Carrier-density-controlled anisotropic spin susceptibility of two-dimensional hole systems

0.21 MB·English

by  T. Kernreiter

Carrier-Density-Controlled Anisotropic Spin Susceptibility of Two-Dimensional Hole
Systems
T. Kernreiter, M. Governale, and U. Zu¨licke
School of Chemical and Physical Sciences and MacDiarmid Institute for Advanced Materials and Nanotechnology,
Victoria University of Wellington, PO Box 600, Wellington 6140, New Zealand
(Dated: July 20, 2012)
We have studied quantum-well-confined holes based on the Luttinger-model description for the
valencebandoftypicalsemiconductormaterials. Evenwhenonlythelowest quasi-two-dimensional
2
(quasi-2D) subband is populated, the static spin susceptibility turns out to be very different from
1
the universal isotropic Lindhard-function lineshape obtained for 2D conduction-electron systems.
0
We discuss how the strongly anisotropic and peculiarly density-dependent spin-related response of
2
2D holes at long wavelengths should make it possible to switch between easy-axis and easy-plane

l magnetization in dilute magnetic quantumwells.
u
J

9
Introduction – The recently achieved ability to make the static spin-susceptibility tensor [12] χ (r ;r ). In a
1 ij α β
bulk semiconductors magnetic [1–3] opens up excit- homogeneouselectronsystemwithspin-rotationalinvari-

] ing possibilities for controlling magnetism in unconven- ance, it has a universalisotropic form that depends only
l
l tional ways. Nanostructuring offers appealing routes to- on the dimensionality of the system [13], rendering the
a
wards realizing magnetic semiconducting devices based impurity-spin interaction (1) to be of Heisenberg-model
h
- on strain-induced anisotropies [4] or wave-function en- type. This case applies to the 2D electron systems re-
s
gineering in heterostructures [5, 6]. In particular, size alized by confining carriers from the conduction band
e
m quantization strongly affects the intrinsic spin-3/2 de- in semiconductor heterostructures as long as inversion
gree of freedom carried by valence-band states [7] and symmetry is not broken by the crystal lattice or due to
.
t is expected to stabilize easy-axis magnetism in a two- structuring of the sample [14–17].
a
m dimensional (2D) hole system with magnetization direc- Here we focus on the properties of 2D hole systems
- tion perpendicular to its plane [8, 9]. Our detailed the- whosechargecarriershaveanintrinsicspin-3/2degreeof
d oretical study shows that the behavior of p-type mag- freedom that is strongly coupled to their orbital motion
n
netic quantum wells is unexpectedly rich. Valence-band evenwheninversionsymmetryisintact[7]. Weadoptthe
o
c mixing turns out to sufficiently weaken easy-axis mag- Luttinger model [18] in axial approximation [7, 19, 20],
[ netism at intermediate densities, triggering a transitions whichprovidesausefuldescriptionoftheupper-mostva-

to an easy-plane magnet. Counterintuitively, valence- lence band of typical semiconductors in situations where
1
band mixing is also responsible for reentrant easy-axis
v
8 magnetism occurring at even higher density. Our results
4 are robust with respect to changes in quantum-well pa-
5 rameters. The density-driven magnetic transitions dis- 4
4
cussed here enable new magneto-electronic device func-
.
7 tionalities [10,11]basedonelectric-fieldmanipulationof 3
0 the magnetization in low-dimensional systems.
0
2 E
Background & Aim – Magnetism is introduced into (cid:144) 2
1 nk
intrinsically nonmagnetic semiconducting materials via E
:
v doping with magnetic ions [1–3] such as Mn, Co, Fe or 1
i
X Gd. One way to generate an exchange interaction be-
r tweenany twolocalizedmagnetic moments embedded in 0
a aconductorisprovidedbytheRuderman-Kittel-Kasuya- -2 -1 0 1 2
Yosida (RKKY) mechanism [12], which gives rise to the kd(cid:144)Π
effective two-impurity spin Hamiltonian
FIG. 1. Lowest three (each of them doubly degenerate) sub-
H = G2 I(α)I(β)χ (r ;r ) . (1)
αβ − i j ij α β bandsofatwo-dimensionalholesystemrealizedbyasymmet-
Xi,j ric hard-wall quantum-well confinement of width d. Disper-
sions are calculated based on the four-band Luttinger-model
(α)
HereI denotesthe ithCartesiancomponentofanim-
i description of bulk valence-band states in axial approxima-
purityspinlocatedatpositionr ,andGisthe exchange
α tion, using band-structure parameters applicable to GaAs
constantforthecontactinteractionbetweenthespinden- confined in [001] direction, and E0 = −π2~2γ1/(2m0d2).
