Physica E 13 (2002) 525 – 528
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Non-linear spin transport in magnetic semiconductor multiple quantum wells David S&ancheza; b; c;∗ , A.H. MacDonaldb; c , Gloria Plateroa a Instituto
de Ciencia de Materiales de Madrid (CSIC), Cantoblanco, 28049 Madrid, Spain of Physics, Indiana University, Bloomington, IN 47405, USA c Department of Physics, The University of Texas at Austin, Austin, TX 78712, USA b Department
Abstract We present a theoretical model for vertical transport in DC-biased magnetically-doped II–VI semiconductor multiple quantum wells. Non-, partly- and fully-polarized spin distributions are taking into account within a self-consistent scheme. We investigate the formation of electric 4eld domains under the combined in5uence of band electron interactions with polarized local moments and spin relaxation processes. ? 2002 Elsevier Science B.V. All rights reserved. PACS: 72.25.Dc; 72.25.Mk; 73.21.Cd; 75.50.Pp Keywords: Magnetic semiconductors; Spin transport; Multiple quantum wells
Novel physical properties and promising technological devices have emerged from the interplay between quantum con4nement and spin-polarized transport [1]. Of special interest are magnetic semiconductor (MS) heterostructures as engineered systems for the manipulation of both the electron charge and the spin [2]. In particular, Mn doped II–VI semiconductor compounds can be manipulated to give rise to strongly spin-polarized electronic systems [3]. The heteroepitaxy of modulation-doped wide-gap II–VI MS quantum wells (QWs) [4] and ZnSe=(Zn,Cd,Mn)Se superlattices [5] results in the fabrication of two-dimensional electron gas (2DEG) systems ferromagnetically coupled to Mn+2 magnetic ions. In DC-biased non-magnetic multiple quantum wells (MQWs) the competition between charge accumulation and resonant tunneling eBects results in ∗ Corresponding author. Tel.: +34-913349000; fax: +34913720623. E-mail address:
[email protected] (David S&anchez).
the formation of electric 4eld domains (EFDs) [6]. As a result, the current voltage (I –V ) characteristics presents sawtooth-like branches in the negative diBerential conductance (NDC) region. Along each branch two nearly uniform electric 4elds regions develop, separated by a domain wall (DW) of accumulated electrons. Including the spin degree of freedom in the study of vertical (i.e., in the growth direction) transport is expected to supply new features to the physics of EFD formation. Here we deal with the formation of EFDs in weakly coupled II–VI MQWs with one magnetic QW. We 4nd rich behavior due to the strong non-linearity of the system and because of the space dependence of the band electron spin polarizations. In II–VI MQWs the conduction electrons and the Mn local moments are coupled by means of an exchange interaction which favors the alignment of the Mn and the band electron spins. Within the mean-4eld and virtual crystal approximation [7] the eBect of this interaction is to make the jth subband energy of the
1386-9477/02/$ - see front matter ? 2002 Elsevier Science B.V. All rights reserved. PII: S 1 3 8 6 - 9 4 7 7 ( 0 2 ) 0 0 1 8 6 - 8
David S2anchez et al. / Physica E 13 (2002) 525 – 528
Ji;i+1 =
2 e 0 Tj 2 2 ˜ j (Ei;1 − Ei+1 j + eVi )2 + (2)2 × d[f( − i ) − f( − i+1 + eVi )]; (1)
where 0 = m∗ =2˜2 is the 2D density of states per spin, Tj are the transmission coeJcients to go from the lowest subband of well i to the jth level of the downstream neighbor (for typical densities ∼1011 cm−2 only the lowest subband is populated), eVi is the voltage drop across the ith barrier which separates both wells, and i is the chemical potential in well i measured from its bottom. In deriving Eq. (1) we have employed a Lorentzian-shape function to represent the in5uence of disorder on quasiparticles. We only keep the imaginary part of the disorder self-energy and take it to be energy independent: =˜=2 scatt ∼1 meV. is a phenomenological parameter which includes any scattering mechanism dominant in the sample (impurities, LO phonons, etc.) [6]. As in weakly coupled superlattices is much larger than the miniband width, the tunneling process is sequential. The exchange coupling between the band electrons and the Mn2+ ions produces a giant shift of the subband energies even in the presence of a small magnetic 4eld [9]: Ej = Ej − s, where s = +(−) for = ↑(↓). The spin splitting is [10] = Jsd NMn SBS (gB BS=kB TeB ), where Jsd is the exchange integral (we take it as a constant), NMn is the volumetric density of Mn2+ ions with spin S = 52 , BS is the Brillouin function, TeB is an eBective temperature which includes a term due to antiferromagnetic interactions between Mn ions, and B is the external
