Pis'ma v ZhETF, vol.102, iss. 4, pp. 250-252
© 2015 August 25
Ferromagnetism mediated by the upper Hubbard band in selectively
doped GaAs/AlGaAs structures
N. V. Agrinskayal\ V. I. Kozub
Ioffe Institute, 194021 S. Petersburg, Russia Submitted 10 June 2015
We consider in detail the indirect exchange between Mn ions imbedded to GaAs/AlGaAs quantum wells where the barriers are doped by acceptor impurity supported by the carriers of the upper Hubbard band supplied by barriers acceptors. A special attention is paid to an interplay between strong derealization of the carriers within the upper Hubbard band (allowing exchange between well separated ions) and relatively weak coupling of these carriers with Mn ions. It is shown, that, despite of the latter factor, the values of Curie temperatures can for such structures be as high as room temperatures.
DOI: 10.7868/S0370274X15160067
Introduction. A lot of attempts were made to fabricate 2D ferromagnetic structure on the base of Mn dopants within GaAs-AlGaAs quantum wells [1]. Since the direct exchange between Mn magnetic ions is negligibly weak, the ferromagnetism was expected to be supported by indirect exchange mediated by holes supplied by Mn acceptors. The important problem here is related to the fact that Mn plays a role in magnetism only being in the substitutional Ga position where Mn is an acceptor. However at significant degree of doping (preferable for high Curie temperatures) many Mn dopants occupy interstitial positions where they play a role of donors thus leading to strong charge disorder. As a result, the holes suffer strong localization and the indirect exchange is weak. In our recent paper, [2], we, however, have shown (both theoretically and experimentally), that at relatively weak degrees of doping, when Mn mostly occupy substitutional positions and the disorder is weak, we deal with virtual Anderson transition, allowing derealization of the holes within the impurity band. As a result, we have observed relatively high Curie temperatures 100 K) at weak degrees of doping. Note, however, that in this case we still need samples with very low degree of any background disorder.
To prevent the localization of the holes by Mn-introduced disorder, in several papers [3] it was suggested to place the Mn delta-layer outside of the quantum well (occupied by holes). In this case the holes were not strongly affected by the charge disorder in Mn delta layer and the indirect exchange was possible. However an inevitable payment in this case is a small tunneling
^e-mail: nina.agrins@mail.ioffe.ru
transparency between the delta-layer and the quantum well.
Here, we will present another realization allowing effective indirect exchange for the case of weak Mn doping (see Fig.l). It is related to the role of the upper Hubbard band of Mn centers which can be occupied provided the AlGaAs barrier is doped by an acceptor impurity. As it is known, the metal-insulator transition in the upper Hubbard band takes place at much weaker concentrations than for single-occipied dopants. Thus we can expect that the indirect exchange (and thus ferromagnetism) can take place for concentrations even less than were used in [2]. Moreover, the limitations on the background disorder are not as strong as implied in [2].
However, it is easily understood that the Coulomb interaction between the hole from the upper Hubbard band with Mn ion (equal to the binding energy of the hole) is much weaker than for the one known for the lower Hubbard band. Indeed, the binding energy for the upper Hubbard band is typically by an order of magnitude lower than for the lower Hubbard band. Thus the indirect exchange via the states of the upper Hubbard band is also to be expected to be weaker than for the lower band. Thus to make a decisive conclusion concerning the role of the effect of the upper Hubbard band on the magnetic properties of the structure we need detailed experimental estimates of all the factors relevant for support of ferromagnetic state.
Discussion. Although earlier [2] we have given semiqualitative estimates for 2D RKKI, here we need somewhat more detailed ones in a view of the factors mentioned above. As it is known, the indirect exchange is supported by a coupling of the potential of magnetic ion j to Friedel oscillations of mobile electron density
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Ferromagnetism mediated by the upper Hubbard band in selectively doped GaAs/AlGaAs structures 251
resulting from the interference of incoming plane wave and scattering state produced by the ion i (and vice versa). Recall that scattering state resulting from the incoming plane wave ipi = expikR scattered by ion i, with a position Rj is given as
where
/=" 1
/
i-iR)1/2
jÇkRi-kR)
d pJe
ikp
(1)
(2)
h2(2'Kk)1/'2
J is the exchange scattering potential for the delocalized mode by the magnetic ion. The result of interference,
ipi'tpl + ip*ips is given as
2 V2/
R1/2
{cos[k(R - Rj) - kR] + sin[k(R - Rj) - kR}} .
