Statistics of the Coulomb blockade peak spacings of a silicon quantum dot F. Simmel1, David Abusch-Magder1,2,∗, D. A. Wharam3, M. A. Kastner2, and J. P. Kotthaus1 (1) Center for NanoScience and Sektion Physik, LMU Mu¨nchen, Geschwister-Scholl-Platz 1, D-80539 Mu¨nchen, 9 (2) Department of Physics, Massachusetts Institute of Technology, Cambridge, MA 02139, 9 (3) Institut fu¨r Angewandte Physik, Universit¨at Tu¨bingen, 9 Auf der Morgenstelle 10, D-72076 Tu¨bingen 1 (February 1, 2008) n a J than expected from RMT, whereas the experiments pre- 6 We present an experimental study of the fluctuations of 2 sentedinRef.[12]yieldsmallerpeakspacingfluctuations, Coulomb blockadepeakpositionsofaquantumdot. Thedot which, however, are still larger than those predicted by is defined by patterning the two-dimensional electron gas of ] RMT. l a silicon MOSFET structure using stacked gates. This per- l The deviations from the RMT predictions have been a mitsvariationofthenumberofelectronsonthequantumdot frequently interpretedas fluctuations in the chargingen- h without significant shape distortion. The ratio of charging - energy to single particle energy is considerably larger than ergy. As the chargingenergy reflects the Coulomb inter- s actions both between the electrons on the dot as well as e in comparable GaAs/AlGaAs quantum dots. The statistical m distributionoftheconductancepeakspacingsintheCoulomb between the dot and its environment, the dependence of the fluctuations on the interaction strength is of funda- blockade regime was found to be unimodal and does not fol- . at low the Wigner surmise. The fluctuations of the spacings mental interest. Numerical studies suggestthat the fluc- m are much larger than the typical single particle level spacing tuations are proportional to the charging energy rather andthusclearlycontradicttheexpectationofrandommatrix than to the single particle level spacing [10,13,14]. This - d theory. is also found theoretically for the classical limit of a n PACS numbers: 73.23.Hk,05.45.+b,73.20.Dx Coulomb glass island [15]. In RPA calculations the fluc- o tuations have been related to fluctuations of the eigen- c functionsofthedot[16]. Anotherapproachbasedonden- [ The spectral properties of many quantum mechanical sityfunctionaltheoryemphasizestheroleoftheCoulomb 1 systems whose classical behavior is known to be chaotic matrix elements of scarredwavefunctions [17]. Recently, v are remarkably well described by the theory of random alsoanon-interactingexplanationfortheGaussianshape 4 matrices (RMT) [1]. This has been experimentally con- of the peak spacing distribution has been given in terms 7 firmed, for example, in measurements of slow neutron of level dynamics due to shape deformation of the quan- 2 resonancesofnuclei[2]andinmicrowavereflectionspec- tum dot [18]. 1 0 tra of billiard shaped cavities [3]. Electron transport ex- Here we present an experimental study of the statis- 9 periments performed on semiconductor quantum dots in tics of Coulomb blockade peak positions of a quan- 9 the Coulomb blockade (CB) regime [4] provide a further tum dot. The dot is defined by patterning the two- / possibility to check RMT predictions. The classical mo- dimensionalelectrongasofasiliconmetaloxidesemicon- t a tionofelectronsinthesestructurescanbeassumedtobe ductor field effect transistor (MOSFET) structure using m chaoticdue to anirregularpotentiallandscape produced stacked gates. These experiments differ significantly in - by impurities, an asymmetric confinement potential [5], twomajorwaysfrompriorexperimentsonquantumdots d and/or electron-electron interactions [6]. The transport definedinGaAs/AlGaAsheterostructures: firstandfore- n propertiesofquantumdotsareinherentlyrelatedtotheir most, due to the different electron density and material o c energyspectraandelectronicwavefunctionsandthusthe propertiesofsilicon,theratioofthechargingenergy,EC, : connection