µ E1 µ
µ µ Energy
ith well, Ei; j , be spin dependent: Ei;j , where = (↑; ↓). To describe current 5ow between two QWs we notice that the elastic and inelastic scattering times which brings non-equilibrium quasiparticle distributions into equilibrium is the shortest time scale ( scatt ∼0:4 ps). scatt does not include those mechanisms which allow for spin relaxation within a QW (see below). Then we can assume that between successive tunneling events the electrons at each QW are in quasi thermal equilibrium and that their charge distributions follow Fermi– Dirac functions. Further, we ignore interwell spin-5ip processes, thus currents being carried by the two spin subsystems in parallel. Hence, the current per spin from well i to well i + 1 is [8]
Energy
526
2∆ E1 E1
2∆
E1 (a)
0
(b) 0
Fig. 1. (a) Schematic view for spin-splitting and density within a well. Dashed lines mark the initial non-equilibrium chemical potentials. The dot-dashed line denotes the 4nal position (t → ∞) of the chemical potentials. Note that the splitting is smaller than ↑ − E1↑ . The abscissa axis represents the bottom of the well (zero of energies). (b) Same for 2 ¿ ↑ − E1↑ . The 4nal position for ↓ is E1↓ (i.e., n↓ → 0 when t → ∞).
magnetic 4eld which lowers the energy for up spins. Here we take the 4eld experienced by the local moments to be B, thus neglecting the contribution of the conduction band spins. However, the mean-4eld approach that we use can be easily extended to include a total eBective magnetic 4eld. Within an isolated QW, thermodynamical considerations leads us to model the spin relaxation which drives the spin subsystems into equilibrium through the following rate equations [11]: dni − iL
0 ; =− i dt sf
(2)
where ni is the 2D density with spin in well i and sf is the intrawell spin-5ip time. Eq. (2) valid for smaller than the chemical potential (see Fig. 1(a)). In addition, this equation assumes a simple relationship between densities and their chemical potentials: ni =
0 (i −Ei;1 ). For greater than the chemical potential, Eq. (2) becomes i↓ − Ei;↓1 dn↓i n↓ dn↑
0 = − i = − i : =− dt sf sf dt
(3)
For large enough splittings, Eq. (3) would lead to full spin polarization (see Fig. 1(b)). From the previous assumptions the evolution of the spin population in N wells is governed by the following discrete continuity
David S2anchez et al. / Physica E 13 (2002) 525 – 528
10000
i = 1; : : : ; N
∆=6 meV
1000 ×5000
for i↑ − Ei;↑1 ¿ 2 (otherwise, Eq. (4) is transformed by using Eq. (3)). The Coulomb interaction between adjacent charged planes in the QWs is accounted for by the application of a simple Hartree mean-4eld theory (Poisson equation) [6,12]. Simple boundary conditions describing the sample contacts and 4xing the total bias voltage close the set of equations to be self-consistently solved (see Ref. [8] for details). As we are interested in steady-state solutions, the total current traversing the sample is J = Ji;i+1 for any well index. We have focused on ZnSe=Zn1−x−y Cd x Mny Se MQWs with N = 9, barrier width b = 5 nm, well thickness w = 10 nm, m∗ = 0:16me , well doping Nw =2×1011 cm−2 and spin relaxation time sf =10 ps. A value of x∼0:2 has been chosen to supply high enough barriers (∼200 meV). For de4niteness, only the central well incorporates Mn2+ magnetic ions within it. This fact is expected to yield stronger effects in the I –V curve. A moderate number of Mn2+ layers has been considered to account for both appreciable spin eBects and suJciently high electron mobility. In the case of non-magnetic MQWs ( = 0) the I –V curve presents a sawtooth behavior in the NDC regions sweeping voltages up. This situation corresponds to an electrostatic con4guration of EFDs due to non-linear charge eBects (see Fig. 2). If increases, new branches in the I –V curve show up. For = 2 meV a new branch develops, at