Observation of a two-dimensional spin-lattice in non-magnetic semiconductor heterostructures

0.48 MB·English

by  Christoph Siegert

Observation of a two-dimensional spin-lattice in non-magnetic semiconductor
heterostructures
Christoph Siegert,1 Arindam Ghosh,1,2,∗ Michael Pepper,1 Ian Farrer,1 and David A. Ritchie1
1Cavendish Laboratory, University of Cambridge,
7 J.J. Thomson Avenue, Cambridge CB3 0HE, United Kingdom.
0 2Department of Physics, Indian Institute of Science, Bangalore 560 012, India.
0 (Dated: February 6, 2008)
2

n
a Tunable magnetic interactions in high-mobility in differential conductance (dI/dV) of mesoscopic de-
J
nonmagnetic semiconductor heterostructures are vices close to zero source-to-drain electric potential

2
centrally important to spin-based quantum tech- (V ) were interpreted to be a consequence of spin-
SD
2
nologies. Conventionally, this requires incor- spin or spin-conduction electron exchange interaction.

] poration of “magnetic impurities” within the Here, we have augmented the nonequilibrium trans-
ll two-dimensional (2D) electron layer of the het- port spectroscopy with perpendicular-field magnetore-
a
erostructures, which is achieved either by dop- sistance measurements, which not only confirm the ex-
h
ing with ferromagnetic atoms [1], or by electro- istence of multiple localized spins within high-mobility
-
s statically printing artificial atoms or quantum GaAs/AlGaAs heterostructures,but also reveals a strik-
e
m dots [2, 3, 4, 5]. Here we report experimen- ing order in the spatial distribution of these spins that
tal evidence of a third, and intrinsic, source of becomes visible over a narrow range of n .
. 2D
t localized spins in high-mobility GaAs/AlGaAs Mesoscopic devices fabricated from Si monolayer-
a
m heterostructures, which are clearly observed in doped GaAs/AlGaAs heterostructures were used, where
the limit of large setback distance ( 80 nm) in the 2D electron layer was formed 300 nm below the sur-
- ≈
d modulation doping. Local nonequilibrium trans- face. Athick( 80nm)spacerlayerofundopedAlGaAs
n port spectroscopy in these systems reveal exis- betweenthedo≈pantsandtheelectronsprovidedaheavily
o
tence of multiple spins, which are located in a compensated dopant layer with a filling factor f 0.9.
c ≈
[ quasiregular manner in the 2D Fermi sea, and Theresultinghighelectronmobility( 1 3 106cm2/V-
mutually interact at temperatures below 100 mil- s)providesalongas-grownelasticme∼anf−ree×path 6 8
1 ∼ −
liKelvin via the Ruderman-Kittel-Kasuya-Yosida µm, which acts as an upper limit to the device dimen-
v
(RKKY)indirect exchange. The presence of such sions, ensuring quasi-ballistic transport. (Micrograph of
9
1 a spin-array, whose microscopic origin appears to a typical device is shown in Fig. 1a.)