sityofdelocalizedchargecarrierswiththeimpurityspins. Gray lines indicate the range of energies and wave vectors
The propertiesofthe chargecarriersmediating the ef- for which only the lowest subband is occupied. This is the
fective impurity-spin exchange coupling are encoded in regime we focus on in this work.
2
itscouplingstotheconductionbandandsplit-offvalence Luttinger-model description of quasi-2D holes – Using
band are irrelevant. Subband k-dot-p theory [21, 22] is the 4 4 Luttinger Hamiltonian in axial approximation
×
employedtoobtainthelowestquasi-2Dholesubbandsfor for the bulk valence band, the bound states of 2D holes
a symmetric hard-wall confinement characterized by its confinedby a potential V(z) aregivenin terms ofspinor
spatial width d. See Fig. 1. We consider the case where wave functions ξ¯ (z) that satisfy the Schr¨odinger equa-
nk
onlythelowest2Dsubbandisoccupiedandcalculatethe tion [H +H +H ]ξ¯ =E ξ¯ , where
0 1 2 nk nk nk
spinsusceptibility. Basedonthisresult,variousmagnetic
~2 5 d2
phases are shown to emerge in magnetically doped hole H0 = 2m (cid:20)γ111−2γ˜1(cid:18)Jˆz2− 411(cid:19)(cid:21)dz2 +V(z) , (5a)
quantum wells depending on the carrier density. 0
Spin susceptibility of a quasi-2D system – The spin ~2k d
H = √2γ˜ ( i) Jˆ,Jˆ + Jˆ,Jˆ , (5b)
susceptibility of 2D charge carriers is given by [13] 1 m0 2 − (cid:16){ z −} { z +}(cid:17)dz
~2k2 5
χij(Ri,z;R∞′,z′)= H2 =−2m0 (cid:20)γ111+γ˜1(cid:18)Jˆz2− 411(cid:19)−γ˜3(cid:16)Jˆ+2 +Jˆ−2(cid:17)(cid:21).
−~Z dt e−ηt h[Si(R,z;t), Sj(R′,z′;0)]i. (2) (5c)
0
Herem istheelectronmassinvacuum,holeenergiesare
Here R and z are components of the position vector in 0
countedasnegativefromthebulkvalence-bandedge,and
the 2D (xy) plane and in the perpendicular (growth)
weusedtheabbreviationsJˆ =(Jˆ iJˆ)/√2, A,B =
direction, respectively. The spin density operator (in ± x± y { }
units of ~) is defined in terms of field operators Ψ, (AB +BA)/2. The constants γ1 and γ˜j are materials-
Ψ† and the Cartesian components Jˆ of the charge related band-structure parameters [25] and depend on
j
the quantum-well growth direction.
carriers’ intrinsic angular momentum as S (R,z) =
j
Straightforward application of the subband k-dot-p
Ψ†(R,z) Jˆ Ψ(R,z). The field operators can be ex-
j method [21, 22] yields the energy dispersions E with
pressed in terms of operators associated with general nk
associated eigenspinors ξ¯ . At k = 0, the eigenspinors
eigenstates [labelled by band index n and in-plane wave nk
are also eigenstates of Jˆ with eigenvalues 3/2 (heavy
vector k = (kx,ky)] of the non-interacting Hamiltonian holes) or 1/2 (light hozles), which are spl±it in energy.