a high voltage corresponding to tunneling between the 4rst subband of a given well and the second one of the neighbor. Another new feature arises as an additional branch at large enough spin splittings ( = 6 meV). It appears at bias voltages below those corresponding to the E1 → E1 case where the electric 4eld drops almost linearly through the system. The subband energies in the non-magnetic wells (E ↑ and E ↓ ) are quasidegenerate. In general, at the voltage where this new branch takes place, there is resonant tunneling between the ground states Ei;↑1 of the (N − 1)=2 wells to the left ↑ of the magnetic one and the ground state E(N +1)=2; 1 of the magnetic QW. Further increase of the voltage, however, results in a decrease of the current since now
2
(4)
Current (A/cm )
equations: Ji−1; dni − iL i − Ji; i+1
0 = − i dt e sf
527
∆=4 meV
100 ×500
∆=2 meV
10 ×25 1
∆=0 meV
0.1 0.01 0
0.1
0.2
0.3 0.4 Voltage (V)
0.5
0.6
Fig. 2. I –V characteristics for =0, 2, 4 and 6 meV. The 4rst peak represents the 4rst → 4rst subband resonant condition. Then there appear several branches each linked to the formation of the DW within each well. The 4rst → second subband transition peak is slightly split at large spin splitting and an additional branch shows up as well for low bias. Curves for = 0 are rescaled for clarity.
↑; ↓ E(N −1)=2; 1 is oB-resonance. Then the current increases
↓ ↓ again since E(N −1)=2; 1 starts to match E(N +1)=2; 1 . The splitting of the additional branch close to the E1 → E2 case has a similar explanation: it results from the ↑ ↑ alignment of E(N −1)=2; 1 with E(N +1)=2; 2 . Subband mismatch gives rise to a sharp reduction of the current, which increases as the voltage does and resonance ↓ ↓ tunneling between E(N −1)=2; 1 and E(N +1)=2; 2 is allowed. When the latter alignment occurs, a large 5ow of down-spin carriers stream towards the magnetic well, causing a sharp decrease of the polarization (see the abrupt drop of the up-spin density at V = 0:45 V in Fig. 3). Another interesting feature can be obtained by sweeping voltages down from a high initial bias. In GaAs=AlGaAs superlattices it has been proven that multistable stationary states show up in this way [6]. We observe in Fig. 2 that for a given voltage diBerent values of the current may be achieved. Spin polarization is strongly modi4ed as well by the Coulomb interaction. The magnetic QW electronic density for up-spin and down-spin electrons as a function of the voltage is plotted in Fig. 3 for = 6 meV. sf is set to 10 ns to amplify eBects (a consequence of spin bottleneck eBects [13]). For both spin densities we observe a sawtooth structure correlated with the discrete jumps of the DW through the sample. Multistability is also observed for both spin densities. Inset
528
David S2anchez et al. / Physica E 13 (2002) 525 – 528
1 0.8 0.6
~ ~
0.4
Spin polarization (%)
Density (in units of Nw)
up-spin 3 98 96
~ ~ 1
94
0.2
0.06 down-spin
0 0
2
by the Spanish DGES Grant No. PB96-00875, by the European Union TMR Contract FMRX-CT98-0180, and by the Indiana 21st century fund, the Welch Foundation and the DARPA=ONR Award No. N00014-00-1-0951. The authors acknowledge valuable assistance from Tomas Jungwirth.
0.1
0.08 0.1 Voltage (V)
0.2 0.3 Voltage (V)
References
0.12
0.4
0.5
Fig. 3. Multistability between distinct polarization states within a superlattice. The inset shows a blow-up with three diBerent states reached at V = 0:08 meV.
of Fig. 3 shows three diBerent values of the magnetic QW spin polarization, P, which can be obtained from diBerent initial biases. Sweeping biases up from very low voltages mean reaching the 4rst value of P. On the other hand, sweeping down voltages from very high values of V would end up in the third state of the polarization. If the sweeping direction is now reversed at around V = 0:1 V (marked with a cross), the second value of P is achieved. Hence a triple stability between distinct permanent physical values of P is observed. We emphasize that this kind of hysteretic phenomena between magnetic states are driven by electric 4elds. One of us (D.S.) thanks Indiana University and The University of Texas at Austin for hospitality while this work was in progress. This work was supported
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