Letter pubs.acs.org/NanoLett
Gate-Induced Carrier Delocalization in Quantum Dot Field Effect Transistors Michael E. Turk,† Ji-Hyuk Choi,§ Soong Ju Oh,§ Aaron T. Fafarman,‡ Benjamin T. Diroll,∥ Christopher B. Murray,∥,§ Cherie R. Kagan,‡,§,∥ and James M. Kikkawa*,† †
Department of Physics and Astronomy, ‡Department of Electrical and Systems Engineering, §Department of Materials Science and Engineering, and ∥Department of Chemistry, University of Pennsylvania, Philadelphia, Pennsylvania 19104, United States S Supporting Information *
ABSTRACT: We study gate-controlled, low-temperature resistance and magnetotransport in indium-doped CdSe quantum dot field effect transistors. We show that using the gate to accumulate electrons in the quantum dot channel increases the “localization product” (localization length times dielectric constant) describing transport at the Fermi level, as expected for Fermi level changes near a mobility edge. Our measurements suggest that the localization length increases to significantly greater than the quantum dot diameter. KEYWORDS: Quantum dots, field-effect transistor, cadmium selenide, delocalization, magnetoresistance, mobility edge
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diameter in response to a gate voltage. Magnetoresistance (MR) measurements reveal an MR component whose sign, magnitude, and temperature dependence agree with that implied by the Anderson model after the Zeeman spin energy is included within a VRH framework. Details of our device preparation are described elsewhere.9,16 Briefly, highly monodisperse, 4 nm diameter CdSe QDs capped with long-chain aliphatic ligands are synthesized, and these ligands are exchanged for compact SCN (thiocyanate) ligands, allowing dispersion in polar solvents. Thin films are prepared by spin-casting SCN-exchanged CdSe QDs dispersed in dimethylformamide (DMF) onto Al2O3 (20 nm)/SiO2 (250 nm)/n+-doped Si substrates, which serve as the dielectric stack, back gate, and support for the device. In/Au electrodes are then thermally deposited on the QD films to define a 40 μm channel length and a 600 μm channel width. Thermal annealing at 250 °C for 10 min promotes indium diffusion and doping and decomposes the SCN ligands, thereby further reducing QD separation.9 A 40 nm thick layer of Al2O3 is evaporated via atomic layer deposition (ALD) to render the devices air stable.16,17 Devices prepared this way have typical room temperature field effect electron mobilities of ∼27 cm2/(V s), and for the present study the characteristic room temperature mobility is 8−12 cm2/(V s) (see Supporting Information). After electrical connections are made, the sample is then transferred into the bore of a 9 T Quantum Design Physical Property Measurement System with the magnetic field B aligned with the sample normal. Temperature-dependent resistance and MR measurements are collected in a ∼ 5 Torr
olloidal semiconductor quantum dots (QDs) are of broad interest due to the composition, size, and shape tunability of their optical and electronic properties.1,2 They are attractive as building blocks for solids whose aggregate function is tailored by manipulating not only the attributes of individual quantum dots, but also their mutual interactions. As such, QD solids have been used to realize solar cells, photodetectors, light-emitting devices, and transistors and circuits.3−7 In all of these examples, charge transport is important, and recent advances have led to high carrier mobilities in QD solids through a combination of strong coupling (via short ligands8−10 and/or thermal annealing8,9,11) and doping to shift the Fermi level (by introducing impurities9 or exploiting nonstoichiometry12). The high mobility in these samples has been discussed in the framework of an Anderson mobility edge,13 ε0, which lies above the Fermi level, εF, and separates low-energy, localized states from higher energy, delocalized, bandlike states.9 Experiment and theory have suggested that such delocalized states may then be accessed through thermal9 or optical9,14 excitation, the latter proceeding through an Auger process of QD ionization.14 In this Letter, we use low-temperature electron transport measurements to study the onset and gate dependence of charge delocalization below the mobility edge in high mobility QD field effect transistors (FETs). By working in the lowtemperature limit (kT much less than the mobility gap, Δε ≡ ε0 − εF) all states participating in conduction lie within the mobility gap and are thus nonextended. We are therefore able to use changes in variable range hopping15 (VRH) to detect enhanced electron delocalization as the Fermi energy is increased, in accord with the presence of an Anderson mobility gap.13 Under any reasonable assumptions, the electron localization length increases significantly beyond the particle © 2014 American Chemical Society
Received: August 1, 2014 Revised: August 26, 2014 Published: August 29, 2014 5948