5 be disorder-bound, simulates a 2D lattice-Kondo At low electron temperatures (T < 100 mK) and
1 system with gate-tunable energy scales. zeromagneticfield,bothequilibriuman∼dnonequilibrium
0
Unintentional magnetic impurities are expected to transport display rich structures well up to linear con-
7
0 be absent in high-quality nonmagnetic semiconductors, ductance G 10 15 (e2/h), as the voltage VG on
∼ − ×
/ such as molecular beam epitaxy-grown GaAs/AlGaAs the gate is increased. For most devices the structures
t
a heterostructures. Contrary to this general belief, re- are strongest at carrier density n 1 3 1010cm−2
2D
m cent observation of a Kondo-like resonance in low- (Fig. 1b), and consist of a repetitiv∼e se−que×nce of two-
- energy density-of-states of one dimensional (1D) quan- types of resonances at the Fermi energy (EF). This is
d
tum wires indicated existence of localized spin in meso- illustrated in the surface plot of dI/dV in Figs. 1c for
n
scopic GaAs/AlGaAs-based devices [6]. Subsequently, device1ofFig.1a. Wedenotethestrongsingle-peakres-
o
c evidence of localized spins was also reported in un- onance at VSD =0 as ZBA-I, which splits intermittently
: confined quasi-ballistic 2D systems [7, 8], instigating to form a double-peaked ZBA with a gap at E , hence-
v F
i the question whether localized spins are intrinsic to forth referred to as ZBA-II. The illustrations of ZBA-
X
GaAs/AlGaAs-based nonmagnetic semiconductors, and I and ZBA-II in the inset of Fig. 1c were recorded at
r if so, what is the microscopic origin of these spins. A points I and II in Fig 1b, respectively. We define ∆
a
fascinating aspect of this problem is the possibility of a as the half-width at half-depth of ZBA-II. While simi-
layer of mutually interacting spins: a system of consid- lar nonequilibrium characteristics was observed in over
erable importance in studying various forms of magnetic 50 mesoscopic devices from 5 different wafers, reducing
ordering, quantum phase transitions and non-Fermi liq- the setback distance below 60 80 nm was generally
∼ −
uid effects (see Ref. [9] for a review). foundto havedetrimentaleffectonthe clarityoftheres-
In the past experiments with ballistic 1D or 2D sys- onance structures (for both ZBA-I and ZBA-II), often
tems[6,7,8],localizedspinsweredetectedwithnonequi- leading to broadening or complete suppression.
librium transport spectroscopy. Pronounced struc- The nature of suppression of such low-energy reso-
tures, commonly known as zero-bias anomaly (ZBA), nances with increasing T and at finite in-plane magnetic
2
field(B )indicateKondo-likeexchange(Figs.1d-1g)[8], which is consistent with the estimate from commensura-
||
and hence presence of localized moments. A complete bility effect. Satellite peaks often appears in the power
description can be obtained with the so-called “two im- spectra, for example those at f∆B ∼= 190 T−1 (orbit c)
purity” Kondomodel, whichembodies the interactionof and50T−1 (orbita),whichcanbe associatedto specific
an ensemble of localized spins within the sea of conduc- stable orbits as indicated in the inset of Fig. 2a. R was
tion electrons [10, 11, 12, 13, 14, 15]. (See Ref. [8] and found to be weakly device-dependent, varying between
Supplementary Information for arguments against alter- 600 800 nm, but insensitive to the lithographic dimen-
−
native explanation of the ZBA.) In the presence of anti- sions of the devices.
ferromagnetic coupling of individual spins to surround- Collectively, the observed nonequilibrium character-
ing conduction electrons, the zero-field splitting of the istics and low-field MR results indicate formation of a
Kondo-resonance(ZBA-II)arisesduetoaV -dependent, quasi-regular2Dspin-latticeembeddedwithinthe Fermi
G
oscillatory inter-impurity exchange J , leading to the sea, where apart from the Kondo-coupling, the conduc-
12
gap ∆ J at E . At certain intermediate values of tion electrons would also undergo potential scattering at
12 F
∼ | |
V one obtains ZBA-I when J k T. For ZBA- the lattice sites. The nature of such potential scatter-
G 12 B
| | ≪
II, nonzero J results in a nonmonotonic suppressionof ing can be cotunneling [12], or scattering off the tun-
12
dI/dV atlowbias(V <∆, asindeedobservedexper- nel barrier at the localized sites [19]. In this framework,
SD
imentally(Figs.1ea|nd1|g∼),whileforZBA-Ithisdecrease the RKKY exchange between the spins naturally leads
is monotonic, and reflects suppression of single-impurity to an oscillatory behavior of J , where range function
12
Kondo-resonance at individual noninteracting spins (see Ψ(2k R) in the interaction magnitude reverses its sign
F
Figs. 1d and 1f) [8]. with a periodicity of π in 2k R, k =√2πn being the
F F 2D
Mesoscopic devices showing clear resonances in the Fermi wave vector. Analytically [20],
nonequilibrium transport also display a characteristic
linear magnetoresistance (MR) over the same range
∆ J E (Jρ )2 Ψ(2k R) (1)
of n2D, when a small magnetic field (B⊥) is applied ∼| |∼ F 2D | F |
perpendicular to the plane of 2D electron layer. As where ρ is the 2D density of states and J is the ex-
2D
shownforadifferentdevice,athighT(=1.4K),theMR changecouplingbetweenanimpurityspinandlocalcon-
∼
consistsof smallpeak-like structuresatspecific valuesof duction electron. Fig. 3a shows the direct confirmation
B⊥ superposedonaparabolicallyincreasingbackground of this, where we have plotted ∆ as a function of 2kFR
(Fig. 2d), while at low T (=30 mK), the magnetotrans- for the device in Fig. 1c. The clear periodicity of π
∼
≈
port breaks into quasi-periodic oscillations irrespective (within 5%)in2k R canbe immediately recognizedas
F
±
of the structure of the ZBA (Fig. 2a and 2b). We note the so-called “2k R-oscillations” in the RKKY interac-
F
that both these evidences indicate electron transport in tion, establishing the spin-lattice picture.