with RMT is readily made [5,7]. to the single particle energy level spacing, ∆ǫ, is consid- v i Indeed,experimentsonthedistributionofconductance erably larger; likewise the dimensionless parameter rs, X peak heights of quantum dots in the Coulomb blockade which characterizes the strength of the Coulomb inter- r regime have shown good agreement with the predictions actions, is larger than in previous studies. Secondly, the a of RMT [8,9]. On the other hand, the distribution of numberofelectronsisvariedbytheapplicationofavolt- the CB peak spacings have been found to deviate from age to a top gate instead of by squeezing the quantum the expectations of RMT [10–12]. The results suggest dot with a plunger gate. We find that the distribution that the peak spacings are not distributed according to of the peak spacings is unimodal and roughly Gaussian. the famous Wigner surmise. Furthermore, there is no Themagnitudeofthefluctuationsis15timeslargerthan indication ofspin degeneracywhich would resultin a bi- that predicted by RMT. modalpeakspacingdistribution[12]. InRefs.[10,11]the Conductionthroughasmallelectronislandcoupledto fluctuations of the peak spacings are considerably larger leadsviatunnelbarriersisnormallysuppressedifkBT ≪ E , where E is the charging energy of the island. This C C 1 effect is known as the Coulomb blockade [4]. The block- lower gate a second layer of silicon dioxide is deposited ade is lifted when the condition µ < µ < µ is satis- (upper oxide), and finally an upper gate is formed; the d dot s fied, where µ , µ , and µ are the chemical potentials upper oxide layer serves to insulate the lower gate from s d dot of the source, drain and dot, respectively. In the linear the upper gate. Application of positive voltages to the responseregime,wheretheseexperimentshavebeenper- upper gate leads to the formation of a two-dimensional formed, µd µs ∆ǫ,kBT. The chemical potential of electron gas (2DEG) at the Si/SiO2 interface; n+ im- | − |≪ thedotisdefinedasµ (N+1)=E(N+1) E(N)where planted regions serve as Ohmic contacts to the 2DEG. dot − E(N) is the total energy of the dot occupied by N elec- Further details about this device may be found else- trons. Inthecasewheretheblockadeisliftedanelectron where [19,20]. The lower gates locally screen the electric can tunnel from the source onto the dot, changing the field created by the upper gate, and a quantum dot is dot’s occupationfromN to N+1, andsequentially tun- formed by applying appropriate negative voltages to the nel off the dot to the drain leaving the dot in its original lower gates. The size of the dot is estimated from the state. The resulting fluctuation of the electron number capacitance to be A 200nm 200nm, which agrees ≈ × onthe dotleads to a finite conductance. Experimentally wellwiththelithographicdimensionsof250nm 270nm × thiscanbeachievedbyappropriatelytuningµ withan when electrostatic depletion at the edge is considered. dot external gate. Sweeping the gate voltage, V , results in The electron density can be varied by changing the up- g the well known conductance oscillations indicating suc- per gate voltage,whereasthe lowergate voltagecontrols cessivefilling ofthe dotwithsingleelectrons. The differ- the tunnel barriersandthe electrostaticconfinement po- ence of µ between two adjacent conductance maxima tential of the quantum dot. This technique allows the dot is thus given by ∆µ =E(N +1) 2E(N)+E(N 1) definition of very small structures which therefore have N − − which can be viewed as the discrete second derivative of low capacitances and high charging energies. For the thequantumdotenergywithrespecttoparticlenumber, quantumdotdiscussedherethese valuesareCΣ 85aF ≈ i.e. the inverse compressibility ∂µ/∂N [13]. andE 1.9meVasobtainedfromtemperaturedepen- C ≈ In the constant interaction (CI) model [4] the energy dence measurements of the conductance resonances [20]. of the dot is