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Coulomb gap and Arrhenius behavior. One would then expect such dramatic responses to gate voltage to represent a redistribution between Coulomb gap and Arrhenius contributions and thus to produce a systematic variation in p, which we do not observe. We are also careful to ensure that the fitted values of p are not distorted by electric field activation.27,28 To this end, we measure the drain−source (VDS) bias dependence of the reduced activation energy for all gate voltages. Supporting Information Figure S3 shows the most severe VDS changes. The signature of metallicity, limT→0 [(d ln W)/(d ln T)] > 0, is seen for VDS > 1 V. Reducing VDS shows this behavior is entirely due to field activation, not a metal insulator transition in our sample, and that limiting behavior in the slope of W, which determines p, is obtained for VDS = 0.1 V. The data shown in Figure 1 are taken under these conditions and cut off when the current drops below 1 × 10−11 A, at least an order of magnitude above the gate leakage current and instrumental resolution. Hopping conduction can be connected to the presence of a mobility gap because VRH depends on both the localization length, a, and the dielectric constant, κ, and these quantities should both increase as Δε decreases. Algebraically, one expects29,30 a ∝ (Δε)−β and κ ∝ (Δε)−δ, where β and δ are critical exponents of order unity.13,31 Since gate bias adjusts εF and thus Δε, the predictions of the Anderson model can be tested. In particular, the slope changes we observe for different VG in Figure 1b arise from changes in T0, which in turn depends on aκ. We denote the latter the “localization product,” and for the p = 2/3 model of Coulomb gap plus thermal broadening, T0 = (3e2√3)/[8πaκϵ0(CVkB)1/2] with e the electron charge, ϵ0 the permittivity of free space, kB the Boltzmann constant, and CV = 3kB the heat capacity of the conduction electron.25 The black squares in Figure 2a show the
helium gas atmosphere. Care is taken to ensure that MR of the Cernox temperature sensor is not convolved with sample MR at temperatures where |dR/dT| is very large. Samples show a temperature- and magnetic field-dependent resistance settling time when modifying VG and, to a lesser extent, VDS. These effects are mitigated by changing VG and VDS at or above 60 K and pausing to settle the sample resistance prior to lowtemperature MR measurements. Bidirectional sweeps of B are binned and averaged to produce all individual MR curves, and no hysteresis is seen in the forward and reverse field sweeps. Temperature-dependent resistance measurements for various gate voltages VG are shown in Figure 1a. All curves demonstrate
Figure 1. (a) Low-temperature resistance, R, of a QD FET for different gate biases, VG, as a function of temperature, T. (b) Resistance for different VG plotted versus 1/T2/3. Data collected with VDS = 0.1 V.
insulating behavior obeying the VRH law, R(T) = R0 exp(T0/ T)p. We note that for zero gate bias the sample resistance exceeds our instrumentation limit below 60 K and is thus not shown. The hopping exponent p and the physical interpretation of the characteristic temperature T0 depend on the transport model. For example, Mott VRH assumes a constant density of states near εF in d dimensions, yielding p = 1/(d + 1).15 Several authors have reported p = 1/4 in QD solids, 18−20 corresponding to 3D Mott VRH. Coulomb gap VRH is described by a vanishing density of states at the Fermi level with a quadratic dispersion in three dimensions, producing p = 1/2.18,21−24 More recently, p = 2/3 has been observed and associated with a Coulomb gap and thermal broadening of individual energy levels.25 Our data show a compellingly linear relationship between log(R) and T−2/3 for all gate biases (Figure 1b). For comparison, clearly nonlinear curves are formed by assuming p = 1/2 instead of 2/3 in a similar plot (Supporting Information, Figure S2a). A more objective procedure for determining p for each gate bias is to plot the reduced activation energy W = −[(d ln R)/(d ln T)] and then extract numerical fits for p and T0. Such analysis is described in the Supporting Information and shown in Figure S2b and yields values of p equal to 0.63, 0.66, 0.68, 0.69, and 0.68 for VG = 20, 29, 37.1, 50, and 65 V, respectively. It has been suggested that p = 2/3 could be observed as an admixture of p = 1/2 and p = 1 in an inhomogeneous sample.26 We have some opportunity to test the relevance of this hypothesis to our samples because gate voltage adjustments have a dramatic influence on our measured resistances all the way down to the lowest measurable temperatures. Over a range of VG from 20 to 65 V, the low-temperature resistance drops by more than 3 orders of magnitude, yet p remains constant. This behavior seems difficult to reconcile with a picture in which the VRH exponent is due to a competition between, for example,
Figure 2. Gate bias dependence of (a) localization product, κa, plotted versus gate bias, VG, for p = 2/3 (black filled squares) and p = 1/2 (blue open circles), as described in the text. (b) Gate bias dependence of localization lengths for κ = 3, 6, and 9 as indicated. Values in (b) computed for p = 2/3.