asΨ(R,z)= d2k eik·R ξ (z) c . The(normal- ±
n (2π)2 nk nk This phenomenon is often referred to as HH-LH split-
ized)spinorsξPnk(zR)andeigenvaluesEnk areobtainedby ting [7, 19]. At finite k, the spinors ξ¯nk are not eigen-
solvingthe multi-bandSchr¨odingerequationforthe con- states of Jˆ anymore. This phenomenon of HH-LH mix-
z
finement in growth direction of the 2D heterostructure. ing arises because a spin-3/2 degree of freedom has a
We can then express the spin susceptibility as much richer structure than the more familiar spin-1/2
case [26]. Figure 1 shows the 2D-subband dispersions
χ (R,z;R′,z′)= d2q eiq·(R−R′) χ (q;z,z′) (3a) obtainedforasymmetrichard-wallconfinementofwidth
ij Z (2π)2 ij
d, using band-structureparametersfor GaAs confinedin
[001] direction. The numerically obtained E and ξ¯
in terms of the 2D Fourier-transformedsusceptibility nk nk
serveasinputforthecalculationofthe2D-holespinsus-
d2k ceptibility according to Eqs. (3) with Eq. (4).
χ (q;z,z′)= Wnl(k,q;z,z′)
ij Z (2π)2 ij Results for the 2D-hole spin susceptibility – Using po-
Xn,l
lar coordinates (q,φ ) for the 2D wave vector q and in-
q
nF(Elk+q)−nF(Enk) ,(3b) troducing the scale χ0 = 2m0/(γ1~2), we find that the
× Elk+q Enk i~η tensor elements of χ (q) have the generic form
− − ij
where n denotes the Fermi function, and
F χ (q)=χ F (q)+G(q) cos(2φ ) , (6a)
xx 0 k q
Wijnl(k,q;z,z′)=hξnk(z)i†·hJˆiξlk+q(z)i χχyy((qq))==χχ0(cid:2)G(cid:2)F(kq()qs)i−n(G2φ(q))c,os(2φq)(cid:3)(cid:3), ((66bc))
† xy 0 q
×hξlk+q(z′)i ·hJˆjξnk(z′)i .(3c) χzz(q)=χ0F⊥(q) . (6d)
Axial symmetry of the 2D system implies Enk Enk ThefunctionsFk,⊥(x)andG(x)dependonmaterialsand
and permits the ansatz [23, 24] ≡ morphological parameters of the 2D hole system, espe-
ciallytheholesheetdensityn k2/(2π),butG(0)=0
2D ≡ F
ξnk(z)=e−iJˆzφk ξ¯nk(z) , (4) generally. Figure2showstypicalresultsobtainedatlow,
intermediate and high densities where still only states in
simplifying calculation of the matrix elements Wnl. In the lowest quasi-2D subband are occupied.
ij
the following, we consider the growth-direction-averaged The density dependence of the 2D-hole spin suscepti-
spin susceptibility χ (q) = dz dz′ χ (q;z,z′) cal- bility follows a generic trend. In the limit of low den-
ij ij
culated at zero temperature. R R sity [Fig. 2(a)], χ (q) χ (q) χ (q), and the line
zz ≫ xx ∼ yy
3
behavior and also the easy-axis response expected from
0.6
aL HH-LH splitting arises because, as the 2D-hole density
Χ
0.5 zz is increased, higher-k states get occupied that are more
k =0.6Π(cid:144)d strongly influenced by HH-LH mixing. At the highest
F
0.4
0 valuesofdensitywherestillonlythelowestquasi-2Dhole
Χ
L(cid:144) 0.3 subbandis populated, easy-axisanisotropyis restoredin
q
H the long-wave-length limit but the response at finite q
Χii 0.2 Χ ~Χ
xx yy becomes as important in strength as that for q 0.