approximated as E(N) = (Ne)2/2CΣ + In contrast to previous experiments on quantum dots N ǫ , where the electrostatic interactions are treated inGaAs/AlGaAsheterostructurestheelectrondensityis Pusiin=g1aisimplecapacitivechargingmodelwithatotaldot considerably higher, ns 2.5 1016m−2. The mobility capacitance CΣ, and the quantum mechanical terms are of the two-dimensional e≈lectron· gas is µ = 0.56m2/Vs, taken into account as single particle energies ǫ . In this and the mean free path l 100nm is comparable to the i ≈ modelthedifferenceofthechemicalpotentialsforsucces- system size. The single particle energy level spacing can siveoccupationnumbers,thesocalledadditionenergy,is be obtained from the estimated dot area, A, via Weyl’s ∆µN =EC+∆ǫN withthechargingenergyEC =e2/CΣ, formula [21] as ∆ǫ= g2mπh¯∗A2 =15µeV, where g is the de- andthe levelspacing∆ǫN =ǫN+1 ǫN. This is mapped generacyofelectronicstatesinthe two-dimensionalelec- togatevoltagesviaeCg∆V =E +−∆ǫ whereC isthe tron gas, and m∗ is the effective mass of the electrons. CΣ g C N g ∗ capacitanceofthedottothegateand∆V thedifference In a 2DEG in a silicon MOS system m = 0.2m , and g e betweenthegatevoltagesatwhichadjacentconductance at B = 0 both the spin and valley degeneracies must be maxima occur. considered and therefore g = 4. While these quantum This final expression motivated the original investiga- dots are smaller than many of the GaAs/AlGaAs quan- tions of the peak spacings in the light of random ma- tum dots studied [10–12],∆ǫ is of the same order due to trix theory. RMT shows that the normalized spacings S the larger effective mass, and to the valley degeneracy. ( S =1)betweenadjacenteigenvaluesofagenerictime- The strength of the electronic interactions char- h i reversal invariant Hamiltonian are distributed according acterized by the dimensionless parameter rs = ∗ to the Wigner surmise g/(2√πn a ) = 2.1 considerably exceeds the values ob- s B PW(S)= πSe−π4S2. (1) ata∗iniesdtihnereecffeencttievxepBeroimhrenrtasd(iuwsh.erSeimrsil≈ar1ly),[1th0e–1r2a];tihoeroef 2 B charging energy to single particle energy level spacing Thefluctuationsofthesespacingsare( S2 S 2)1/2 E /∆ǫ 125, another measure of the relative impor- 0.52 S . However, experiments have(cid:10)sh(cid:11)ow−nh tihat th≈e taCnce of≈electron-electron interactions, is larger than in h i combined CI-RMT model is not capable of describing the experiments performed on GaAs/AlGaAs quantum the observedpeak spacing distribution correctly [10–12]. dots. TheCoulombblockademeasurementsonwhichthefol- The measurements were performed in a 3He refriger- lowinganalysis is basedhavebeen performedona quan- ator at a temperature of T = 320mK using standard tumdotdefinedinasiliconMOSFETstructure. Wehave lock-in techniques at low frequencies and bias. The con- utilized a stacked gate structure to pattern the electron ductance oscillations were measured as a function of the gas as shown in Fig. 1. First, a gate oxide is grown upper gate voltage. Consequently, the electron density on a p-type silicon substrate (lower oxide), and then a wasvariedwithoutdrasticallychangingothersystempa- lower metal gate is deposited and patterned using elec- rameters such as charging energy, single particle energy, tronbeamlithographyandlift-offtechniques. Abovethe 2 and dot shape. This also contrasts with former experi- theinfluenceofspindegeneracyontheadditionspectrum ments onthe statistics ofconductance oscillationswhere is washed out for stronger electron-electron interactions the shape of the quantum dot was distorted by plunger (r >1) [14,16]. s gates [9,12]. Inconclusion,wehaveinvestigatedtheCoulombblock- The following analysis is based on a series of more ade peak spacing distribution of a quantum dot fabri- than 100 conductance peaks occurring in the upper gate cated in the 2DEG of a silicon MOSFET structure. In voltage range from 12.1 V to 13.5 V (see Fig. 2 