localization product extracted from this equation and its dependence on VG. A positive VG increases the electron concentration, thereby raising εF and decreasing Δε. As VG is applied, a strong response is seen with the correct sign; the localization product increases as VG (Δε) increases (decreases). In addition, Figure 2b shows the dependence of a on VG implied by assuming constant values of κ within a range expected for this system (see ref 32 and ellipsometry measurements in the Supporting Information, Figure S4). A visual extrapolation of a toward the particle diameter at zero bias appears reasonable for κ somewhere between 3 and 6. As noted above, changes in both a and κ may be expected as the 5949
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(T ≥ 20 K). Accordingly, for a given VG we average the data from 20, 40, and 60 K to obtain this univeral line shape (Figure 4b, red curve) and find that its amplitude at those three
gate bias is applied, so precise determination of a is not possible from this data. That being said, the range of κ implied if the localization length is held fixed to the particle diameter is quite large and physically hard to justify without a concomitant change in a (Supporting Information, Figure S5). Thus, our data suggest that gate bias expands the localization length beyond the particle diameter. We emphasize that our data strongly support the assignment of p = 2/3. Nevertheless, for comparative purposes we also compute the localization product for the Coulomb gap model according to T 0 = (2.8e 2 )/(4πaκϵ 0 kB ),33 where T 0 is determined self-consistently from (poor) fits of our data to p = 1/2. The result is shown in Figure 2a and Supporting Information, Figure S5, and shows that our qualitative conclusions are not model-dependent. Magnetoresistance, defined numerically as Δρ/ρ0 [R(B) − R(B = 0)]/[R(B = 0)], generally provides additional insight into transport mechnisms at low temperature. As shown by representative MR curves in Figure 3, the QD FET displays
Figure 3. (a,b) Temperature and gate bias dependence of magnetoresistance of a QD FET at VG = 50 V and T = 7.5 K, respectively. Plots are vertically offset for clarity. Data collected at VDS = 1 V to improve signal-to-noise. No VDS dependence of these large-field features was observed up to VDS = 10 V.
both positive and negative MR contributions whose magnitudes depend on both temperature (Figure 3a) and gate bias (Figure 3b). A more comprehensive cross-section of the data appears in Supporting Information, Figure S6. We note that a low-field (|B| ∼ 100 mT) positive MR component is barely visible in the data presented here but strengthens with high drain−source bias. This latter feature and its VDS dependence have been extensively discussed elsewhere and assigned to random nuclear field mixing of spin singlets and triplets.34 Here we focus on the more dominant features seen across entire field range. The list of mechanisms that can produce positive and negative MR, even in nominally nonmagnetic systems such as these, is extensive. Negative MR can arise from weak localization35,36 and Zeeman modulation of the mobility gap.37 Positive MR can arise from wave function shrinkage,33 electron spin blockade,34,38 and classical disorder induced magnetoresistance.39 Additionally, quantum corrections in the hopping regime can produce both positive and negative MR.40,41 The hopping nature of conduction in these samples makes weak localization physically inappropriate, and the spin blockade mechanism is that associated with the weak, low-field positive MR signal discussed above. To help filter the remaining possibilities, we now show that the MR data can be deconstructed as the superposition of two relatively simple line shapes with inverse power law temperature dependences. We first note that we observe a universal line shape describing negative MR for higher temperatures
Figure 4. (a) Magnetoresistance of a QD FET at VG = 50 V and T = 3 K. (b) Decomposition of (a) into negative (red) and positive (blue) components as described in text. Green shows a parabolic fit to the positive component. (c) Exponents of temperature dependence, x1 (red ×) and x2 (blue +), describing power law temperature dependence, MR ∝ T−x, for negative and positive components of the MR. Point sizes indicate error bars. (d) Mobility gap Δε extracted using p = 2/3.
