Pauli spin susceptibility of a strongly correlated two-dimensional electron liquid

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by  A. A. Shashkin

Pauli spin susceptibility of a strongly correlated two-dimensional electron liquid
A. A. Shashkin∗, S. Anissimova, M. R. Sakr†, and S. V. Kravchenko
Physics Department, Northeastern University, Boston, Massachusetts 02115, U.S.A.
V. T. Dolgopolov
Institute of Solid State Physics, Chernogolovka, Moscow District 142432, Russia
6
0 T. M. Klapwijk
0 Kavli Institute of Nanoscience, Delft University of Technology, 2628 CJ Delft, The Netherlands
2
Thermodynamic measurements reveal that the Pauli spin susceptibility of strongly correlated
n
two-dimensional electrons in silicon grows critically at low electron densities — behavior that is
a
J characteristic of the existenceof a phase transition.

3
PACSnumbers: 71.30.+h,73.40.Qv
2

]
l Presently, theoretical description of interacting elec- the critical density nc for the zero-field metal-insulator
e
tron systems is restricted to two limiting cases: (i) weak transition (MIT) in our samples. The nature of the low-
-
r electron-electroninteractions(smallratiooftheCoulomb density phase (n < n ) still remains unclear because
s χ
t
s and Fermi energies rs = EC/EF ≪ 1, high electron even in the cleanest of currently available samples, it is
.
t densities) and (ii) very strong electron-electron inter- masked by the residual disorder in the electron system.
a actions (r ≫ 1, very low electron densities). In the
m s MeasurementsweremadeinanOxforddilutionrefrig-
first case, conventional Fermi-liquid behavior [1] is es-
eratoronlow-disordered(100)-siliconsamples with peak
d- tablished, while in the second case, formation of the electron mobilities of 3 m2/Vs at 0.1 K and oxide thick-
n Wigner crystal is expected [2] (for recent developments, ness 149 nm. These samples are remarkable by the ab-
o see Ref. [3]). Numerous experiments performed in both senceofabandtailoflocalizedelectronsdowntoelectron
c three- (3D) and two-dimensional (2D) electron systems densities n ≈ 1×1011 cm−2, as inferred from the coin-
[ at intermediate interaction strengths (1 . r . 5) have s
s cidence of the full spin polarization field obtained from
3 not demonstrated any significant change in properties parallel-field magnetotransport and from the analysis of
v compared to the weakly-interacting regime (see, e.g., Shubnikov-de Haas oscillations (the former is influenced
0 Refs. [4, 5]). It was not until recently that qualitative
bypossiblebandtailoflocalizedelectrons,while the lat-
0 deviations from the weakly-interacting Fermi liquid be-
terisnot;formoredetails,seeRefs.[6,8,9]). Thisallows
1
havior (in particular, the drastic increase of the effec-
9 one to study properties of a clean 2D electron system
tiveelectronmasswithdecreasingelectrondensity)have
0 without admixture of local moments [8, 9, 10]. The sec-
been found in strongly correlated 2D electron systems
4 ond advantage of these samples is a very low contact re-
0 (rs & 10) [6]. However, these findings have been based sistance (in “conventional” silicon samples, high contact
t/ solely on the studies of a kinetic parameter (conductiv- resistancebecomesthemainexperimentalobstacleinthe
a ity), which, in general, is not a characteristic of a state
lowdensity/lowtemperaturelimit). Tominimizecontact
m of matter.
resistance, thin gaps in the gate metalization have been
- introduced, which allows for maintaining high electron
d The2Delectronsysteminsiliconturnsouttobeavery
density near the contacts regardless of its value in the
n convenient object for studies of the strongly correlated
o regime due to the large interaction strengths (r > 10 main part of the sample.