in- accordancewithexperimentsonGaAs/AlGaAsquantum set). In this range the electron density changes from dotsthedistributiondiffersfromtheWignersurmiseand 2.4 1016m−2 to 2.6 1016m−2. The quantum point is roughlyGaussian. The fluctuations areapproximately × × contacts connecting the quantum dot to the leads are 0.06 E . Due to the large ratio of chargingenergyE C C × tuned into the tunneling regime by applying voltages of to single particle energy ∆ǫ this strongly suggests that 4.5V and 8.0V to the left and right pair of lower the fluctuations scale with E and not with ∆ǫ. This C − − gates,respectively. Thepositionofeachpeakisobtained clearlycontradictsthe predictionsofRMT andindicates by fitting the peak by a thermally broadened line shape that the fluctuations are dominated by electron-electron −2 cosh (eα(V V0)/2kBT)[4]where αandV0 serveas interactions in this system. ∝ − the fit parameters. The gate voltage spacings ∆V are *newaddress: BellLaboratories,LucentTechnologies, g calculated from the peak positions. 600 Mountain Ave., Murray Hill NJ 07974 The mean value ∆V of the spacings as a func- g h i tion of V is obtained from a linear fit as ∆V (V ) g g g 12.2 3.5·10−2 (V 12.5V) mV. The smallness o≈f ACKNOWLEDGMENTS h − V × g − i the slope ofthis fit showsthatthe influence ofthe upper TheauthorswouldliketothankthestaffoftheMicro- gate on the capacitance and therefore on the size of the electronicsLabatLincolnLaboratoryfortheirhelpwith dot is rather weak. Accordingly, the shape deformation device fabrication, and U. Sivan, M. Stopa, H. Baranger which has been postulated to explain the distribution of for fruitful discussions. This work was supported by the ∆V [18]playsnosignificantroleinthisexperiment. The g Alexander-von-Humboldt Foundation, the DFG Sonder- normalized peak spacings forschungsbereich348, and by the Army Research Office ∆V ∆V undercontractDAAH04-94-0199andcontractDAAH04- g g δ = −h i (2) 95-0038. Work at Lincoln Laboratory was sponsored by ∆V g h i the U.S. Air Force. DA-M gratefully acknowledges sup- are displayed in Fig. 2. The fluctuations of δ are com- portfromtheAlexander-von-HumboldtFoundation,and puted to be δ2 1/2 0.06. The fluctuations in the ad- the Bell Labs Foundation. (cid:10) (cid:11) ≈ dition energy are, therefore, roughly 115µeV, which is 7.5 times the mean level spacing ∆ǫ, and thus 15 times larger than the fluctuations expected from RMT. This supports the view that the fluctuations of the addition energy scale with the Coulomb energy rather than with the kinetic energy. However, the proportionality factor [1] M. L. Mehta, Random Matrices, 2nd ed. (Academic 0.06 is smaller than that suggested by numerical calcu- Press, London, 1991). lations (0.1 - 0.2) [14]. It should be noted that in these [2] R.U.Haq,A.Pandey,andO.Bohigas, Phys.Rev.Lett. experiments ∆ǫ kBT. We expect that the effect of 48, 1086 (1982). ≈ thermal broadening would be to reduce the fluctuations [3] H.D.Gr¨af,H.L.Harney,H.Lengeler,C.H.Lewenkopf, in peak spacing. A simple model [12] predicts that the C. Rangacharyulu, A. Richter, P. Schardt, and H. A. fluctuations expected within RMT would be reduced by Weidenmu¨ller, Phys. Rev.Lett. 69, 1296 (1992). a factor of 2 – 3. If we incorporate this correction into [4] There are several recommendable review articles about RMTthenthefluctuationswefindinourexperimentare the Coulomb blockade phenomenon in semiconductor 30 – 45 times larger than those predicted by RMT. quantum dots, e.g. H. van Houten, C. W. J. Beenakker, The distribution of the peak spacings normalized to andA.A.M.StaringinSingle Charge Tunneling,edited an area of unity is shown in Fig. 3. The distribution is by H. Grabert, J. M. Martinis, and M. H. Devoret unimodalandroughlyhastheshapeofaGaussian. Inthe (Plenum,NewYork,1991);U.MeiravandE.B.Foxman, inset of Fig. 3 the experimental distribution is depicted Semicond. Sci. Technol. 