temperatures obeys a scaling law T −x1. The values of x1 are plotted in Figure 4c (red ×). This temperature dependence is then extrapolated to lower temperatures and subtracted from the MR data (Figures 4a,b), yielding a remaining positive MR contribution (Figure 4b, blue curve) with a predominantly parabolic form. Remarkably, the temperature dependence of the positive component is T −x2 with x2 very close to 2.0 (Figure 4c, blue +). Thus, our MR data set is well-described by (1) a negative MR contribution with a characteristic line shape shown in Figure 4b (red curve) that has a temperature dependence T −x1 and (2) a positive, parabolic MR contribution proportional to T −2. The temperature dependence of (2) can be compared to theoretical predictions for the positive MR mechanisms described above. The associated exponent x2 = −2 is wellreceived by all mechanisms with known temperature 5950
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predictions. Quantum corrections in VRH40,41 and wave function shrinkage in the presence of a transverse field33 both scale as40,41 T −3p = T −2, and spin blockade scales as34 T −2. We note that the latter is conventionally associated with nuclear spin disorder, and saturates at very low fields, whereas (2) does not. Classical disorder induced MR does not have a clear temperature prediction for our system, but its characteristic signature is a transition to a linear behavior at high fields which we do not observe here.42 The temperature dependence of (1) is more telling. As we said, weak localization is inapplicable. Quantum corrections to hopping interference are mostly negative but, as discussed above, predict T −2, which is far from what we observe here. Zeeman energy modulation of the mobility gap predicts37 2
Δρ 1 = − (p(β + δ)d)2 ρ0 2
( 12 gμBB) Δε
2
⎛ T0 ⎞2p ⎜ ⎟ ⎝T ⎠
dielectric constant measurements and calculations, and MR curves for all VG and T used in calculations. This material is available free of charge via the Internet at http://pubs.acs.org.
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Corresponding Author
*E-mail: [email protected]. Notes
The authors declare no competing financial interest.
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ACKNOWLEDGMENTS All aspects of this work supported by the U.S. Department of Energy Office of Basic Energy Sciences, Division of Materials Science and Engineering, under Award No. DE-SC0002158.
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REFERENCES
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(1)
where g ∼ 1.2 is the electron g-factor for CdSe,43 μB is the Bohr magneton, d is the dimensionality of the system, and β, δ and Δε are defined above. Note that here we modify the expression found in ref 37 to include the divergence of the dielectric constant, κ. The associated temperature dependence for p = 2/3 is T −1.33, which is shown as a dotted line in Figure 4c and for which the values of x1 show relatively good agreement. Thus, Zeeman modulation of the mobility gap fits the negative MR signal best, and in many ways it is a magnetic counterpart to the application of a gate bias. In contrast to a gate bias, however, the exact variation in the gap is known from the g-factor. Thus, eq 1 gives a value for the mobility gap, which we can obtain for each value of VG. Figure 4d shows the inferred mobility gap taking β ∼ 1 and δ ∼ 1,13,31 setting d = 3, and using p = 2/3. Values of T0 are those collected synchronously with MR data. With increasing gate bias the calculated mobility gap clearly decreases, and the trend of the gate bias dependence indicates that for sufficiently high values of VG the devices may transition to a metallic state characterized by delocalized states. In conclusion, by combining the analysis of the temperaturedependent resistance and MR, a cohesive picture of electrical transport in these QD FETs emerges. The low-temperature transport studied in this report is consistent with a Coulomb gapped density of states with thermal broadening in an Anderson localized insulator. The negative MR feature is consistent with that expected from Zeeman splitting in the presence of Anderson localization. The data provide strong evidence that gate bias delocalizes electrons in this system beyond the particle diameter, a behavior consistent with the existence of a mobility gap. Magnetoresistance shows both positive and negative components, with the latter bearing a temperature dependence and magnitude also consistent with a mobility gap that acquires a magnetic field dependence through the spin Zeeman energy. These data provide compelling motivation to pursue high gate biases in an effort to observe an insulator−metal transition in these systems.
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AUTHOR INFORMATION
ASSOCIATED CONTENT
S Supporting Information *
Details regarding room temperature mobility measurements, calculation of parameters in the Coulomb gap models, reduced activation energy W, R vs T−1/2, R and W vs T for varied VDS, 5951
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(28) Apsley, N.; Hughes, H. P. Philos. Mag. 1975, 31, 1327−1339. (29) Abrahams, E.; Anderson, P. W.; Licciardello, D. C.; Ramakrishnan, T. V. Phys. Rev. Lett. 1979, 42, 673−676. (30) Kawabata, A. Prog. Theor. Phys. Suppl. 1985, 84, 16−46. (31) Hess, H. F.; DeConde, K.; Rosenbaum, T. F.; Thomas, G. A. Phys. Rev. B 1982, 25, 5578−5580. (32) Wang, L.-W.; Zunger, A. Phys. Rev. B 1996, 53, 9579−9582. (33) Shklovskii, B. I.; Efros, A. L. Electronic Properties of Doped Semiconductors; Springer-Verlag: Berlin, 1984. (34) Guyot-Sionnest, P.; Yu, D.; Jiang, P.-h.; Kang, W. J. Chem. Phys. 2007, 127, 014702. (35) Bergmann, G. Phys. Rep. 1984, 107, 1−58. (36) Kawabata, A. Solid State Commun. 1980, 34, 431−432. (37) Fukuyama, H.; Yoshida, K. J. Phys. Soc. Jpn. 1979, 46, 102−105. (38) di Vittorio, S. L.; Dresselhaus, M. S.; Endo, M.; Nakajima, T. Phys. Rev. B 1991, 43, 12304−12315. (39) Parish, M. M.; Littlewood, P. B. Nature 2003, 426, 162−165. (40) Sivan, U.; Entin-Wohlman, O.; Imry, Y. Phys. Rev. Lett. 1988, 60, 1566−1569. (41) Nguen, V. L.; Spivak, B. Z.; Shklovskii, B. I. J. Exp. Theor. Phys. 1985, 62, 1021. (42) Parish, M. M.; Littlewood, P. B. Phys. Rev. B 2005, 72, 094417. (43) Gupta, J. A.; Awschalom, D. D.; Efros, A. L.; Rodina, A. V. Phys. Rev. B 2002, 66, 125307.