s
c
canbe easilyreached)andhighhomogeneityofthe sam- For measurements of the magnetization, the paral-
:
v ples estimated (from the width of the magnetocapaci- lel magnetic field B was modulated with a small ac
Xi tance minima in perpendicular magnetic fields) atabout field Bmod in the range 0.01 – 0.03 T at a frequency
4 × 109 cm−2 [7]. In this Letter, we report measure- f = 0.45 Hz, and the current between the gate and the
r
a ments of the thermodynamic magnetization and density two-dimensionalelectronsystemwasmeasuredwithhigh
of states in such a low-disordered, strongly correlated precision (∼ 10−16 A) using a current-voltage converter
2D electron system in silicon. We concentrate on the and a lock-in amplifier. The imaginary (out-of-phase)
metallic regime where conductivity σ ≫ e2/h. We have currentcomponentisequaltoi=(2πfCBmod/e)dµ/dB,
foundthatinthis system, the spinsusceptibility ofband where C is the capacitance of the sample and µ is the
electrons (Pauli spin susceptibility) becomes enhanced chemical potential. By applying the Maxwell relation
by almost an order of magnitude at low electron den- dM/dn = −dµ/dB, one can obtain the magnetization
s
sities, growing critically near a certain critical density M from the measured i. A similar technique has been
n ≈ 8×1010 cm−2: behavior that is characteristic in appliedbyPruset al.[11]toa2Delectronsysteminsili-
χ
the close vicinity of a phase transition. The density n conwith highlevelofdisorder,inwhich casethe physics
χ
is coincident within the experimental uncertainty with of local moments has been mainly studied. As discussed
2
3 0.4
-15i (10A) 012 insulator metal 112mM (10/cm)B 01..01550insulatornmetals2 (10114 cm-26) 0001...482mmd/dB ()B md/dB1 m B (a) B654 =TTT 7 Tm)/dB (/teslaB0.2 (b) cc/0012340ns (n12c011 c4136m. 5TT -T2)6
3 T md 0
1
-1 the onset of complete -0.4 2 T -B 7 T 4 T
spin polarization 1.5 T 6 T 3 T
5 T 2 T
0 1 2 3 4 5 6 7
n (1011 cm-2) -0.2
0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7
s
n (1011 cm-2) n (1011 cm-2)
s s
FIG. 1: Imaginary current component in the magnetization
experimentasafunctionoftheelectrondensityinamagnetic FIG. 2: (a) The experimental dµ/dB as a function of elec-
field of 5 T and T = 0.4 K. Grey area depicts the insulating tron density in different magnetic fields and T =0.4 K. The
phase. Magnetizationversusnsisdisplayedintheinset. Note curvesareverticallyshifted forclarity. Grey areadepictsthe
that the maximum M is coincident within the experimental insulatingphase. Notethattheonsetoffullspinpolarization
uncertainty with µBns. in our experiment always takes place in the metallic regime.
(b)Scalingofthedµ/dBcurves,normalizedbymagneticfield
magnitude, at high electron densities. The dashed line rep-
below, the data analysis and interpretation is not quite resents the “master curve”. Spin susceptibility obtained by
correct in Ref. [11]; in particular, Prus et al. do not dis- integrating the master curve (dashed line) and raw data at
tinguish between the Pauli spin susceptibility of band B= 1.5, 3, and 6 T is displayed in the inset.
electrons and the Curie spin susceptibility of local mo-
ments.
For measurements of the thermodynamic density of at which the electrons start to fill the upper spin sub-
states, a similar circuit was used with a distinction that band — is given by the condition dµ/dB = 0 (M(ns)
the gate voltage was modulated and thus the imaginary reaches a maximum), as indicated by the black arrow in
current component was proportional to the capacitance. thefigure. Itisimportantthatovertherangeofmagnetic
Thermodynamic density of states dn /dµ is related to fieldsusedintheexperiment(1.5–7tesla),themaximum
s
magnetocapacitance via 1/C = 1/C0 +1/Ae2(dns/dµ), M coincides within the experimental uncertainty with
where C0 is the geometric capacitanceandAis the sam- µBns thus confirming that all the electrons are indeed
ple area. spin-polarizedbelownp. Note howeverthat the absolute
Atypicalexperimentaltraceofi(ns)inaparallelmag- value of dµ/dB at ns .nc is reduced in the experiment.
netic field of 5 T is displayed in Fig. 1. The inset We attribute this to smearing of the minimum in i(ns)
shows magnetization M(n ) in the metallic phase ob- caused by possible influence of the residual disorder in
s
tained by integrating dM/dn = −dµ/dB with the in- the electron system, which is crucial in and just above
s
tegration constant M(∞) = Bχ0, where χ0 is the Pauli the insulating phase, in contrast to the clean metallic
spin susceptibility of non-interacting electrons. A nearly regime we focus on here. Another reason for the reduc-
anti-symmetric jump of i(n ) about zero on the y-axis tion in dµ/dB is the electron-electron interactions (due
s
(markedbytheblackarrow)separatesthehigh-andlow- to, e.g., the enhanced effective mass).
density regions in which the signal is positive and neg- InFig.2(a),weshowasetofcurvesfortheexperimen-
ative (M(n ) is decreasing and increasing), respectively. tal dµ/dB versus electron density in different magnetic
s
Such a behavior is expected based on simple considera- fields. Experimental results in the range of magnetic
tions. Atlowdensities,allelectronsarespin-polarizedin fields studied do not depend, within the experimental
amagneticfield,soforthesimplecaseofnon-interacting noise,ontemperaturebelow0.6K(downto0.15Kwhich
2D electrons one expects dµ/dB = −µ (at n → 0, wasthelowesttemperatureachievedinthisexperiment).