a mesoscopic, quasi-regular array of antidots, which has The absolute magnitude of J , and hence ∆, for a
12
beenstudiedextensivelyinartificiallyfabricatedantidot- 2D distribution of spins may differ widely from the sim-
lattices [16, 17, 18]. In the classicalregime (high T), the ple two-impurity RKKY interaction, and would be af-
peaksinMRcorrespondtocommensurablecyclotronor- fectedbyfrustratedmagneticorderingorspinglassfreez-
bits enclosing fixed number of antidots [16], while at ing [21], as well as deviation from perfect periodicity in
low T, quantum interference leads to the phase coherent the spin arrangements [22]. Nevertheless, a framework
oscillations in magnetoconductance, arising from trans- for relative comparison of ∆ in different samples can be
port along multiple connected Aharonov-Bohm rings as obtainedfromEq.1bynormalizing∆withE . Asshown
F
theinelasticscatteringlengthexceedssampledimensions in Fig. 3b, adjusting for the experimental uncertainty
[17]. in k and R, ∆/E for four different devices with vari-
F F
The magnetotransportdata shown in Fig. 2 allows an ous lithographic dimensions can be made to collapse on
estimationoftheinter-antidotdistanceR. InFig.2d,on thesolidlineproportionaltomodulusofpairwiseRKKY
subtractingthebackground,signatureofcommensurable range function over a wide range of 2k R [20].
F
orbits at B 0.08, 0.05 and 0.03 T corresponding to With known inter-spin distance, we shall now discuss
⊥
≈
cyclotron radius of the electron encircling one, two and twooutstandingissuesofthis paper: (1)the microscopic
four antidots respectively (see inset). While this gives originoftheuniformarrayofantidots,andsubsequently,
R 500nm (using the n 1.31 1010 cm−2), a more (2) emergence of the localized spins. A structurally in-
2D
∼ ≈ ×
accurate estimate of R was obtained by Fourier trans- trinsic origin of antidots in 2D Fermi sea of modulation-
forming the low-T phase-coherent oscillations of Fig. 2a dopedhigh-mobilityGaAs/AlGaAsheterostructurescan
and2b. InFig.2c,thepowerspectracalculatedoverthe arise from long-range potential fluctuations in the con-
range 0 to 0.065 T at both values of V show a strong duction band. Direct experimental imaging of disorder
G
peak at the frequency f eR2/h 105 10 T−1, insimilarsystemsreportedtypicaldistancebetweenfluc-
∆B
≈ ≈ ±
corresponding to one flux quantum through unit cell of tuation to be 0.5 1 µm, in excellent agreement to
∼ −
theantidotlattice(orbitb). ThisgivesR 670 30nm, the magnitude of R in our devices [23]. In presence of
≈ ±
3
strong correlation in the dopant layer at large f, theo- magnetic. Science 281, 951-956 (1998).
retical investigations have also indicated a well-defined [2] Goldhaber-Gordon, D. et al. Kondo effect in a single-
length scale in the spatial distribution of the potential electron transistor. Nature 391, 156-159 (1998).