(3)
Here we have assumed that the effective radius of J is small. To obtain local electron density at the point R = = Rj we should sum Eq. (3) over k. In particular, the result of summation over the angle variable f of the cosine term gives
/•2-7T
/ d(pReexp(ikRij (cos (p — 1)) = cos(kRij) Jo(\kRij |),
Jo
(4)
where f is Z(k, Rj¿), Jo is the Bessel function. As for the sin term, we note that at small kR it can be neglected. Then, in what follows we will show that the dominant contribution to the final result corresponds to kR = 7T. As it can be seen, the result of integration of the sin term over f is equal to zero. Thus, in what follows we will restrict ourselves with cosine term. Thus we obtain for the modification of the electron density in a position of the ion J resulting from the ion i by summation over all of the modes. Then, the coupling energy between ions i and j is
U-- S-S __—
J2 a4 x
x / dkg(k)fk cos(kRij)Jo(\kRij\),
(5)
where fk is the distribution function while g is the density of states while J is the magnitude of the exchange integral between the localized spin and the delocalized mode and a is the effective radius of the corresponding exchange potential. In our case all of the hole states within the band are occupied and / = 1. Unfortunately, we have no detailed information concerning the delocalized states resulting from the virtual Anderson transition. Although we expect that the corresponding modes
(or their wave packets) can be described by some plane waves with isotropic spectrum, we do not know the energy and wave vector dependence of the effective mass and of density of states. However, we can conclude that the upper limit for k is given by ty/R where R is a typical distance between Mn atoms. Correspondingly, for the exchange between the neighboring magnetic ions the maximal value of the factor kRij in the exponent is estimated as ir. For such cl CclSG the integral over k in 5 for g(k) = Ck (which is typical for isotropic spectrum of plane waves in 2D where C = 2-k~2) gives ~ 2.5C/R2 > 0 which corresponds to a ferromagnetic type of the exchange. Actually, one can expect that the density of states decays near the limiting value of k and thus the upper limit for kR can be even less than ir. However for the angular factor given by Eq. (4) (which is positive except of a small interval of k) one still expects a positive sign of the integral over k almost irrespective to the form of g(k).
The final estimate gives
Hi-
21/2 .2.5
J2
(2tt)5/2 № '
(6)
Basing on the estimates given in [4] for Tc of square plane lattice (the number of neighbors equal to 4),
Tr
2 U Ïn2
we finally obtain
Tr
2.5
n
J
2(7t)3/2 In 2 R2h2
(7)
(8)
One notes that the first factor is ~ 0.5. Thus assuming a ~ R r^ 10~6 cm we obtain as a very rough estimate Tc ~ J21015 erg. Correspondingly, even for J ~ 6meV (which is reasonable estimate for the binding energy of the upper Hubbard band) we can obtain Tc ~ 300 K. Thus one can conclude that the kinetic factor resulting from effective derealization of carriers within the upper Hubbard band can dominate over the relative weakness of the exchange coupling of these carriers within the Mn ion.
Note that in our considerations we have assumed that all barrier acceptors supply their holes to form A+ centers. However, in general it is not cl CclSG. As it was noted in [5], if the barrier acceptor is close enough to the interface between the well and the barrier, there is a possibility to form so-called A0 center which is just a complex of the acceptor within the barrier and hole in the well coupled to this acceptor. The energy of such a complex can be lower than the energy of A+ center. It
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252
N. V. Agrinskaya, V. I. Kozub
can be shown that for dilute structures the relation between A+ and A0 centers is controlled by the following inequality:
UA+ + ^ + + ^
Here, dw is a width of the well (it is assumed that the dopants are within the center of the well), db is a distance of the negatively charged barrier acceptor {A~ center) from the interface, r is a distance between the two acceptors along the plane of the structure, £/ 4+ is the binding energy of A+ center while U ^0 is the binding energy of A0 center.
< dw ,
1 r ( >
J r
/ /
Barrier Well Barrier
Fig. 1. Schematic picture of the structure discussed above
When this inequality holds, then A+ center is formed, otherwise we deal with A0 center. In principle, basing on this inequality we could easily calculate densities of states for A+ and A0 centers. However, in realistic situations one can not restrict himself by only interactions within A+ and A0 complexes due to partial overlapping of the potentials of the corresponding complexes. Nevertheless, qualitatively we can conclude that for large db the binding energy of A0 complex is too small and almost all barrier acce
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