Abstract
We calculate the characteristic current-voltage curve of a tunneling device based on semimagnetic semiconductor materials. The device is a heterostructure with layers of Cd1<FONT FACE=Symbol>-</FONT> x Mn x Te in which the magnetic ions Mn2+ interact strongly with the conducting electrons via the s-d exchange interaction. Thermal fluctuations of Mn2+ magnetic moments cause spin-dependent electron scattering that modifies the characteristic current-voltage curve. Our calculation shows how this electron-ion scattering is expected to affect the spin dynamics in transport measurements.
Spin-flip Scattering Contribution to Resonant-Tunneling Current in Semimagnetic Semiconductor Heterostructures
V. A. Chitta, M. Z. Maialle, S. A. Leão, and M. H. Degani
Grupo de Simulação Computacional de Sistemas Semicondutores
Universidade São Francisco, Faculdade de Engenharia,
13251-900 Itatiba, São Paulo, Brazil
Received February 6, 1999
We calculate the characteristic current-voltage curve of a tunneling device based on semimagnetic semiconductor materials. The device is a heterostructure with layers of Cd1- x MnxTe in which the magnetic ions Mn2+ interact strongly with the conducting electrons via the s-d exchange interaction. Thermal fluctuations of Mn2+ magnetic moments cause spin-dependent electron scattering that modifies the characteristic current-voltage curve. Our calculation shows how this electron-ion scattering is expected to affect the spin dynamics in transport measurements.
I Introduction
Recent advances in the growth of semimagnetic semiconductor (SMS) quantum structures have renewed the interest in the long-standing study of carriers interacting with magnetic ions.[1] SMS quantum structures are versatile systems in which to conduct such study because they make possible the adjust of the spatial overlap between the carrier wave functions and the magnetic ions.[2]
Optical spectroscopy with polarized light have contributed greatly to the understanding of the carrier spin dynamics in undoped SMS quantum structures.[3] That is because angular momentum conservation at the absorption of a polarized photon allows creation of carriers in definite spin states. Furthermore, detection of the polarization of the luminescence reveals the spin states of the carriers at the moment they recombine; from this it is possible to study the role of different spin relaxation mechanisms on the spin population initially photoexcited. On the other hand, most of the transport measurements do not give access to the spin state of the conducting carriers, which complicates the study of the spin dynamics. Also, such measurements usually require doped SMS samples that only recently have been grown with reliable quality. Despite these difficulties, preliminary investigations have shown interesting behaviors that are attributed to the spin polarization of the conducting carriers.[4]
In this work we investigate a tunneling device made out of layers of Cd1-xMnxTe materials, where the magnetic moments arise from the half-filled d shell of the ions Mn2+. An external magnetic field B is applied along the growth axis (z-axis) to align the localized moments and to create a magnetization proportional the average ion spin á Sz ñ , which acts on the electrons via the s-d exchange interaction. This effect can be seen, in the mean-field approximation, as a change in the potential profiles; electrons with spins up see a different potential than electrons with spins down.[2] Several devices have been proposed[5] based on this difference in potential profiles to create tunneling current with some degree of spin polarization.
II Spin-dependent potential profiles
The coupling between electrons and the magnetic ions is given by the s-d exchange interaction term in the electron Hamiltonian, where
is the electron-spin operator, Si's are the ion spins, and Js-d the exchange coupling constant. As discussed above, in the mean-field approximation, the Hamiltonian can be written as
with the signs corresponding to the electron spin components |sp =
1/2 ñ . m is the electron effective mass and V(z) is the band-edge potential of the SMS heterostructure. The quantized motion in the xy-plane yields the Landau level ladder with El=
eB(l +
)/m. The Zeeman spin splitting is gmBB and the average spin á Sz ñ gives the so-called giant-Zeeman splitting that cause the aforementioned change in the potential profile. Also, in Eq. (1), N0=1/W0, with W0 as the volume of the primitive cell, a= á uc|Js-d| uc ñ , where uc is the periodic part of the conducton band Bloch function, and x is the Mn2+ molar fraction.[6]
The eigenstates of H0± that we are concerned with are those involved in the tunneling current. We use a short-handed notation for these eigenstates | p ñ =| pz,p||,sp ñ ,
where
pz(z), with pz = (Ep, kzp), gives the solution of the z-direction problem for an electron with longitudinal energy Ep, propagating to the right (kzp > 0) or left (kzp < 0). To obtain![](https://minio.scielo.br/documentstore/1678-4448/jry9CfDp3J3jtSYmvDGDSNb/990cc898620d348d36ba8574c20fee4653b79f4d.gif)
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An electron in a state | p ñ , when tunneling potential barriers of SMS material, has a transmission coefficient , which can be easily calculated from the eigestates
pz(z) and depends on the longitudinal energy
and spin sp. By considering the thermal fluctuations of the Mn2+ moments (see next section), we allow for transitions from the initial state | p ñ to other states | q ñ with a probability rate per second that we call Wpq. The resulting transmission TEp,sp is then obtained using a semi-classical Boltzmann equation[7] in the first Born approximation
where L is the system size, which includes the well and barrier regions, and
is the electron wave vector at the emitter, where z = -L/2.