10, 255 (1995); L. P. Kouwen- hoven, C. M. Marcus, P. L. McEuen, S. Tarucha, R. M. togetherwiththeWignersurmise(Eq.1);forcomparison Westervelt, and N. S. Wingreen, in Mesoscopic Electron toδ wehaverescaledthepredictionsofRMTtakinginto Transport, edited by L. L. Sohn, L. P. Kouwenhoven, account the experimental values of E and ∆ǫ. As in C and G. Sch¨on, NATO ASI Series E (Kluwer, Dordrecht, previous experiments there is no evidence of a bimodal 1997). additionspectrumasispredictedbytheCI-RMTmodel. [5] R. A. J. Jalabert, A. D. Stone, and Y. Alhassid, Phys. This is in agreementwith the theoreticalpredictionthat Rev. Lett.68, 3468 (1992). 3 [6] L.Meza-MontesandS.E.Ulloa,Phys.Rev.E55,R6319 FIG.3. Histogram showingthedistributionofthenormal- ized peak spacings from Fig. 2. The area of the histogram (1997). [7] C. W. J. Beenakker, Rev.Mod. Phys. 69, 731 (1997). is normalized to unity. A Gaussian fit with standard devi- ation of σ = 0.06 is also displayed. The inset shows the [8] A.M.Chang,H.U.Baranger,L.N.Pfeiffer,K.W.West, and T. Y. Chang, Phys. Rev.Lett. 76, 1695 (1996). same histogram alongside the Wigner surmise, the distribu- tion predicted by RMT, for ∆ǫ = 15µeV. The experimental [9] J. A. Folk, S. R. Patel, S. F. Godijn, A. G. Huibers, S. distribution is muchbroader than expected from RMT. M. Cronenwett, C. M. Marcus, K. Chapman, and A. C. Gossard, Phys.Rev. Lett. 76, 1699 (1996). [10] U. Sivan,R. Berkovits, Y. Aloni, O. Prus, A. Auerbach, and G. Ben-Yoseph,Phys. Rev.Lett. 77, 1123 (1996). [11] F. Simmel, T. Heinzel, and D. A. Wharam, Europhys. Lett.38, 123 (1997). [12] S. R. Patel, S. M. Cronenwett, D. R. Stewart, A. G. Huibers, C. M. Marcus, C. I. Duru¨oz, J. S. Harris Jr., K.Champman,andA.C.Gossard, Phys.Rev.Lett. 80, 4522 (1998). [13] R.BerkovitsandB.L.Altshuler,Phys.Rev.B55, 5297 (1997). [14] R.Berkovits, Phys. Rev.Lett. 81, 2128 (1998). [15] A.A.Koulakov,F.G.Pikus,andB.I.Shklovskii,Phys. Rev.B 55, 9223 (1997). [16] Y. M. Blanter, A. D. Mirlin, and B. A. Muzykantskii, Phys.Rev.Lett. 78, 2449 (1997). [17] M. Stopa, Physica B 249-251, 228 (1998). [18] R. O. Vallejos, C. H. Lewenkopf, and E. R. Mucciolo, Phys.Rev.Lett. 80, 677 (1998). [19] David Abusch-Magder, M.A. Kastner, C.L. Dennis, W.F. DiNatale, T.M. Lyszczarz, D.C. Shaver, and P.M. Mankiewich, in Quantum Transport in Semiconductor Submicron Structures, edited by B. Kramer, NATO ASI Series E 326 (Kluwer,Dordrecht, 1996), p. 251. [20] David Abusch-Magder, Ph.D. thesis, Massachusetts In- stituteof Technology, 1997. [21] H. P. Baltes and E. R. Hilf, Spectra of Finite Systems: A Review of Weyl’s Problem, Bibliographisches Institut, Zu¨rich (1976). FIG.1. Schematic representation of the device design. A crosssectionofthedeviceisshownin(a). Twooxideandtwo gatelayersareformedontopofp-typesiliconsubstrate. The lower oxidelayer hasa thicknessof roughly 20nm, while the upperoxideisapproximately80nmthick. Thevoltageonthe upper gate is used to vary the electron density in the 2DEG inducedat theinterface of thelower oxideandthesilicon. A top view of the device is shown in (b). The pattern in the lower gates defines a quantum dot in the induced electrons; notethattheuppergatecoversalloftheareashowin(b)and overlaps the source and drain. The lithographic dimensions of the quantumdot are 250 nm× 270 nm. FIG.2. The normalized peak spacings δ obtained from an uppergatevoltage sweep. Thefluctuationsaround themean value 0 are much larger than expected from the CI-RMT model. The inset shows Coulomb blockade conductance os- cillations as a function of theuppergate voltage. 4 (a) upper gate upper oxide layer lower gate lower oxide layer 2DEG silicon (b) Ohmic contact electron 2DEG island drain source lower gate 0.2 0.1 0.0 -0.1 1.5 ) h d -0.2 2/ 1.0 e 3 - 0 0.5 -0.3 1 ( g 0.0 12.50 12.55 12.60 12.65 -0.4 V (V) g 12.2 12.4 12.6 12.8 13.0 13.2 13.4 V (V) g 12 100 75 10 ) 50 d( P 8 25 ) d( 0 P 6 -0.2 -0.1 0.0 0.1 0.2 d 4 2 0 -0.2 -0.1 0.0 0.1 0.2 0.3 d