→
0.1 2D-hole-mediated magnetism – In a magnetically
doped p-type quantum well, holes can mediate a mag-
0
netic interaction between localized moments via the
0 0.5 1 1.5 2 2.5 3
RKKY mechanism. The corresponding spin Hamil-
q(cid:144)kF tonian is given by Eq. (1), with χij(rα;rβ)
≡
χ (R ,z ;R ,z ) being the spin susceptibility of the
ij α α β β
2
bL 2D hole system calculated here. For a random but on
averagehomogeneousdistributionofmagneticions,stan-
Χ
1.5 yy Χ kF=Π(cid:144)d dardmean-fieldtheory[12]yieldstheCurietemperatures
zz
0
Χ
L(cid:144)q 1 χ (q)
HΧii TC(Mj F) =8π T0 (cid:12) jχj (cid:12) (7a)
0.5 Χxx (cid:12)(cid:12) 0 (cid:12)(cid:12)q=0
(cid:12) (cid:12)
forferromagneticorderwithmagnetizationdirectionpar-
0 allel to the j axis. The temperature scale
0 0.5 1 1.5 2 2.5 3
I(I +1) G2 n m
q(cid:144)k T = I 0 (7b)
F 0 12 k d π~2γ
B 1
1.2
cL depends onthe impurity-spin magnitude I and the aver-
1 age 3D density n of magnetic impurities, and its func-
I
kF=1.6Π(cid:144)d tional form is that obtained for 2D charge carriers with
0.8 Χ
0 zz parabolicsubbanddispersion[9,27] correspondingto an
Χ
L(cid:144)q 0.6 effective mass m0/γ1.
HΧii 0.4 Χxx Figure 3 shows the mean-field Curie temperatures for
perpendicular-to-plane ( ) and in-plane ( ) magnetiza-
⊥ k
0.2 Χ tion directions calculated with the same input param-
yy
eters used for obtaining the subbands given in Fig. 1.
0
0 0.5 1 1.5 2 2.5 3
q(cid:144)k
F
50
FIG.2. q-dependentspinsusceptibilityforφq=0andvalues 40 T ´
ofthe2D-holeFermiwavevectorkF asindicated intheindi- C
vidual panels. Calculations are based on the band structure 0
T 30
shown in Fig. 1. (cid:144)
L
´
¦H,C 20 TC¦
T
shape of χ (q) is similar to the universal 2D-electron 10
zz
(Lindhard) result [13]. The strong easy-axis response is
expected [6, 8, 9] as a result of HH-LH splitting, which 0
favors a spin-3/2 quantization axis perpendicular to the 0.6 0.8 1 1.2 1.4 1.6 1.8
2D plane [7]. Interestingly, the behavior of the spin sus- k d(cid:144)Π
F
ceptibility changes as density is increased. For interme-
diate values of hole density, results like that shown in FIG. 3. Mean-field Curie temperatures for 2D-hole-mediated
Fig. 2(b) are obtained, exhibiting easy-plane anisotropy easy-axis (TC⊥) and easy-plane (TCk) magnetism as a function
in the long-wave-length limit and a nontrivial structure of the 2D-hole Fermi wave vector kF, obtained for a hard-
developing at wave vectors comparable to k . The sig- wallconfinementwithwidthdandband-structureparameters
F
nificant deviation from both the universal-2D-electron applicable to GaAs.
4
The density dependence of χ (0) is directly reflected
xx,zz 0.006
k,⊥
in that of T . At the mean-field level, the ordered
C
state associated with the maximum transition temper- 0.005 Χxx
ature will be established. Our results suggest that the
0.004
0
type of magnetic ordering can be modified by changing ŽΧ L
tinheac2cDu-mhuolleatdioenn-sliatyye,red.ge.v,icbeysa[2d8ju].stEinagsyt-haxeisgamteagvnoelttiasgme HL(cid:144)RΧii 00..000023 1(cid:144)2HΧxx+Χyy
prevailsatlowandhighdensities,whereasanunexpected
Χ
easy-planemagneticorderemergesatintermediatevalues 0.001 Χyy zz
of the density. Also for high densities, a local maximum 0
appearsinχxx(q)atq ≈0.6kF,whichalmostreachesthe 0.6 0.8 1 1.2 1.4 1.6 1.8
value of χ (q = 0). See Fig. 2(c). Even after averag-
zz k d(cid:144)Π
ing over the polar angle, we find that for k 1.5 π/d F
F
∼
the in-plane susceptibility canhavea maximum atq =0
which is as large as χzz(q =0). Thus it may be poss6ible FsyIsGte.m4,.plRoettael-dspinacuenistpsionf-sχ˜u0sc=ep2tπib2imlit0y/(χγi1j~(2Rd)2)oafsaaf2uDnchtioolne
that the high-density easy-axisferromagneticstate must
of kFd for fixed φR =0 and kFR=10.
coexist (or compete) with helical magnetism [12].