The tunneling current emitter-to-collector is calculated as the product of the electron change (e < 0) and the probability current, summed over the occupied states | p ñ at the emitter that are moving toward the collector, yielding
where fe is the emitter Fermi distribution function, and Ep,sp is given in Eq. (3) with the substitution
®
(1-fc(Eq)) to account for the restriction due to the exclusion principle on the occupation of the collector states. A similar result is found for the collector-to-emitter current Jc® e, and the net current is then obtained as J=Je® c-Jc® e.
III Scattering by thermal fluctuations
An additional contribution of the s-d exchange interaction to the electron motion comes from the thermal fluctuations of the magnetic moments. Although the average magnetization components normal to B vanish, since á Sx ñ = á Sy ñ =0, thermal fluctuations allow a finite normal magnetization component proportional to that is varying in time. Fluctuations of the longitudinal component áSz ñ is also expected. The time-dependent Hamiltonian accounting for these contributions is written as
where the electron-spin raising and lowering operators
![](https://minio.scielo.br/documentstore/1678-4448/jry9CfDp3J3jtSYmvDGDSNb/650998fdb8903225e2bdeb016deae33ce3425f62.gif)
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where á ñ av means an average over ensembles accounting for the random dependence of H1(t) on time, as well as the average on the ion positions in the lattice sites. We assume an exponential decay Gpq(t) µ exp(-| t| /tc), with tc setting the time scale of the ion-spin auto-correlations Gpq(t) µ åi,j á Sa,i(t)Sa¢,j(0) ñ that Eqs. (6) and (7) imply. The transition rate is then
which is a Lorentzian function of the transferred energy Eq-Ep between the initial and final scattering states of the transition.
The matrix element á p| H1| q ñ in Eq. (8) is then calculated as follows. The integral of the spatial parts can be performed using the short-range nature of J(r) and the periodicity of the Bloch functions. Using the states Eq. (2),
where the magnetic ion positions are Ri=(Ri||,Zi) . Calculating the spin-dependent part we obtain
where DSz,i(t)=Sz,i(t)- á Sz,i ñ and the upper (lower) signs holding for the spin-up to spin-down transitions (vice-versa)
To proceed with the calculation of Gpq(t) å i, j á Sa,i(t)Sa¢, j(0) ñ , we assume in this work no correlation between spins of different ions, i.e.
with a = x, y, z. This neglects the fact that pairing of spins in antiferromagnetic states is actually expected in the systems investigated; which is however accounted for in an approximated manner in the effective concentration x.[6] The transition probability Eq. (8) then becomes
with wpq=(Eq-Ep)/ and
for the spin-flip (sf) scattering (sq=-sp) and
for the spin-conserving (sc) scattering (sq=sp).
The summation over the final states, given in Eq. (3), allows us to perform the integration over the planar spin distribution
where A is the system area and Nn is the number of ions in the nth layer.
Finally, using the exponential decay of the spin correlations Gpq(t) exp(-| t| /tc), Eq. (13) gives
For the spin-conserving scattering, a similar expression is obtained by changing [S(S+1)-áSz2ñ ± áSzñ ]« [ áSz2ñ- áSzñ 2]. Note that is proportional to the product of probabilities of finding the electron in a primitive unit cell of the nth layer, i.e.
| yp(Zn)| 2, for the initial and final states, times the number Nn of such cells containing a magnetic ion in that layer. The sum over the Landau orbits gives the number of states A(eB/h) in the degenerate levels lq. The thermal equilibrium averages used above are áSzñ =-SBS(y) and áSz2 ñ = áSzñ 2+S2
IV Results
The tunneling device we investigate has the potential profile depicted in Fig. 1, where the material compositions are indicated and the percentages shown are the Mn2+ molar fraction x. The levels drawn in the wells A and B of the potential profile are the energies of the peaks of the transmission coefficients corresponding to the quasi-bound spin-up (+) and spin-down (-) states in these wells. For zero magnetic field B these levels are degenerate (solid line in Fig. 1). They split for B ¹ 0 due to the s-d exchange term in the Hamiltonian. We have chosen the system parameters in such a way that the resonant conditions for the tunneling through the levels of similar spins, A--B- and A+-B+, occur at the same gate voltage. In this way, when scattering is neglected, only one peak is expected in the characteristic current-voltage (I-V) curve of this device, exactly at the voltage that align those levels. By including scattering by the thermal fluctuations, alignments of the levels of opposite spins, A+-B- and A+-B-, are also allowed to contribute to the tunneling current. This is seen in Figs.2(a) and 2(b) for different values of magnetic field, where the main peaks are due to spin-conserving tunneling and the smaller peaks at lower and higher voltages appear because of the spin-flip scattering.[9] Experimental data [3] indicate long correlation times tc ( ~ 250 ps). Since the width of the transmission coefficient in Eq. (3) is much larger than the Lorentzian width /tc in the expression Eq. (8) for Wpq, in this work the Lorentzian was used as a d-function.
