Variation of the hopping exponent in disordered silicon MOSFETs
T. Ferrus, R. George, C. H. W. Barnes, N. Lumpkin, D. J. Paul, M. Pepper
aa r X i v : . [ c ond - m a t . s t r- e l ] S e p Variation of the hopping exponent in disorderedsilicon MOSFETs
T Ferrus, R George, C H W Barnes, N Lumpkin, D J Paul andM Pepper
Cavendish Laboratory, University of Cambridge, J J Thomson avenue, CB3 0HE,Cambridge, United KingdomE-mail: [email protected]
Abstract.
We observe a complex change in the hopping exponent value from 1/2 to 1/3 as afunction of disorder strength and electron density in a sodium-doped silicon MOSFET.The disorder was varied by applying a gate voltage and thermally drifting the ions todifferent positions in the oxide. The same gate was then used at low temperature tomodify the carrier concentration. Magnetoconductivity measurements are compatiblewith a change in transport mechanisms when either the disorder or the electron densityis modified suggesting a possible transition from a Mott insulator to an Andersoninsulator in these systems.PACS numbers: 71.55 Eg, 71.55 Jv, 72.15 Rn, 72.20 Ee, 73.40 Qv
Submitted to:
J. Phys.: Condens. Matter ariation of the hopping exponent in disordered silicon MOSFETs
1. Introduction
The crossover from Mott to Efros-Shklovskii (ES) variable-range hopping (VRH) hasbeen extensively investigated either to confirm the existence of a Coulomb gap in 2D[1] or to experimentally verify its theoretical formalism [2]. It has only been observed inthe temperature dependence of conductivity of insulating devices close to the metal-to-insulator transition. Effectively, in that region, the crossover temperature, a density ofstate dependent entity is a slowly varying function of the electron density and disorder.This prevents a change in the value of the hopping exponent as a function of the twoprevious parameters from being observed. However, the existence of impurity bandtails in disordered insulators [3, 4, 5] allows such a study to be performed at a fixedtemperature and in the insulating regime. In this paper we show that these experimentalconditions are achieved by using sodium ions in the oxide of a silicon MOSFET. Whilethe gate controls the electron density at low temperature, the in-situ modification ofthe position of the ions in the oxide by the simple application of a gate voltage at roomtemperature allows the disorder strength to be changed. We obtained similar resultsto Pepper on small silicon MOSFETs [6] when the ions are close to the interface anda comparable electronic situation to Ghosh’s GaAs/AlGaAs devices with a fixed silicon δ -doped layer [7] when they are close to the metal gate. The study is extended tointermediate positions by analysing the variation of the source-drain conductivity intemperature and in magnetic field.
2. Experiments
All measurements were performed on identically processed silicon MOSFETs (Fig. 1a).Such devices have been widely used because of the ability to continuously vary theelectron density and thus the Fermi energy by use of a metal gate. The geometry ofthe devices was circular (Corbino) to avoid leakage current paths around the source anddrain contacts. The devices were fabricated using a high resistivity (10 Ω.cm) (100) p -silicon wafer to minimize the scattering with boron acceptor impurities, especially closeto the silicon-oxide interface. A 40- nm thick gate thermal oxide was grown at 950 ◦ C in adry, chlorine-free oxygen atmosphere. Contacts were realized by implanting phosphorousat high dose and sputtering aluminium. Sodium ions were introduced onto the oxidesurface by immersing the device in a 10 − N solution ( ∼ × cm − ) of high-puritysodium chloride in deionized water for 30 s. The surface of the chip was then dried withnitrogen gas and an aluminium gate subsequently evaporated. The effective gate lengthof the primary Corbino MOSFET was 2 µ m and the diameter of the interior contactwas 110 µ m. The Na + ions were drifted through the SiO by applying +4 V dc at 65 ◦ Cfor 10 mins but did not diffuse into silicon[8]. The device was then slowly cooled downto about 300 mK. Once the temperature lower than about 150 K, the ions remain frozen in position allowing consistency in temperature dependence measurements. A series ofdrifts (D i for the i th drift) were performed and allowed modifying the position of the ions ariation of the hopping exponent in disordered silicon MOSFETs ◦ C for 1 h. The undrifted device(D0) were used as a reference and for estimating the ion concentration in the oxide.Standard low-noise lock-in techniques with an amplifier gain of 10 V/A were used tomeasure the source to drain conductivity. The ac excitation amplitude was V ac=10 µ Vat a frequency of 23 Hz. The dc offset of the amplifier was suppressed using a blockingcapacitor. The gate voltage was controlled by a high resolution digital to analogueconverter and the temperature measured by a calibrated germanium thermometer. Allexperiments were performed in an He cryostat where the magnetic field was appliedperpendicular to the Si-SiO interface.
