω 2 ω 2 0) 2 (2ω L ω) 2 n 2 2 = 1+4πNα zz = 1 4πNe2 /m ω 2 ω0 2 β = i4πnα xy = 8πNe 2 ω L ω/m (ω 2 ω0) 2 2 (2ω L ω). 2 As with a biaxial crystal, to find k for each polarization within the media in terms of the incident wavevector k 0, we substitute these equations for D into k 2 E 0 (k E 0 ) k k 2 0 D 0 =0 If k 0 is parallel or anti-parallel to B, this is the canonical Faraday effect geometry, and the solutions for E are E 0z =0(E k.), and the two solutions of (k 2 n 2 1k0) 2 E 0x + iβk0 2 E 0y = 0 iβk0 2 E 0x +(k 2 n 2 1k0) 2 E 0y = 0. Solving for the positive roots of k gives k = k 0 n β and k = k 0 n 2 1 β. Plugging these values into the above equations for E 0x and E 0y give left and right rotating circular waves travelling through the medium with indices defined by k = n L k 0 and k = n R k 0, respectively. To include the damping term γ, the real parts of the dielectric constants become n 2 1 = 4πNe 2 (ω 2 ω 2 1 0)/m (ω 2 ω0) 2 2 (2ω L ω) 2 +(γω 3 /ω0) 2 2 n 2 2 = 4πNe 2 (ω 2 ω 2 1 0)/m (ω 2 ω0) 2 2 (2ω L ω) 2 +(γω 3 /ω0) 2 2 β = 8πNe 2 ω L ω/m (ω 2 ω0) 2 2 (2ω L ω) 2 +(γω 3 /ω0), 2 2 Note that while the Larmor frequency appear in the denominator of these terms, it s effect on the indices of refraction comes through the numerator of β, and so the interaction will lead to a symmetric splitting in energy around ω 0. 3 Time Resolved Faraday Rotation Measurements It has been known for some time that Faraday measurements on semiconductor samples can provide information about carrier concentrations [3]. By combining Faraday rotation with time resolved pump-probe spectroscopic techniques, David Awschalom s group at

4 UCSB has been able to monitor the coherent dynamics of photoexcited spins [4], and has interacted with them using tipping pulses [5]. The experimental arrangements are shown schematically in Figure 2. A complete description of these experiments is beyond the scope of this report, and the reader is referred to the aforementioned papers for a complete description. Figure 2: (a) The time-resolved Faraday rotation experiment. The circularly polarized pump excites electron-hole pairs. The holes rapidly dephase, but the electrons coherently precess around the applied magnetic field. The projection of the magnetic moment of these spins along the k-vector of the probe beam causes a Faraday rotation, which is then measured in a balanced polarometer. By varying the time-delay between pump and probe, the spin dynamics are mapped out. (b) Modifying the experiment for tipping pulses. The experiment is the same as in (a), but with an additional tipping pulse tuned below the band-gap to a virtual level. The repulsion between the real energy levels and the virtual levels causes a Stark field, which rotates the spins. The essence of the first experiment is that the applied magnetic field causes the energy bands of the semiconductor quantum wells to split into spin-up and spin-down levels, which are addressed by the circularly polarized pump beam. The field dependence of this splitting is shown in Figure 3, from experiments performed in [4]. The Voigt geometry is the case in which the Magnetic field is applied transversely to the incident wave-vector. The electron spins precess coherently in this field, and the magnetic moment they induce causes a Faraday rotation in the probe beam. Experimental results from this are shown in Figure 4. The second experiment introduces a tipping pulse to control the spins. By tuning a second pump off-resonance, a dynamic magnetic field is introduced, which rotates the spins. This experiment is similar to the π-pulses in photon-echo experiments, and experimental data from [5] is shown in Figure 5. Currently, this group is working to understand the strength of their tipping pulses. The tipping angle, Θ tip, is an integration of the impulse provided by the tipping probe, but they were unable to exactly quantify this. 4 Conclusions As of the publication date of the tipping paper, this research group believes that spin echoes could occur if they can overcome limitations due to inhomogeneous spin-dephasing. The promise of this work is that an all-optical technique can manipulate a robust quantum variable, which may justify some of the hyperbole around this field.

5 Figure 3: On the left are normalized absorption spectra of a magnetic doped quantum well sample at T=5K. In the Faraday geometry (B growth axis observation direction), the σ + (σ ) polarization state is right- (left-) circularly polarized. In the Voigt geometry, (transverse magnetic field), the two absorption resonances are linearly polarized orthogonal to and parallel to the applied magnetic field. Shown in the top right figure are the measured energy dependence of the absorption resonances (Solid lines for the Faraday geometry, dashed for the Voigt geometry). Below this are shown calculated band splittings and selection rules in the Voigt geometry. The spins of the heavy- (H) and light- (L) hole mixed valence bands are labelled in a basis along ˆx, which becomes exact only in the limit of large Zeeman splittings.

6 Figure 4: On the left is shown time-resolved Faraday rotation in small longitudinal magnetic fields, showing that on short time scales the excitons preferentially scatter to the σ + state. Shown below, at longer times, the excitons recombine and the magnetic sublattice dynamically warms. On the right, time resolved Faraday rotation in transverse magnetic fields (Voigt geometry), showing the strong electron spin beats. Below is shown the dependence of the Larmor frequency vs fild in the different widths of quantum wells. Besides this is shown the measured electron and hole g-factors as a function of cell width. Figure 5: Illustration of the net tipping angle with two co- and counter-polarized tipping pumps.

7 References [1] S. A. Wolf et al, Science 294, 1488 (2001). [2] Radiation and Optics: An Introduction to the Classical Theory, John M. Stone, McGraw-Hill, New York, 1963 [3] Optical Properties of Semiconductors, Paul M. Amirtharaj and David G. Seile, in Handbook of Optics, Vol. 2, 2nd edition (ed. Michael Bass), McGraw-Hill, New York [4] S. A. Crooker, et al., Phys. Rev. B., 56, 7574 (1997) [5] J. A. Gupta, et al., Science, 292, 2458 (2001)