Effective potential profile. Solid line is the potential for electrons in zero magnetic field. In nonzero field the spin degeneracy is lifted, such that there are different potential profiles for spin-up (dashed line) and spin-down (dotted line) electrons. Levels in the wells A and B indicate the energies of the quasi-bound states of spins up (+) and down (-). The materials in the layers of the heterostructure are Cd1-xMnxTe, with x being given by the percentages shown. For this figure the bias voltage is zero and Fermi energy is set to EF=30 meV in both emitter and collector, corresponding to an In doping concentration 1×1018cm-3.
It may be worthwhile mentioning that we had better resolved the spin-flip scattering contributions, as in Fig. 2, when the spin splitting was large (large values of B), otherwise the overlap with the main resonant tunneling peak obscures the spin-flip peaks. The effect of temperature is twofold on the results of Fig. 2. First, temperature enters the Fermi distribution functions f(E) in Eq. (5) for the emitter and collector.[10] Second, the alignment of the magnetic moments and their thermal fluctuations depend on T, which affects the observation of the spin-flip peaks in the I-V curve. For small temperatures the spin splittings are large, but the thermal fluctuations are reduced, and consequently the spin-flip scattering is weakened. For high temperatures the thermal fluctuations are enhanced, but the spin splittings are smaller and the I-V peaks are broadened. Some important contributions to the tunneling current, which are however beyond the scope of this paper, have not been treated in our model calculation, such as phonon-assisted and sequential tunneling, as well as charge accumulation effects and higher-order electron-magnetic-ion interactions.[1]
Acknowledgments
This work was supported by FAPESP and PROPEP-USF (Brazil).
References
[1] C.B. Duke, Tunneling in Solids, Suppl. 10, Solid State Physics, ed. by F. Seitz, D. Turnbull, and H. Ehrenreich (Academic Press, New York, 1969) p. 294.
[2] J.K. Furdyna, J. Appl. Phys. 64, R29 (1988).
[3] S.A. Crooker, D.D. Awschalom, J.J. Baumberg, F. Flack, and N. Samarth, Phys. Rev. B 56, 7576 (1997).
[4] I.P. Smorchkova, N. Samarth, J.M. Kikkawa, and D.D. Awschalom, Phys. Rev. Lett. 78, 3571 (1997).
[5] J.C. Egues, Phys. Rev. Lett. 80, 4578 (1998). R. Wessel and I.D. Vagner, Superlattices and Microstructures 8, 443 (1990). In the latter work the transverse magnetization was assumed nonzero and independent of time. This does not account properly for thermal fluctuation effects.
[6] Actually, an effective Mn2+ molar fraction xeff=x(1-x)12 is used to account for the antiferromagnetic pair formation.
[7] B. Vinter and F. Chevoir, Resonant Tunneling in Semiconductors, Edited by L.L. Chang et al. (Plenum Press, New York, 1991) p. 201.
[8] C.P. Slichter, Principles of Magnetic Resonance (Harper & Row, New York, 1963) p. 190.
[9] As a conservative approximation that underestimates the spin-flip scattering we have taken xeff=x(1-x)12 as the molar fraction, assuming the antiferromagnetic pair states not taking part on the scatterings. See, e.g., G. Bastard and L. L. Chang, Phys. Rev. B 41, 7899 (1990).
[10] This is a minor effect for the triple-barrier SMS system because, at relatively low temperatures, the condition for resonant tunneling is satisfied only for states that are deep into the emitter Fermi sea and away form the collector Fermi level (see Fig. 1).
- [1] C.B. Duke, Tunneling in Solids, Suppl. 10, Solid State Physics, ed. by F. Seitz, D. Turnbull, and H. Ehrenreich (Academic Press, New York, 1969) p. 294.
- [2] J.K. Furdyna, J. Appl. Phys. 64, R29 (1988).
- [3] S.A. Crooker, D.D. Awschalom, J.J. Baumberg, F. Flack, and N. Samarth, Phys. Rev. B 56, 7576 (1997).
- [4] I.P. Smorchkova, N. Samarth, J.M. Kikkawa, and D.D. Awschalom, Phys. Rev. Lett. 78, 3571 (1997).
- [5] J.C. Egues, Phys. Rev. Lett. 80, 4578 (1998).
- R. Wessel and I.D. Vagner, Superlattices and Microstructures 8, 443 (1990). In the latter work the transverse magnetization was assumed nonzero and independent of time. This does not account properly for thermal fluctuation effects.
- [7] B. Vinter and F. Chevoir, Resonant Tunneling in Semiconductors, Edited by L.L. Chang et al (Plenum Press, New York, 1991) p. 201.
- [8] C.P. Slichter, Principles of Magnetic Resonance (Harper & Row, New York, 1963) p. 190.
- [9] As a conservative approximation that underestimates the spin-flip scattering we have taken xeff=x(1-x)12 as the molar fraction, assuming the antiferromagnetic pair states not taking part on the scatterings. See, e.g., G. Bastard and L. L. Chang, Phys. Rev. B 41, 7899 (1990).
Publication Dates
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Publication in this collection
23 Feb 2001 -
Date of issue
Dec 1999
History
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Received
06 Feb 1999