3. Results and discussion
When the ions are not drifted (e.g. close to the metal gate), the variation of the source-drain conductivity with gate voltage is similar to sodium free MOSFETs and, in theregion of study (subthreshold region), the conductivity depends exponentially on gatevoltage. Following a drift of the ions, the onset voltage for conduction is shifted towardsnegative gate voltages indicating the presence of positive charges in the oxide (Fig. 1b).Fluctuations in the conductivity also appear and strengthen as the ions approach theSi-SiO interface. These fluctuations are reproducible both in position and height withtime and with thermal cycling up to 120 K. This result is in agreement with Pepper’sresults on short channel length silicon MOSFETs with low-doped substrate [6]. Theabsence of an impurity band as seen by many authors [3, 5] may indicate the presenceof strong potential fluctuations at the interface and a large oxide charge concentrationoften associated with a decrease in the impurity band visibility [9]. Once at the vicinityof the interface, any further drift becomes inefficient. By neglecting the ion distributionin the oxide, we can estimate the position of the ions for intermediate drifts from theexperimental threshold voltage shift ∆ V t i = V t i − V t = e ( d − h i ) N ox / ( ǫ ǫ ), where N oxis the effective oxide charge density, d the oxide thickness and h i and V t i respectivelythe distance of the ions to the Si-SiO interface and the threshold voltage after driftD i (Fig. 1c). Threshold voltages were determined by the linear extrapolation to zeroof σ (cid:16) V g (cid:17) at 290 mK. Allowing for this approximation, we find an upper bound for N ox ∼ . × cm − for the ion density by supposing that all the ions reached theinterface after drift D5 ( h ∼ d Na ∼ N ox accounts for the active charges (e.g. not neutralized at the interface)and represent about 45 % of the total oxide charge [10]. It is an overestimate of thereal ion concentration because it is plausible that some ions have already reached theSi-SiO boundary before performing drift D5. Subsequent drifts would then change thedistribution of ions at the interface by pulling the remaining ions in the oxide to theinterface. ariation of the hopping exponent in disordered silicon MOSFETs -0.6 -0.4 -0.2 0.2 0.4 0.6 0123456 a) ( - e / h ) (D0)(D2)(D3)(D1)(D4) V g (V)b) (D5)-10 0 10 20 30 40 50-0.2-0.10.00.10.20.30.40.50.60.7 V t i ( V ) h i (nm) AlSiO Si c) Na + p-SiSiO Al GateDrain Source
Figure 1. a) Cross-section view of a Corbino MOSFET used in the experiment afterdrift (D5). b) σ (cid:0) V g (cid:1) for sodium ions close to the metal gate (D0) to ions close tothe Si-SiO interface (D5) at 290 mK. c) Estimate position of the ions in the oxide fordifferent drifts from ∆ V t i . s = 0.50 s = 0.49 s = 0.50 T s ( e / h . K s ) (D0) 0.2 0.4 0.6 0.8 1.0 1.2 s = 0.50 s = 0.46 s = 0.41 s = 0.35 (D2) T -s (K -s ) s = 0.44 s = 0.37 s = 0.33 s = 0.31 (D4) Figure 2. σ ( T ) when the ions were close to the Al-SiO interface (D0) for V g = 0 . after drift D2for V g = 0 . interface after drift D4 for V g = 0 . We studied the influence of the position of the ions in the oxide on the electronicproperties of the device by measuring the temperature dependence of the conductivityafter the different drifts. In all cases, the device showed an insulating behaviour in theregion of study and the conductivity was well fitted to the generalized hopping formulaover several orders of magnitude between 1 K and 20 K typically (Fig. 2) [11]: σ ( T ) = σ T − ps e − ( T s /T ) s (1) ariation of the hopping exponent in disordered silicon MOSFETs -0.4 -0.3 -0.2 -0.1 0.0 0.10.260.280.300.320.340.360.380.400.420.440.460.480.500.52 (D5)(D2) (D3) (D4)(D1)(D0) V t0 = 0.65 V V t1 = 0.62 V V t2 = 0.51 V V t3 = 0.47 V V t4 = 0.35 V V t5 = -0.13 V s V g -V ti (V) Figure 3.