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Gate-Induced Carrier Delocalization in Quantum Dot Field Effect Transistors Michael E. Turk,† Ji-Hyuk Choi,§ Soong Ju Oh,§ Aaron T. Fafarman,‡ Benjamin T. Diroll,∥ Christopher B. Murray,∥,§ Cherie R. Kagan,‡,§,∥ and James M. Kikkawa*,† †
Department of Physics and Astronomy, ‡Department of Electrical and Systems Engineering, §Department of Materials Science and Engineering, and ∥Department of Chemistry, University of Pennsylvania, Philadelphia, Pennsylvania 19104, United States S Supporting Information *
ABSTRACT: We study gate-controlled, low-temperature resistance and magnetotransport in indium-doped CdSe quantum dot field effect transistors. We show that using the gate to accumulate electrons in the quantum dot channel increases the “localization product” (localization length times dielectric constant) describing transport at the Fermi level, as expected for Fermi level changes near a mobility edge. Our measurements suggest that the localization length increases to significantly greater than the quantum dot diameter. KEYWORDS: Quantum dots, field-effect transistor, cadmium selenide, delocalization, magnetoresistance, mobility edge
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diameter in response to a gate voltage. Magnetoresistance (MR) measurements reveal an MR component whose sign, magnitude, and temperature dependence agree with that implied by the Anderson model after the Zeeman spin energy is included within a VRH framework. Details of our device preparation are described elsewhere.9,16 Briefly, highly monodisperse, 4 nm diameter CdSe QDs capped with long-chain aliphatic ligands are synthesized, and these ligands are exchanged for compact SCN (thiocyanate) ligands, allowing dispersion in polar solvents. Thin films are prepared by spin-casting SCN-exchanged CdSe QDs dispersed in dimethylformamide (DMF) onto Al2O3 (20 nm)/SiO2 (250 nm)/n+-doped Si substrates, which serve as the dielectric stack, back gate, and support for the device. In/Au electrodes are then thermally deposited on the QD films to define a 40 μm channel length and a 600 μm channel width. Thermal annealing at 250 °C for 10 min promotes indium diffusion and doping and decomposes the SCN ligands, thereby further reducing QD separation.9 A 40 nm thick layer of Al2O3 is evaporated via atomic layer deposition (ALD) to render the devices air stable.16,17 Devices prepared this way have typical room temperature field effect electron mobilities of ∼27 cm2/(V s), and for the present study the characteristic room temperature mobility is 8−12 cm2/(V s) (see Supporting Information). After electrical connections are made, the sample is then transferred into the bore of a 9 T Quantum Design Physical Property Measurement System with the magnetic field B aligned with the sample normal. Temperature-dependent resistance and MR measurements are collected in a ∼ 5 Torr
olloidal semiconductor quantum dots (QDs) are of broad interest due to the composition, size, and shape tunability of their optical and electronic properties.1,2 They are attractive as building blocks for solids whose aggregate function is tailored by manipulating not only the attributes of individual quantum dots, but also their mutual interactions. As such, QD solids have been used to realize solar cells, photodetectors, light-emitting devices, and transistors and circuits.3−7 In all of these examples, charge transport is important, and recent advances have led to high carrier mobilities in QD solids through a combination of strong coupling (via short ligands8−10 and/or thermal annealing8,9,11) and doping to shift the Fermi level (by introducing impurities9 or exploiting nonstoichiometry12). The high mobility in these samples has been discussed in the framework of an Anderson mobility edge,13 ε0, which lies above the Fermi level, εF, and separates low-energy, localized states