[3] Cronenwett, S. M., Oosterkamp, T. H., Kouwenhoven,
fluctuations [24].
L.P.ATunableKondoEffectinQuantumDots.Science
A disorder-templatedlocalizedmoment formation can 281, 540-544 (1998).
then be envisaged through local depletion of electrons, [4] Jeong, H., Chang, A. M., Melloch, M. R. The Kondo
analogous to moment formation in metal-semiconductor Effect in an Artificial Quantum Dot Molecule. Science
Schottky barriers [19]. We discuss this on the basis of 293, 2221-2223 (2001).
[5] Craig, N. J. et al. Tunable Nonlocal Spin Control in
three common representations of disorder and screening
a Coupled-Quantum Dot System. Science 304, 565-567
in high-mobility GaAs/AlGaAs systems, as schematized
(2004).
in Fig. 4a-c [25]. At high n , the background disorder
2D [6] Cronenwett,S.M.etal.Low-TemperatureFateofthe0.7
is linearly screened at all points, with local charge fluc- Structure in a Point Contact: A Kondo-like Correlated
tuations δn2D n2D (Fig.4a). Withdecreasingn2D, State in an Open System. Phys. Rev. Lett. 88, 226805
| |≪h i
linearscreeningwillbreakdownlocally atthe maximaof (2002).
slow potential fluctuations, where n becomes smaller [7] Ghosh, A., Ford, C. J. B., Pepper, M., Beere, H. E.,
2D
Ritchie, D. A. Possible Evidence of a Spontaneous Spin
thanthelocal(rapid)densityfluctuationsofthedopants,
Polarization in Mesoscopic Two-Dimensional Electron
resulting in the formation of single-particle localized
Systems. Phys. Rev. Lett. 92, 116601 (2004).
states. Assuming a random distribution of the dopants,
[8] Ghosh,A.et al.Zero-BiasAnomalyandKondo-Assisted
this is expected to occur at nc2D ∼ [(1−f)nδ/π]1/2/ξ, Quasiballistic2DTransport.Phys.Rev.Lett.95,066603
wherenδ =2.5 1012cm−2 isthebaredopantdensityin (2005).
×
our devices, and ξ is the localization length. From f [9] Stewart, G. R. Non-Fermi-liquid behavior in d- and f-
∆B
oforbitainFig.2aweestimateξ 150nm, whichgives electron metals. Rev. Mod. Phys. 73, 797-855 (2001).
nc 1.9 1010 cm−2, which inde∼ed marks the onsetof [10] Jayaprakash, C., Krishnamurthy, H. R., Wilkins, J. W.
2D ≈ × Two-ImpurityKondoProblem.Phys.Rev.Lett.47,737-
strong ZBA (see Fig. 1b).
740 (1981).
On further lowering of n2D, the system crosses over [11] Affleck,I.,Ludwig,A.W.W.Exactcriticaltheoryofthe
to the strongly localized regime (G e2/h), and the two-impurity Kondo model. Phys. Rev. Lett. 68, 1046-
≪
2D electron system disintegrates into puddles, which 1049 (1992).
[12] Pustilnik,M.,Glazman,L.I.KondoEffectinRealQuan-
are often interconnected through quantum point con-
tum Dots. Phys. Rev. Lett. 87, 216601 (2001).
tacts (Fig. 4c). While local spins can form at the
[13] Vavilov, M. G., Glazman, L. I. Transport Spectroscopy
point contacts [6, 26], the commensurability effect and
ofKondoQuantumDotsCoupledbyRKKYInteraction.
phase-coherent magnetoconductance oscillations shown Phys. Rev. Lett. 94, 086805 (2005).
in Fig. 2 cannot be explained in such a picture, as they [14] Golovach,V.N.,Loss,D.Kondoeffectandsinglet-triplet
require extended and uninterrupted electron orbits. splitting in coupled quantum dots in a magnetic field.