Variation of s (cid:0) V g (cid:1) after after drifts D i . Dotted lines represent the value ofthe hopping exponent in the Mott regime ( s = 1 /
3) and in the ES regime ( s = 1 / s are determined within an error ± . Below 1 K the conductivity is mostly determined by scattering with acousticphonons. The fitting procedure used to find the exponent p and s is a standard one basedon the minimization of the chi-square from the non-Arrhenius plot of ln( σT ps ) versus1/ T s . Best fits are obtained with p = 2 in all cases. Such a value for p has already beenobserved in similar devices [3] as well as in Si:As [12] and theoretically predicted [13].When the ions are close to the metal gate, s ∼ /
2, indicating the presence of ES VRH[1] (Fig. 3). For intermediate oxide positions, we find a smooth change from s ∼ / s ∼ / V g ∼ .
47 V and 0.45 V respectively. This indicates theCoulomb gap progressively diminishes as the electron density is decreased. Effectively,the value of T / experimentally increases when the electron density is decreased, suchthat, at some point, the hopping energy in the ES regime ∼ (cid:16) T T / (cid:17) / becomes greaterthan the Coulomb gap ∼ T / /T / [15, 16]. The ES VRH then changes to Mott VRHfor T > T ∗ ∼ ξN (cid:16) E F (cid:17) . Because of the presence of band tails in our device, boththe density of states at the Fermi energy N (cid:16) E F (cid:17) and the localization length ξ aredecreasing functions of the electron density, indicating that the ES regime is more likelyto be present at high electron density. This result is inconsistent with the screening ofthe Coulomb interaction by metallic gate as previously reported in similar devices [3, 17]and in other 2D systems where the Mott regime is expected at high electron density [18].After the drift D4, the crossover is still present although 0 . < s < .
44. Finally, whenthe ions are close to the Si-SiO interface (drift D5), hopping transport is still active but s (cid:16) V g (cid:17) becomes non-monotonic (0 . < s < .
32) due to the presence of fluctuations inthe conductivity. In this specific case, the conductivity was first averaged over a range δV g ∼
20 mV that was larger than the average fluctuations in gate voltage and centredaround the chosen value for V g. Minimization procedures were then performed. We didnot observe any substantial variation of s in this case but a gate voltage independent ariation of the hopping exponent in disordered silicon MOSFETs s ∼ /
3, indicating non-interacting hopping. This clearly shows thatthe crossover observed when varying V g is progressively disappearing as the ions getcloser to the Si-SiO interface. The presence of impurities in the silicon oxide effectivelycreates disorder at the Si-SiO in the form of potential fluctuations whose amplitudeis proportional to 1 /h i and whose length scale varies as h i [19]. Close to the Si-SiO interface, these fluctuations are short-range and are pinning strongly the electrons at thepotential minima. However, this is the long-range character of the Coulomb interactionthat is responsible of the formation of the soft Coulomb gap in the density of states[20]. Thus, hopping is not-interacting and Mott law prevails in this case. The variationof s with V g shows a notable change, especially with the ratio W/U where W is thedisorder strength and U the on-site energy. The presence of Mott VRH at high disorderstrength is compatible with W > U characteristic of Anderson localization [21]. But,when
W < U , a transition to a Mott insulator is expected [22]. So it is legitimateto investigate the possibility of an Anderson-like transition in this device [23]. In thiscase, we would expect an increase of the localization length at the transition. Still, thedetermination of ξ from σ ( T ) is difficult because of the variation in s with position ofthe ions in the oxide and the dependence of the density of state at the Fermi level with V g. However, studies at constant temperature and in magnetic field allow getting accessto the variation of ξ independently. In all cases, the magneto-conductivity (MC) is negative up to 10 T, as expected forinsulating materials [24, 16] and is well described by the general equationln σ ( B ) ∼ − ( B/B ) γ with l B = (¯ h/eB ) / (2)The use of the exponent γ is a convenient way of assessing the non-linearity of theMC. Its value ranges from 0.6 to 2 depending on the gate voltage range and positionof the ions in the oxide. It is interesting to notice that, at specific values for V g, thequadratic behaviour ( γ = 2) may extend up to 10 T whereas at others, the linear MC( γ = 1) may be present down to 0 T (Fig. 4). The quadratic term is associated with anincrease of the localization due to the shrinkage of the electron wave functions centredat the impurity site as well as interference effects due to backward electron paths [25]. Ithas already been observed in similar devices where the two-dimensional impurity bandwas formed in the inversion layer [26]. It is classically associated with closed orbitsand present as long as the hops between the initial and final states are enclosed in acyclotron orbit, e.g. ξ ≪ l B /r (3)where l B = ¯ h/eB and r is the hopping length.This condition generally applies at low magnetic field but equivalently in a stronglocalized regime. As for the linear term, it has been observed experimentally by many ariation of the hopping exponent in disordered silicon MOSFETs -5 -4 -3 -2 -1 ( e / h ) ~ 1.3 ~ 1.8 ~ 1.9 (D0) 0 2 4 6 8 10 B (T) ~ 1.4 ~ 0.9 ~ 1.4 (D2) 0 2 4 6 8 10 ~ 1.0 ~ 2.1 ~ 0.7 (D4) Figure 4. σ ( B ) at 290 mK for V g=0.7 V, 0.58 V and 0.55 V (from top to bottom)after D0, V g=0.6 V, 0.47 V and 0.33 V (from top to bottom) after D2 and V g=0.46 V,0.27 V and 0.2 V (from top to bottom) after D4. Best fits are shown with dotted lines.Dashed lines indicate the noise level. (D0) (D2) (D3) -0.2 -0.1 0.0 0.19101112131415 l B ( n m ) -0.2 -0.1 0.0 0.1 V g - V t i (V) -0.2 -0.1 0.0 0.1 Figure 5. γ (cid:0) V g (cid:1) and l B (cid:0) V g (cid:1) at 290 mK for D0, D2 and D3. authors [27, 28] and explained by the presence of strong random potential fluctuationsin a two-dimensional electron gas [29, 30] leading to incomplete orbit scattering. Inthe hopping regime, this gives rise to sublinear terms in the MC under appropriateconditions [31]. In our device both phenomena contribute to the MC, and the variationof γ (cid:16) V g , W (cid:17) reflects the relative strength of the two (Fig. 5). Close to the metal-oxide interface, there is a smooth variation from a quasilinear MC for V g > .