from higher energy, delocalized, bandlike states.9 Experiment and theory have suggested that such delocalized states may then be accessed through thermal9 or optical9,14 excitation, the latter proceeding through an Auger process of QD ionization.14 In this Letter, we use low-temperature electron transport measurements to study the onset and gate dependence of charge delocalization below the mobility edge in high mobility QD field effect transistors (FETs). By working in the lowtemperature limit (kT much less than the mobility gap, Δε ≡ ε0 − εF) all states participating in conduction lie within the mobility gap and are thus nonextended. We are therefore able to use changes in variable range hopping15 (VRH) to detect enhanced electron delocalization as the Fermi energy is increased, in accord with the presence of an Anderson mobility gap.13 Under any reasonable assumptions, the electron localization length increases significantly beyond the particle © 2014 American Chemical Society
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Coulomb gap and Arrhenius behavior. One would then expect such dramatic responses to gate voltage to represent a redistribution between Coulomb gap and Arrhenius contributions and thus to produce a systematic variation in p, which we do not observe. We are also careful to ensure that the fitted values of p are not distorted by electric field activation.27,28 To this end, we measure the drain−source (VDS) bias dependence of the reduced activation energy for all gate voltages. Supporting Information Figure S3 shows the most severe VDS changes. The signature of metallicity, limT→0 [(d ln W)/(d ln T)] > 0, is seen for VDS > 1 V. Reducing VDS shows this behavior is entirely due to field activation, not a metal insulator transition in our sample, and that limiting behavior in the slope of W, which determines p, is obtained for VDS = 0.1 V. The data shown in Figure 1 are taken under these conditions and cut off when the current drops below 1 × 10−11 A, at least an order of magnitude above the gate leakage current and instrumental resolution. Hopping conduction can be connected to the presence of a mobility gap because VRH depends on both the localization length, a, and the dielectric constant, κ, and these quantities should both increase as Δε decreases. Algebraically, one expects29,30 a ∝ (Δε)−β and κ ∝ (Δε)−δ, where β and δ are critical exponents of order unity.13,31 Since gate bias adjusts εF and thus Δε, the predictions of the Anderson model can be tested. In particular, the slope changes we observe for different VG in Figure 1b arise from changes in T0, which in turn depends on aκ. We denote the latter the “localization product,” and for the p = 2/3 model of Coulomb gap plus thermal broadening, T0 = (3e2√3)/[8πaκϵ0(CVkB)1/2] with e the electron charge, ϵ0 the permittivity of free space, kB the Boltzmann constant, and CV = 3kB the heat capacity of the conduction electron.25 The black squares in Figure 2a show the
helium gas atmosphere. Care is taken to ensure that MR of the Cernox temperature sensor is not convolved with sample MR at temperatures where |dR/dT| is very large. Samples show a temperature- and magnetic field-dependent resistance settling time when modifying VG and, to a lesser extent, VDS. These effects are mitigated by changing VG and VDS at or above 60 K and pausing to settle the sample resistance prior to lowtemperature MR measurements. Bidirectional sweeps of B are binned and averaged to produce all individual MR curves, and no hysteresis is seen in the forward and reverse field sweeps. Temperature-dependent resistance measurements for various gate voltages VG are shown in Figure 1a. All curves demonstrate
Figure 1. (a) Low-temperature resistance, R, of a QD FET for different gate biases, VG, as a function of temperature, T. (b) Resistance for different VG plotted versus 1/T2/3. Data collected with VDS = 0.1 V.
