Elec. Spin Polariza. Determin. from Lumines. Polari.

Electron Spin Polarization Determination from Luminescence Polarization

Figure 1. Geometry and directions. The measurement of the circular polarization of the luminescence generated when electrons recombine across the band gap in, for example, GaAs, provides an accurate determination of the spin polarization of the electron density at the time of recombination. Recently there has been considerable interest in the spin polarization of electrons which have been injected into GaAs from a ferromagnet, for example through the Schottky barrier between a ferromagnet and a GaAs-based quantum well light emitting diode, or by tunneling from a ferromagnetic tip through the vacuum barrier in an STM experiment. Here we discuss 1) potential sources of confusion in relating the sign of electron spin polarization in the semiconductor to the electron spin polarization in the ferromagnet; and 2) factors which relate the circular polarization of the detected light to the electron spin polarization in the semiconductor.

Potential sources of confusion in relating the sign of electron spin polarization in the semiconductor to the electron spin polarization in the ferromagnet include i) differences in nomenclature convention between the magnetic metals and semiconductor communities; and ii) imprecise specification of experimental quantities in a consistent reference frame.

In the semiconductor community one refers to the orientation of the carrier spin, while in the magnetic metals community one refers to the orientation of the electron magnetic moment. Since the electron spin and magnetic moment are antiparallel in magnetic metals, the terms "majority spin" and "spin up", which are synonymous for a ferromagnet, refer to an electron with moment parallel to the magnetization and spin antiparallel to the magnetization. Of course another potential source of confusion is that the terms majority and minority in semiconductors are usually used to refer to the carrier type, a different usage entirely.

Confusion can be avoided in communicating experimental results and relating the sign of the electron spin polarization in the semiconductor to that of the ferromagnet by carefully specifying quantities in terms of angular momentum in a common reference frame. A widely used geometry is that shown in Fig. 1 where the magnetization axis is parallel to the light propagation along the surface normal (out-of-plane), taken to be the + z-axis. The angular momenta of the light, and of the electrons in the ferromagnet and semiconductor are shown in Fig. 2. The electron states in the semiconductor are designated by their total angular momentum quantum number m_j relative to the quantization axis +z. In the conduction band, the spin quantum number m_s is identical to the total angular momentum quantum number m_j. Thus the spin polarization in the conduction band at the time of recombination is $P_s = {{n_{s \uparrow } - n_{s \downarrow } } \over {n_{s \uparrow } + n_{s \downarrow } }}$ where n_s↑ and n_s↓ refer to the number of electrons whose spins are parallel (m_j = +1/2) and antiparallel (m_j = -1/2) to +z.

Figure 2. Angular momenta for light emission from GaAs. The third column shows the angular momenta (in units of h bar ) of light and electrons in the ferromagnet and semiconductor. The second column gives the moment of the same electrons in the ferromagnet and the final column gives the helicity of the light. At the bottom, the third column is expanded to give the angular momentum of states in the semiconductor at the conduction band minimum and valence band maximum, and illustrates the relevant heavy hole transitions.

The circular polarization of the recombination radiation is related to P_s through the matrix elements. In the light emitting diode geometry of Fig. 1, the recombination typically occurs in a quantum well in the semiconductor. The quantum well confinement lifts the light and heavy hole degeneracy at the top of the valence band as shown in the lower part of Fig. 2. Here a -1/2 to -3/2 transition is indicated by the heavy line resulting in Δm_j = -1. To conserve angular momentum, the photon generated must have angular momentum +1 and hence is σ⁺ light when propagating in the +z direction. Just the opposite is true for the transition shown by the dashed line. The circular polarization of the light is defined as $P_{{\rm{circ}}} = {{I(\sigma^+ ) - I(\sigma^- )} \over {I(\sigma^+ ) + I(\sigma^- )}}$ , where I(σ⁺) is the intensity of light when analyzed for positive helicity, where positive helicity σ⁺ light has angular momentum in the direction of light propagation. Circular dichroism in the ferromagnetic film can change the circular polarization of light passing through it. When it is negligible or P_circ is suitably corrected, we see by inspection of Fig. 2 that P_circ = - P_s. A positive P_circ means that there are a greater number of (m_j = -1/2) electrons than (m_j = +1/2) electrons in the conduction band of the quantum well. The m_j = -1/2 electrons have spins that are parallel to those of the majority electron in the ferromagnet (since a majority spin electron has moment parallel to the magnetization and spin in the -z direction or antiparallel to the magnetization.)