60 V to aquadratic MC for V g < .
57 V. At low electron density, Eq. 3 is fulfilled and γ = 2. Athigher electron density, the system is no longer strongly localized and ξ increases suchthat it invalidates Eq. 3. The distance between scattering regions diminishes giving riseto the linear or sublinear terms in the MC. When the ions are close to the silicon-oxideinterface, no clear variation of γ with V g is observed. Interference effects also contributeto the fluctuations in the MC, so that even the use of averaging procedures, as describedpreviously, cannot provide a smooth variation for γ (cid:16) V g (cid:17) , but surprisingly, l B (cid:16) V g (cid:17) is ariation of the hopping exponent in disordered silicon MOSFETs s , as observedfor D0, is not present. Instead a minimum is observed in γ (cid:16) V g (cid:17) in case of drift D2for V g ∼ .
47 V, a value close to the crossover voltage in s (cid:16) V g (cid:17) . A similar but weakerminimum is also present in the case of the drift D3 at V g ∼ .
50 V but none for driftD1. In presence of a magnetic field, single hops occur between impurity sites locatedwithin a cigar-shaped area of dimension r and ( rξ ) / [32]. Because of the dependenceof the hopping length on the localization length, the magnetic flux through the hoppingarea is proportional to ξ α where α >
0. The magnetic field length l B as defined in Eq.2 is thus essentially dependent on the electron density n e- and the localization length[30] so that a non-monotonic variation in l B (cid:16) V g (cid:17) could be attributed to a variation in ξ (cid:16) V g (cid:17) . In the case of drift D0, l B is nearly gate voltage independent indicating thatthe dependence of the hopping length on n e- is weak and so ξ ∼ n / e- [30] whereas ξ is maximum at the crossover in case of drift D2. Such an increase has been associatedto the presence of an impurity band in low-doped devices [3] but in the present device,we did not observe any enhancement of the conductivity. Alternatively, it is possibleto interpret an increase of ξ by the occurrence of a phase transition. Because of thedependencies observed in Figs. 3 and 5, the physical process that is responsible for theseis likely to rely on both the electron-electron interaction and disorder relative strengths[33].
4. Conclusion
We have analysed the electronic properties of sodium-doped silicon MOSFETs bytransport measurements in both temperature and magnetic field. We have observeda complex change in transport mechanism from a Mott hopping to an ES hoppingregime as a function of the position of the ions in the oxide and the electron density. Ingate voltage, the variation of the hopping exponent is attributed to the replenishmentof the Coulomb gap due to the decrease of the localization length with electron density.This is possible because of the presence of a long impurity band tails in the densityof states. This crossover only appears when the ions are within a certain distance tothe silicon-oxide interface. Close to the metal gate, no crossover is observed and ESVRH is present at all sub-threshold gate voltages whereas when the ions are pinnedat the silicon-oxide interface, the transport is governed by the Mott non-interactinghopping. Magnetotransport measurements showed an increase in the localization lengthat the crossover, suggesting the presence of a possible disorder-driven transition in thesesystems, like the Mott-Anderson transition.
Acknowledgement
We would like to thank Drs F. Torregrosa and T. Bouchet from Ion Beam Services-France device processing as well as funding from the U.S. ARDA through U.S. ARO ariation of the hopping exponent in disordered silicon MOSFETs
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