Finite-temperatureeffects –Thermalexcitationofspin
waves (magnons) suppresses the magnitude of the mag-
Fig. 2(c)], we find ν = 2 and ε¯ > 0. In this case a fi-
netization below its mean-field value M . This effect is 0
0 nite spin-wave-related critical temperature is obtained.
captured by the relation [29, 30]
Our discussion of finite-temperature effects was based
M(T) 1 on the spin susceptibility obtained for a non-interacting
=1 d2q n T , (8)
M − 4π2n dZ q 2Dhole systeminaninversion-symmetricquantumwell.
0 I (cid:0) (cid:1) However, recent studies [30] of magnetism mediated by
wherenq(T)istheoccupation-numberdistributionfunc- 2D conduction-band carriers showed that the interplay
tion of magnon modes at temperature T. A spin-wave- of structural inversionasymmetry and Coulomb interac-
related critical temperature is defined by the condition tionscandramaticallyincreasethe stability offerromag-
M(T(SW)) = 0 because, for T > T(SW), too many netic order. As the same mechanisms exist for 2D holes,
C C
magnon excitations will have been excited to sustain moredetailed investigationofsucheffects willbe needed
a finite magnetization. In equilibrium, nq(T) is given toconclusivelyestablishthestabletypesofconfined-hole-
by the Bose-Einstein distribution function nB(εq) = mediated magnetic order.
1/(eεq/[kBT] 1), which depends on the spin-wave en- Real-spacespinresponse–FouriertransformingEq.(6)
−
ergy dispersion εq. The latter’s expression in terms of yields the spin susceptibility in real space χ (R), which
ij
the charge carriers’ q-dependent spin susceptibility de- is relevant, e.g., for the proposed use of RKKY interac-
pends on the type of magnetic order (Heisenberg, Ising, tions to couple quantum-dot-confined spins as part of
or helical) [29], but the parameterisation quantum-information-processing protocols [14, 16, 17].
n q ν Figure 4 shows the density dependence of χij(R) for
ε =IG2 I χ ε¯ +c¯ (9) a 2D hole system. In the low-density limit, we find
q 0 0 ν
d (cid:20) (cid:18)k (cid:19) (cid:21)
F χ χ χ 0, reflecting the strong easy-axis
zz ≫ xx ∼ yy ∼
generally holds for the relevant energy range. The di- anisotropy arising due to HH-LH splitting [8, 9]. As the
mensionlessquantitiesε¯ , ν, c¯ dependonthefunctional density of 2D holes in the lowest subband increases, in-
0 ν
form of the q-dependent spin susceptibility and can es- plane components are observed to become relevant. For
sentially be read off Fig. 2. Stability of the magnetic kF ∼1.6π/d we find χxx to be the dominant component
order requires both coefficients ε¯0 and c¯ν to be posi- of the spin susceptibility tensor while χzz ∼ χyy ∼ 0.
tive. Inserting Eq. (9) into Eq. (8) we obtain an implicit Switchingofthe effectiveHamiltonianthatdescribesthe
equationforthecriticaltemperature. Specializingtoour couplingoftwolocalizedspinsatafixeddistanceisthere-
situation of 2D-hole-mediated magnetism, we see from fore possible by simply adjusting the density of the 2D
Fig. 2(a) that the easy-axis magnetism expected at low hole system in which they are embedded.
hole-sheet densities is actually destabilized by magnons Conclusions – Our work illustrates the crucial impor-
because ε < 0. Considering the easy-plane magnetism tance of valence-band mixing for the spin-related re-
q
at intermediate densities, Fig. 2(b) reveals that the as- sponse in confined hole systems, which turns out to
sociated magnon dispersion is characterized by ν = 2 be much more strongly affected than the density re-
and ε¯ = 0, which again implies destabilization of this sponse [31–33]. It also clarifies the impact of band-
0
magnetic order due to spin-wave excitations. For the structure effects on hole-mediated magnetism in 2D sys-
easy-axis (Ising) magnet expected at high densities [cf. tems which had previously been only considered for the
5
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