insulating behavior obeying the VRH law, R(T) = R0 exp(T0/ T)p. We note that for zero gate bias the sample resistance exceeds our instrumentation limit below 60 K and is thus not shown. The hopping exponent p and the physical interpretation of the characteristic temperature T0 depend on the transport model. For example, Mott VRH assumes a constant density of states near εF in d dimensions, yielding p = 1/(d + 1).15 Several authors have reported p = 1/4 in QD solids, 18−20 corresponding to 3D Mott VRH. Coulomb gap VRH is described by a vanishing density of states at the Fermi level with a quadratic dispersion in three dimensions, producing p = 1/2.18,21−24 More recently, p = 2/3 has been observed and associated with a Coulomb gap and thermal broadening of individual energy levels.25 Our data show a compellingly linear relationship between log(R) and T−2/3 for all gate biases (Figure 1b). For comparison, clearly nonlinear curves are formed by assuming p = 1/2 instead of 2/3 in a similar plot (Supporting Information, Figure S2a). A more objective procedure for determining p for each gate bias is to plot the reduced activation energy W = −[(d ln R)/(d ln T)] and then extract numerical fits for p and T0. Such analysis is described in the Supporting Information and shown in Figure S2b and yields values of p equal to 0.63, 0.66, 0.68, 0.69, and 0.68 for VG = 20, 29, 37.1, 50, and 65 V, respectively. It has been suggested that p = 2/3 could be observed as an admixture of p = 1/2 and p = 1 in an inhomogeneous sample.26 We have some opportunity to test the relevance of this hypothesis to our samples because gate voltage adjustments have a dramatic influence on our measured resistances all the way down to the lowest measurable temperatures. Over a range of VG from 20 to 65 V, the low-temperature resistance drops by more than 3 orders of magnitude, yet p remains constant. This behavior seems difficult to reconcile with a picture in which the VRH exponent is due to a competition between, for example,
Figure 2. Gate bias dependence of (a) localization product, κa, plotted versus gate bias, VG, for p = 2/3 (black filled squares) and p = 1/2 (blue open circles), as described in the text. (b) Gate bias dependence of localization lengths for κ = 3, 6, and 9 as indicated. Values in (b) computed for p = 2/3.
localization product extracted from this equation and its dependence on VG. A positive VG increases the electron concentration, thereby raising εF and decreasing Δε. As VG is applied, a strong response is seen with the correct sign; the localization product increases as VG (Δε) increases (decreases). In addition, Figure 2b shows the dependence of a on VG implied by assuming constant values of κ within a range expected for this system (see ref 32 and ellipsometry measurements in the Supporting Information, Figure S4). A visual extrapolation of a toward the particle diameter at zero bias appears reasonable for κ somewhere between 3 and 6. As noted above, changes in both a and κ may be expected as the 5949
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(T ≥ 20 K). Accordingly, for a given VG we average the data from 20, 40, and 60 K to obtain this univeral line shape (Figure 4b, red curve) and find that its amplitude at those three
gate bias is applied, so precise determination of a is not possible from this data. That being said, the range of κ implied if the localization length is held fixed to the particle diameter is quite large and physically hard to justify without a concomitant change in a (Supporting Information, Figure S5). Thus, our data suggest that gate bias expands the localization length beyond the particle diameter. We emphasize that our data strongly support the assignment of p = 2/3. Nevertheless, for comparative purposes we also compute the localization product for the Coulomb gap model according to T 0 = (2.8e 2 )/(4πaκϵ 0 kB ),33 where T 0 is determined self-consistently from (poor) fits of our data to p = 1/2. The result is shown in Figure 2a and Supporting Information, Figure S5, and shows that our qualitative conclusions are not model-dependent. Magnetoresistance, defined numerically as Δρ/ρ0 [R(B) − R(B = 0)]/[R(B = 0)], generally provides additional insight into transport mechnisms at low temperature. As shown by representative MR curves in Figure 3, the QD FET displays
Figure 3. (a,b) Temperature and gate bias dependence of magnetoresistance of a QD FET at VG = 50 V and T = 7.5 K, respectively. Plots are vertically offset for clarity. Data collected at VDS = 1 V to improve signal-to-noise. No VDS dependence of these large-field features was observed up to VDS = 10 V.
both positive and negative MR contributions whose magnitudes depend on both temperature (Figure 3a) and gate bias (Figure 3b). A more comprehensive cross-section of the data appears in Supporting Information, Figure S6. We note that a low-field (|B| ∼ 100 mT) positive MR component is barely visible in the data presented here but strengthens with high drain−source bias. This latter feature and its VDS dependence have been extensively discussed elsewhere and assigned to random nuclear field mixing of spin singlets and triplets.34 Here we focus on the more dominant features seen across entire field range. The list of mechanisms that can produce positive and negative MR, even in nominally nonmagnetic systems such as these, is extensive. Negative MR can arise from weak localization35,36 and Zeeman modulation of the mobility gap.37 Positive MR can arise from wave function shrinkage,33 electron spin blockade,34,38 and classical disorder induced magnetoresistance.39 Additionally, quantum corrections in the hopping regime can produce both positive and negative MR.40,41 The hopping nature of conduction in these samples makes weak localization physically inappropriate, and the spin blockade mechanism is that associated with the weak, low-field positive MR signal discussed above. To help filter the remaining possibilities, we now show that the MR data can be deconstructed as the superposition of two relatively simple line shapes with inverse power law temperature dependences. We first note that we observe a universal line shape describing negative MR for higher temperatures
Figure 4. (a) Magnetoresistance of a QD FET at VG = 50 V and T = 3 K. (b) Decomposition of (a) into negative (red) and positive (blue) components as described in text. Green shows a parabolic fit to the positive component. (c) Exponents of temperature dependence, x1 (red ×) and x2 (blue +), describing power law temperature dependence, MR ∝ T−x, for negative and positive components of the MR. Point sizes indicate error bars. (d) Mobility gap Δε extracted using p = 2/3.