Figure 3. Geometry of the STM experiment.

For the general geometry exemplified by the STM experiment in Fig. 3, additional factors must be considered in connecting the polarization of the detected light to the electron spin density in the semiconductor: i) the angle of the light emitted in the GaAs with respect to the quantization axis; and ii) the change in circular polarization of the light passing through the interface.

The ratio of the emitted light polarization to the electron spin density, depends on the optical matrix elements and the angle of the emitted light, θ_i, with respect to the quantization axis, given by the magnetization direction, which is also the surface normal here. For bulk GaAs the light and heavy holes are degenerate at the valence band maximum, and a 100% polarized electron spin density would recombine to give 50% circularly polarized light along the surface normal (and 50% unpolarized light). For light emitted off-normal from recombination of a conduction electron spin density with polarization P_s, the circular polarization of the light in the GaAs is P_i = P_s / 2cosθ_i.

On passing through the interface, from Snell’s law the angle θ_i of the light with respect to the normal to the interface is related to the angle in vacuum θ_f as θ_i = sin^-1[(sinθ_f) / n_i]. The large index of refraction n_i = 3.4 of GaAs means that for an emission angle in vacuum of 60°, for example, the light is incident in the GaAs at about 15°. From the Fresnel formulae [1], the ratio of the circular polarization of the light in vacuum, P_f, (P_circ in the symmetric geometry discussed above) to the circular polarization in GaAs, P_i, is ${{P_f } \over {P_i }} = {{2\cos (\theta _i - \theta _f )} \over {\cos ^2 (\theta _i - \theta _f ) + 1}}$ . For the specific case of θ_i = 15° and θ_f = 60°, P_f / P_i = 0.94.

The luminescence measurement gives the polarization of the electron density at the time of recombination in the semiconductor, and we have seen how with care to avoid confusion the sign of the polarization in the semiconductor can be related to the majority spin direction in the ferromagnet. However, the magnitude of the measured electron polarization in the semiconductor is not simply related to the magnitude of the polarization in the ferromagnet. Transport across the barrier may be spin dependent and there is spin-flip scattering in the semiconductor. Moreover, there are several polarizations that can be defined in a ferromagnet. The polarization relevant to the bulk magnetization is $P_m = {{n_{maj} - n_{\min } } \over {n_{maj} + n_{\min } }}$ , where n_maj (n_min) is the number of majority (minority) spin electrons integrated over the filled states. Another commonly discussed polarization is defined in terms of the density of states at the Fermi level which can differ from P_m in sign and magnitude. A third, the polarization of the conductivity, describes the transport in bulk ferromagnets, and a fourth, the polarization of the interface conductance, describes the transport through the interface. The polarization P_m of the bulk density is positive by definition, but the other polarizations need not be. For example, the polarization of the Fermi level density of states for Fe is positive, and the polarization of the conductivity is negative. Just the opposite is true for Co and Ni where the polarization of the Fermi level density of states is negative while the polarization of the conductivity is positive. As can be seen, relating the spin polarization measured in the semiconductor to spin-dependent materials properties of the ferromagnet is by no means trivial and in general can only be inferred via a model dependent calculation.

Spin Nomenclature for Semiconductors and Magnetic Metals
Spin Polarization of Injected Electrons

Daniel T. Pierce
Mark D. Stiles

S. F. Alvarado - IBM Zurich Research Laboratory
J. A. C. Bland - University of Cambridge
R. A. Buhrman - Cornell University
J. M. Byers - Naval Research Laboratory
William F. Egelhoff, Jr. - NIST
A. T. Hanbicki - Naval Research Laboratory
J. F. Gregg - University of Oxford
M. B. Johnson - Naval Research Laboratory
B. T. Jonker - Naval Research Laboratory
David P. Pappas - NIST

References:
[1] M. Born and E. Wolf, Principles of Optics, (Cambridge University Press, 1980) 6th ed., p. 40.

Online: September 2004
Last Updated: February 2008

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