Europhys. Lett. 62, 83-89 (2003).
Our experiments thus outline a new microscopic
[15] Hofstetter, W.,Schoeller, H.QuantumPhase Transition
mechanism of local moment formation in high-mobility
in a Multilevel Dot.Phys. Rev. Lett. 88, 016803 (2002).
GaAs/AlGaAs-based semiconductors with remote mod- [16] Weiss,D.etal.Electronpinballandcommensurateorbits
ulationdoping. Thishasseriousimplicationsonthepos- in a periodic array of scatterers. Phys. Rev. Lett. 66,
sibility of a gate-tunable static spontaneous spin polar- 2790-2793 (1991).
ization in mesoscopic devices at low temperatures, and [17] Schuster, R., Ensslin, K., Wharam, D., Khn, S., Kot-
hinged on the Kondo-coupling of the localized moments thaus, J. P. Phase-coherent electrons in a finite antidot
lattice. Phys. Rev. B49, 8510-8513 (1994).
to the surrounding conduction electrons. To verify such
[18] Weiss,D.etal.Quantizedperiodicorbitsinlargeantidot
a coupling, we have estimated the ratio ǫ/Γ for localized
arrays. Phys. Rev. Lett. 70, 4118-4121 (1993).
statesusingexperimentallyobservedKondotemperature
[19] Wolf, E. L., Losee, D. L. Spectroscopy of Kondo and
TK, and that TK (EF/kB)exp(πǫ/2Γ) in the U Spin-Flip Scattering: High-Field Tunneling Studies of
∼ → ∞
limit,whereǫandΓaretheenergyofsingle-electronstate Schottky-Barrier Junctions. Phys. Rev. B 2, 3660-3687
(with respect to E ), and level broadening respectively (1970).
F
(Fig. 4b), and U e2/ξ E , is the on-site Coulomb [20] B´eal-Monod, M. T. Ruderman-Kittel-Kasuya-Yosida in-
F
∼ ≫ direct interaction in two dimensions. Phys. Rev. B 36,
repulsion. Taking the measured T 265 mK at ZBA-I
K ≈ 8835-8836 (1987).
at point I in Fig. 1c as an example, we find ǫ/Γ 2.1,
≈− [21] Hirsch, M. J., Holcomb, D. F., Bhatt, R. N., Paala-
which confirms the Kondo regime.
nen, M. A. ESR studies of compensated Si:P,B near
themetal-insulatortransition.Phys.Rev.Lett.68,1418-
1421 (1992).
[22] Roche, S., Mayou, D. Formalism for the computation of
the RKKY interaction in aperiodic systems. Phys. Rev.
∗ Electronic address: [email protected] B 60, 322-328 (1999).
[1] Ohno, H. Making Nonmagnetic Semiconductors Ferro- [23] Finkelstein, G., Glicofridis, P. I., Ashoori, R. C.,
4
Shayegan,M.TopographicMappingoftheQuantumHall D. A., Anomalous spin-dependent behavious of one-
LiquidUsingaFew-ElectronBubble.Science289,90-94 dimensional subbands. Phys. Rev. B 72, 193305 (2005).
(2000).
[24] Grill, R., D¨ohler, G. H. Effect of charged donor correla- Acknowledgement: We acknowledge discussions with
tionandWignerliquidformation onthetransportprop- C. J. B. Ford, G. Gumbs, M. Stopa, P. B. Littlewood,
erties of a two-dimensional electron gas in modulation
H. R. Krishnamurthy, B. D. Simons, C. M. Marcus, D.
δ-doped heterojunctions. Phys. Rev. B 59, 10769-10777
Goldhaber-Gordon and K. F. Berggren. The work was
(1999).
supported by an EPSRC funded project. C.S. acknowl-
[25] Efros, A. L., Pikus, F. G., Burnett, V. G. Density of
states of a two-dimensional electron gas in a long-range edges financialsupport from Gottlieb Daimler- andKarl
random potential. Phys. Rev. B 47, 2233-2243 (1993). Benz-Foundation.