B s
deepintheinsulatingregime,the capacitanceofthesys- Theonsetoffullspinpolarizationshiftstohigherelectron
tem vanishes and, therefore, the measured current ap- densities with increasing magnetic field. Grey area de-
proaches zero). At higher densities, when the electrons pictstheinsulatingphase,whichexpandssomewhatwith
start to fill the upper spin subband, M(n ) starts to de- B (formoreonthis,seeRef.[12]). Notethattherangeof
s
crease, and dµ/dB is determined by the renormalized magnetic fields used in our experimentis restricted from
Pauli spin susceptibility χ and is expected to decrease below by the condition that dµ/dB crosses zero in the
with n due to reduction in the strength of electron- metallic regime. In Fig. 2(b), we show how these curves,
s
electron interactions. Finally, in the high-density limit, normalized by magnetic field, collapse in the partially-
the spin susceptibility approaches its “non-interacting” polarizedregime onto a single “master curve”. The exis-
value χ0, and dµ/dB should approach zero. The onset tence of such scaling verifies proportionality of the mag-
of complete spin polarization — the electron density n netization to B, confirming that we deal with Pauli spin
p
3
624 7
(a) )0.008 (b) 0.6 10
C (pF)623 B7 T= 9.9 T (0) - C(B)] / C(0000...000000246 99876. 9TTTT T cc/03456 mB (meV)Bc00..2400 0.5nnsc n(c110111 .c5m-22) 2.502468B (tesla)c
4 T C 5 T 2
0 T [ 0 4 T n
622 1 c
1 2 3 4 1 2 3 4 0.5 1 1.5 2 2.5 3 3.5 4
n (1011 cm-2) n (1011 cm-2) n (1011 cm-2)
s s s
FIG. 3: (a) Magnetocapacitance versus electron density for FIG. 4: Dependenceof the Pauli spin susceptibility on elec-
different magnetic fields. (b) Deviation of the C(ns) depen- trondensityobtained byall threemethodsdescribed intext:
dences for different magnetic fields from the B =0 reference integral of the master curve (dashed line), dµ/dB = 0 (cir-
curve. The traces are vertically shifted for clarity. The onset cles), and magnetocapacitance (squares). The dotted line is
of full spin polarization is indicated by arrows. aguidetotheeye. Alsoshownbyasolidlineisthetransport
data of Ref. [7]. Inset: polarization field as a function of the
electron density determined from the magnetization (circles)
and magnetocapacitance (squares) data. Thesymbolsize for
susceptibility of band electrons, and establishes a com-
themagnetization datareflectstheexperimentaluncertainty,
mon zero level for the experimental traces. Integration
andtheerrorbarsforthemagnetocapacitancedataextendto
of the master curve over n yields the spin susceptibil-
s the middle of the jump in C. The data for Bc are described
ity χ = M/B, as shown in the inset to Fig. 2(b). Also
byalinearfitwhich extrapolatestoadensitynχ closetothe
shown is the spin susceptibility obtained by integration critical density nc for the B=0 MIT.
of raw curves at B = 1.5, 3, and 6 tesla, which, within
the experimental error, yield the same dependence.
Thismethodofmeasuringthespinsusceptibility,being
larization at the beginning of the interaction-broadened
the most direct, suffers, however,from possible influence
jump, as indicated by arrows in the figure. In case the
of the unknown diamagnetic contribution to the mea-
residualdisorderdoescontributetothejumpbroadening,
sureddµ/dB,whicharisesfromthefinitewidthofthe2D
we extend error bars to the middle of the jump, which
electronlayer[13]. Toverifythatthisinfluenceisnegligi-
yieldsanupper boundaryforthe onsetoffullspinpolar-
ble in our samples, we employ another two independent
ization.