temperatures obeys a scaling law T −x1. The values of x1 are plotted in Figure 4c (red ×). This temperature dependence is then extrapolated to lower temperatures and subtracted from the MR data (Figures 4a,b), yielding a remaining positive MR contribution (Figure 4b, blue curve) with a predominantly parabolic form. Remarkably, the temperature dependence of the positive component is T −x2 with x2 very close to 2.0 (Figure 4c, blue +). Thus, our MR data set is well-described by (1) a negative MR contribution with a characteristic line shape shown in Figure 4b (red curve) that has a temperature dependence T −x1 and (2) a positive, parabolic MR contribution proportional to T −2. The temperature dependence of (2) can be compared to theoretical predictions for the positive MR mechanisms described above. The associated exponent x2 = −2 is wellreceived by all mechanisms with known temperature 5950
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predictions. Quantum corrections in VRH40,41 and wave function shrinkage in the presence of a transverse field33 both scale as40,41 T −3p = T −2, and spin blockade scales as34 T −2. We note that the latter is conventionally associated with nuclear spin disorder, and saturates at very low fields, whereas (2) does not. Classical disorder induced MR does not have a clear temperature prediction for our system, but its characteristic signature is a transition to a linear behavior at high fields which we do not observe here.42 The temperature dependence of (1) is more telling. As we said, weak localization is inapplicable. Quantum corrections to hopping interference are mostly negative but, as discussed above, predict T −2, which is far from what we observe here. Zeeman energy modulation of the mobility gap predicts37 2
Δρ 1 = − (p(β + δ)d)2 ρ0 2
( 12 gμBB) Δε
2
⎛ T0 ⎞2p ⎜ ⎟ ⎝T ⎠
dielectric constant measurements and calculations, and MR curves for all VG and T used in calculations. This material is available free of charge via the Internet at http://pubs.acs.org.
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Corresponding Author
*E-mail: [email protected]. Notes
The authors declare no competing financial interest.
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ACKNOWLEDGMENTS All aspects of this work supported by the U.S. Department of Energy Office of Basic Energy Sciences, Division of Materials Science and Engineering, under Award No. DE-SC0002158.
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REFERENCES
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(1)
where g ∼ 1.2 is the electron g-factor for CdSe,43 μB is the Bohr magneton, d is the dimensionality of the system, and β, δ and Δε are defined above. Note that here we modify the expression found in ref 37 to include the divergence of the dielectric constant, κ. The associated temperature dependence for p = 2/3 is T −1.33, which is shown as a dotted line in Figure 4c and for which the values of x1 show relatively good agreement. Thus, Zeeman modulation of the mobility gap fits the negative MR signal best, and in many ways it is a magnetic counterpart to the application of a gate bias. In contrast to a gate bias, however, the exact variation in the gap is known from the g-factor. Thus, eq 1 gives a value for the mobility gap, which we can obtain for each value of VG. Figure 4d shows the inferred mobility gap taking β ∼ 1 and δ ∼ 1,13,31 setting d = 3, and using p = 2/3. Values of T0 are those collected synchronously with MR data. With increasing gate bias the calculated mobility gap clearly decreases, and the trend of the gate bias dependence indicates that for sufficiently high values of VG the devices may transition to a metallic state characterized by delocalized states. In conclusion, by combining the analysis of the temperaturedependent resistance and MR, a cohesive picture of electrical transport in these QD FETs emerges. The low-temperature transport studied in this report is consistent with a Coulomb gapped density of states with thermal broadening in an Anderson localized insulator. The negative MR feature is consistent with that expected from Zeeman splitting in the presence of Anderson localization. The data provide strong evidence that gate bias delocalizes electrons in this system beyond the particle diameter, a behavior consistent with the existence of a mobility gap. Magnetoresistance shows both positive and negative components, with the latter bearing a temperature dependence and magnitude also consistent with a mobility gap that acquires a magnetic field dependence through the spin Zeeman energy. These data provide compelling motivation to pursue high gate biases in an effort to observe an insulator−metal transition in these systems.
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AUTHOR INFORMATION
ASSOCIATED CONTENT
S Supporting Information *
Details regarding room temperature mobility measurements, calculation of parameters in the Coulomb gap models, reduced activation energy W, R vs T−1/2, R and W vs T for varied VDS, 5951
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