[26] Graham, A. C., Pepper, M., Simmons, M. Y., Ritchie,
5
FIG. 1: Nonequilibrium characteristics: a, Picture of a set of two typical devices and electrical connections. Active area of
a device is defined by the gate-covered region of the etched mesa. The data shown here was obtained from device 1, with
device 2 and other side-gates kept grounded. b, Typical linear conductance, G, vs. gate voltage VG (and electron density
n2D) for device 1 at 30 mK and zero external magnetic field. c, Surface plot of differential conductance dI/dV of device 1 in
VSD−VG plain. EachdI/dV −VSD traceataparticularVG wasvertically shifted forleveling. Theinsetsillustrate ZBA-Iand
ZBA-II-type resonances at points I and II in Fig. 1b, respectively. d, Surface plot of ZBA-I in in-plane magnetic field (B ).
||
Thedashedlineshowsalinearmonotonicsplittingwitheffectiveg-factor|g∗|≈0.5,whichconfirmstheroleofspin. e,Surface
plot of ZBA-II in B||. The single-impurity Kondo-behavior dominates above B|| ∼ ∆/g∗µB (the horizontal line), where ∆ is
thehalf-gapdefinedintext. f,Monotonictemperature(T)suppressionofZBA-I.g,T-dependenceofdI/dV atZBA-II.dI/dV
is nonmonotonic in T for |VSD|<∼∆.
Siegert et al.
a b d
ZBA-II
30 mK 1.4 K
G G
12
)
ZBA-I
30 mK k
0.0 0.1 -0.1 0.0 0.1 ( R4
4
nit) B (T) R B (Tb) c R 8
u
rb. a b c c B (T)
a a
(
r 0.00 0.05 0.10 0.15 0.20
e
w
o
p
4
0 210/D0B 400 0.0 0.1 0.2 0.3 0.4 0.5
^ -1
f (T ) B (T)
B
FIG. 2: Quantum and classical magnetotransport in perpendicular magnetic field (B ). Typical linear magnetoconduc-
⊥
tance oscillation at a, a single-peak resonance (ZBA-I), and b, a double-peak resonance (ZBA-II). c, Power spectra of the
magnetoconductance oscillations. The filled markers (blue: ZBA-I and red: ZBA-II) represent spectra obtained from the
range |B | ≤ 0.065 T, while the empty (blue) marker represents the spectrum (vertically scaled for clarity) from 0.065 T
⊥
<B <0.15 T.Theorbitscorresponding tothepeaksareindicatedin theschematic. d,Four-probelinearmagnetoresistance
⊥
at 1.4 K for five electron densities from 1.22 (topmost trace) to 1.39×1010 cm−2 (bottom trace), where well-defined ZBA’s
appear at low temperatures. Inset: Magnetoresistance after subtracting the parabolic background. The dashed lines, which
denote variouscommensurate orbits, are computed using the average density of 1.31×1010 cm−2, and R≈500 nm.
6
et al.
Siegert

p p p a
0.)08
V
m
(

2

0.00
2k R 40
F
b
0.F1
E
/

2
0.0
40 60
2k R
F
FIG.3: RKKYindirectexchangeand“2kFR-oscillations”: a,2∆from Fig.1casafunction2kFR,wherekF istheFermiwave
vector, and R is the inter-spin distance obtained from magnetoconductance oscillations. b, 2∆/EF as a function of 2kFR for
four different devices. Solid line is proportional to the pairwise RKKY range function (see text). Local disorder governs the
experimentally attainable range of 2kFR within a given mesoscopic device.
7
FIG.4: Schematicofbackgrounddisorderandlocalmomentformation. Whiletherapidfluctuationsarisefromdopantdensity
fluctuations,theslowquasi-regularfluctuationsindicateseffectofstrongcorrelation inthedonorlayeratlargefilling. a,large,
b, intermediate, and c, low, carrier density regimes are shown. Ourexperimental results conform tothe island-in-sea scenario
of Fig. 4b.