methodstodetermineχ. Thesecondmethodisbasedon
markingtheelectrondensitynp atwhichdµ/dB =0and In Fig. 4, we show the summary of the results for the
which corresponds to the onset of complete spin polar- Pauli spin susceptibility as a function of n , obtained
s
ization, as mentioned above. The so-determined polar- using all three methods described above. The excel-
ization density np(B) can be easily converted into χ(ns) lentagreementbetweenthe resultsobtainedbyallofthe
via χ = µBnp/B. Note that in contrast to the value methods establishes that a possible influence of the dia-
of dµ/dB, the polarization density np is practically not magnetic shift is negligible [14] and, therefore, the valid-
affected by possible influence of the diamagnetic shift. ityofthedataincludingthoseatthelowestelectronden-
The third method for measuring n and χ, insensitive sities is justified. There is also good agreement between
p
to the diamagnetic shift, relies on analyzing the magne- these results and the data obtained by the transport ex-
tocapacitance, C. Experimental traces C(n ) are shown periments of Ref. [7]. This adds credibility to the trans-
s
in Fig. 3(a) for different magnetic fields. As the mag- port data and confirms that full spin polarizationoccurs
netic field is increased, a step-like feature emerges on at the field B ; however,we note againthat evidence for
c
the C(n ) curves and shifts to higher electron densities. the phase transition can only be obtained from thermo-
s
This feature corresponds to the thermodynamic density dynamic measurements. The magnetization data extend
ofstatesabruptlychangingwhenthe electrons’spinsbe- to lower densities than the transport data, and larger
come completely polarized. To see the step-like feature values of χ are reached, exceeding the “non-interacting”
moreclearly,inFig.3(b)wesubtracttheC(ns)curvesfor value χ0 by almost an order of magnitude. The Pauli
different magnetic fields from the reference B =0 curve. spin susceptibility behaves critically close to the critical
The fact that the jumps in C (as well as in dµ/dB) are density n for the B = 0 metal-insulator transition [15]:
c
washed out much stronger than it can be expected from χ∝n /(n −n ). This is in favorof the occurrence of a
s s χ
possibleinhomogeneitiesintheelectrondensitydistribu- spontaneousspinpolarization(eitherWignercrystal[16]
tion(about4×109cm−2 [7])pointstotheimportanceof or ferromagnetic liquid) at low n , although in currently
s
electron-electron interactions. Since the effects of inter- available samples, the formation of the band tail of lo-
actions are different in the fully- and partially-polarized calized electrons at n . n conceals the origin of the
s c
regimes, it is natural to mark the onset of full spin po- low-density phase. In other words, so far, one can only
4
reach an incipient transition to a new phase. dence characteristicof local moments, and the extracted
The dependence B (n ), determined from the magne- spin susceptibility in their sample has a Curie tempera-
c s
tization and magnetocapacitance data, is represented in turedependence[9]. Thisisthecaseevenathighelectron
the inset to Fig. 4. The two data sets coincide and are densities, where metallic behavior might be expected in-
describedwellbyacommonlinearfitwhichextrapolates stead. Such effects are absent in our samples: the spin
to a density n close to n . We emphasize that in the susceptibility (in the partially-polarized system) is inde-
χ c
low-field limit (B <1.5 tesla), the jump in dµ/dB shifts pendent of the magnetic field and temperature, confirm-
to the insulating regime, which does not allow us to ap- ing that we deal with Pauli spin susceptibility of band
proach closer vicinity of n : based on the data obtained electrons.
χ
in the regime of strong localization, one would not be
In summary,the Paulispin susceptibility has been de-
abletomakeconclusionsconcerningpropertiesofaclean
termined by measurements of the thermodynamic mag-
metallic electronsystem which we are interested in here.
netization and density of states in a low-disordered,
Clearly, the fact that the linear B (n ) dependence per-
c s strongly correlated 2D electron system in silicon. It is
sistsdowntothelowestelectrondensitiesachievedinthe
foundto behavecriticallynearthe zero-fieldMIT,which
experiment confirms that we always deal with the clean
is characteristic of the existence of a phase transition.
metallic regime.
Finally, we would like to clarify the principal differ- We gratefully acknowledge discussions with S.
ence between our results and those of Ref. [11]. In the Chakravarty, D. Heiman, N. E. Israeloff, R. S.
sample usedby Pruset al., the criticaldensity n forthe Markiewicz,andM.P.Sarachik. Oneofus(SVK)would
c
B = 0 MIT was considerably higher than in our sam- liketothankB.I.Halperinforsuggestingthismethodto
ples caused by high level of disorder, and the band tail measure spin susceptibility. We would also like to thank
of localized electrons was present at all electron densi- A. Gaidarzhy and J. B. Miller for technical assistance
ties [11]. As a result, the crucial region of low electron andC.M.MarcusandP.Mohantyforanopportunityto
densities, in which the critical behavior of the Paulispin use their microfabrication facilities. This work was sup-
susceptibility occurs, falls within the insulating regime ported by NSF grant DMR-0403026, PRF grant 41867-
where the physics of local moments dominates [8, 9, 10]. AC10,the RFBR, RAS, and the Programme“The State
Indeed, Prus et al. have found sub-linear M(B) depen- Support of Leading Scientific Schools”.
[*] Permanent address: Institute of Solid State Physics, dalov, Phys.Rev.B 67, 205407 (2003).
Chernogolovka, Moscow District 142432, Russia. [12] A. A. Shashkin, S. V. Kravchenko, and T. M. Klapwijk,
[†] Presentaddress: DepartmentofPhysicsandAstronomy, Phys. Rev.Lett. 87, 266402 (2001).
UCLA,Los Angeles, CA 90095, U.S.A. [13] To deal with this problem, the “subtraction of the
[1] L. D.Landau, Sov.Phys. JETP 3, 920 (1957). diamagnetic contribution” was suggested in Ref. [11].
[2] E. Wigner, Phys. Rev. 46, 1002 (1934). The diamagnetic contribution ∆ was determined in the
[3] B. Tanatar and D. M. Ceperley, Phys. Rev. B 39, 5005 partially-polarized regime as the difference between the
(1989);G.Benenti,X.Waintal,andJ.-L.Pichard,Phys. direct magnetization data (“Mag”) and the data ob-
Rev. Lett. 83, 1826 (1999); C. Attaccalite, S. Moroni, tained from Shubnikov-de Haas oscillations (“SdH”):
P.Gori-Giorgi, andG.B.Bachelet,Phys.Rev.Lett. 88, ∆ =Mag−SdH. The experimental data were then cor-
256601 (2002). rected by ∆. We find this procedure meaningless as it
[4] H. von L¨ohneysen, Advances in Solid State Physics 30, essentiallyresultsinreplacingthemagnetizationdataby
95 (1990). Shubnikov-de Haas data: Mag− ∆ = Mag −(Mag−
[5] T. Okamoto, K. Hosoya, S. Kawaji, and A. Yagi, Phys. SdH)=SdH. In fact, the difference between magnetiza-
Rev.Lett. 82, 3875 (1999); J. Zhu,H.L.Stormer, L.N. tion and Shubnikov-deHaas data in their experiment is
Pfeiffer, K. W. Baldwin, and K. W. West, Phys. Rev. likelytobeduetothepresenceofabandtailoflocalized
Lett.90, 056805 (2003). electrons at all electron densities in their sample.
[6] S. V. Kravchenko and M. P. Sarachik, Rep. Prog. Phys. [14] Since the diamagnetic shift decreases with increasing ns
67, 1 (2004); A. A. Shashkin, Physics-Uspekhi 48, 129 and/or decreasing B, it may in principle be noticeable
(2005). at the lowest ns and highest B used in the experiment.
[7] A.A.Shashkin,S.V.Kravchenko,V.T.Dolgopolov,and Comparinglow-andhigh-fieldcurvesshowninFig.2(a),
T. M. Klapwijk, Phys. Rev.Lett. 87, 086801 (2001). weestimatethatatns ∼1×1011 cm−2 andB=7T,the
[8] V. T. Dolgopolov and A. Gold, Phys. Rev. Lett. 89, contribution of the diamagnetic shift is less than 0.2µB.
129701 (2002). [15] The critical density for the MIT was determined from
[9] A.GoldandV.T.Dolgopolov,J.Phys.:Condens.Matter transport measurements (see Refs. [6, 12]).
14, 7091 (2002). [16] The fact that Bc →0 as ns →nc speaks in favor of the
[10] N.F.MottandE.A.Davis,ElectronicProcessesinNon- strongly correlated liquid being close to thecrystal [17].
Crystalline Materials (Clarendon Press, Oxford, United [17] B. Castaing and P. Nozi`eres, J. Phys. (Paris) 40, 257
Kingdom, 1971). (1979).
[11] O. Prus, Y. Yaish, M. Reznikov, U. Sivan, and V. Pu-