Using the Seebeck coefficient to determine charge

Using the Seebeck coefficient to determine charge carrier concentration, mobility, and relaxation time in InAs nanowires Volker Schmidt, Philipp F. J. Mensch, Siegfried F. Karg, Bernd Gotsmann, Pratyush Das Kanungo, Heinz Schmid , and Heike Riel Citation: Applied Physics Letters 104, 012113 (2014); doi: 10.1063/1.4858936 View online: http://dx.doi.org/10.1063/1.4858936 View Table of Contents: http://scitation.aip.org/content/aip/journal/apl/104/1?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Theoretical interpretation of the electron mobility behavior in InAs nanowires J. Appl. Phys. 116, 174505 (2014); 10.1063/1.4900980 A comprehensive study of thermoelectric and transport properties of β-silicon carbide nanowires J. Appl. Phys. 114, 184301 (2013); 10.1063/1.4829924 The influence of charged InAs quantum dots on the conductance of a two-dimensional electron gas: Mobility vs. carrier concentration Appl. Phys. Lett. 99, 223510 (2011); 10.1063/1.3665070 Thermoelectric properties for single crystal bismuth nanowires using a mean free path limitation model J. Appl. Phys. 110, 053702 (2011); 10.1063/1.3630014 Diameter-dependent conductance of InAs nanowires J. Appl. Phys. 106, 124303 (2009); 10.1063/1.3270259

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APPLIED PHYSICS LETTERS 104, 012113 (2014)

Using the Seebeck coefficient to determine charge carrier concentration, mobility, and relaxation time in InAs nanowires Volker Schmidt,a),b) Philipp F. J. Mensch,b) Siegfried F. Karg, Bernd Gotsmann, Pratyush Das Kanungo,c) Heinz Schmid, and Heike Riel IBM Research Zurich, S€ aumerstrasse 4, 8803 R€ uschlikon, Switzerland

(Received 28 October 2013; accepted 6 December 2013; published online 10 January 2014) A method for determining charge carrier concentration, mobility, and relaxation time in semiconducting nanowires is presented. The method is based on measuring both the electrical conductivity and the Seebeck coefficient of the nanowire. With knowledge on the bandstructure of the material, Fermi level and charge carrier concentration can be deduced from the Seebeck coefficient. The ratio of measured conductivity and inferred charge carrier concentration then leads to the mobility, and using the Fermi level dependence of mobility one can finally obtain the relaxation time. Using this approach we exemplarily analyze the characteristics of an n-type InAs C 2014 Author(s). All article content, except where otherwise noted, is licensed under a nanowire. V Creative Commons Attribution 3.0 Unported License. [http://dx.doi.org/10.1063/1.4858936]

The Seebeck coefficient (or thermopower) S, is an extremely useful quantity for characterizing semiconductor materials. It contains more information than just quantifying the induced voltage in response to a temperature difference.1,2 From its sign, the prevalent type of charge carrier can directly be inferred. Its magnitude is determined by the density of states, the energy dependence of the carrier relaxation time, and the Fermi level g. Thus, for a material of known band-structure and known energy dependence of relaxation time, the Fermi level g can be deduced from the Seebeck coefficient. The Fermi level position g can then be used to determine the charge carrier concentration n.3 Defining mobility l as the ratio of electrical conductivity r to charge carrier concentration n and elemental charge e, i.e., l ¼ r/(en), the mobility can be deduced with ease and fair accuracy from Seebeck coefficient and conductivity data obtained at the same experimental conditions. The main advantage of this procedure is that is does not require estimating the gate capacitance, as is the case if the charge carrier mobility is deduced directly from the gate voltage dependence of the conductivity. In particular for back-gated nanowire measurements, a sufficiently precise value of the gate capacitance is difficult to assess as the capacitance not only depends on the properties of the oxide covering the substrate but also on the charge distribution within the nanowire. By deducing mobility from Seebeck and conductivity data this problem is circumvented. It will be further demonstrated that via the energy dependence of relaxation time, mobility acquires a dependence on the Fermi level. While this dependence is negligible in the Boltzmann limit, the dependence on the Fermi level can not be neglected if a degenerate semiconductor is considered. It will be shown that the Fermi level dependence of the mobility can be used to consistently determine the relaxation time.

a)

Electronic mail: [email protected] V. Schmidt and P. F. J. Mensch contributed equally to this work. c) Present address: Paul Scherrer Institute, 5232 Villigen, Switzerland. b)

0003-6951/2014/104(1)/012113/4

Gate-voltage- and temperature-dependent measurements of both the electrical conductivity and the Seebeck coefficient were conducted on a single indium-arsenide (InAs) nanowire. The InAs nanowires were synthesized by metalorganic chemical vapor deposition using Au-colloid particles of 20 nm diameter as catalysts. The resulting nanowires have a length of about 4.5 lm and their shape is slightly tapered. The as-grown nanowires were dispersed in isopropanol by sonication and drop casted onto a thermally oxidized highly doped Si substrate with an oxide thickness tox, as measured by ellipsometry, of 158 nm. Subsequently, the InAs nanowires were localized by scanning electron microscopy (SEM) with respect to predefined markers and contact pads. Leads and contacts to the nanowire were processed using electron-beam lithography, followed by a short dip in hydrofluoric acid before thermal evaporation of nickel and lift-off. A schematic of the final nanowire device structure together with corresponding scanning electron micrographs is shown in Fig. 1. The nanowire radius R was determined after the electrical characterization using high-resolution SEM. The nanowire portion used for the calculations of this paper had a diameter of 30 6 4 nm, including a presumably present oxide shell. Four contacts were made to the nanowire, with three of them serving as 4-point contacted resistive thermometers. The thermometers were calibrated by performing a temperature sweep and measuring their resistance under isothermal conditions in a four point manner. The temperaturedependent electrical conductivity of the nanowire is obtained in a similar fashion. By using the resistive heater, structured in close vicinity to the nanowire (c.f. Fig. 1), a temperature gradient of about 1 K is created along the nanowire length. By measuring both the temperature difference between two of the resistive thermometers and the corresponding thermal voltage, the Seebeck coefficient is obtained. All measurements were performed in a cryo-probe-station under high vacuum conditions (pressure 1 104 Pa) with the temperature of the sample measured by a calibrated diode clamped onto the substrate. An 8 24 switching matrix (Keithley

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FIG. 1. (a) Schematic of the fabricated device structure used for measuring electrical conductivity and Seebeck coefficient of an InAs nanowire. (b) Scanning electron micrograph of a contacted InAs nanowire, contact design as sketched in (a); (c) close-up of the center part of (b).

7172) in combination with a 16-pin probe-card was used to address the different leads. For the Seebeck measurement, a current source together with a voltage meter (Keithley, DMM196) was used to apply and measure the heater current/voltage; a Keithley 6221 current source and 2182A nanovoltmeter were used to characterize the resistivity of the thermometers as well as for measuring the thermal voltage. Conductivity measurements were performed using a highresolution electrometer (Keithley 6517). Both Seebeck coefficient S and electrical conductivity r of an InAs nanowire were measured at a multitude of temperatures T and gate-voltages VG. The highly doped silicon substrate acted as back-gate, by means of which the charge carrier concentration in the nanowire was modified. Overall a complete array of Seebeck and conductivity data were obtained for 13 different temperatures and 17 different gate voltages. Fig. 2(a) shows the gate voltage dependence of the electrical conductivity for four different temperatures. Applying a positive back-gate bias, only a moderate change of the conductivity (about a factor of four) can be induced. The high conductivity of around 200 X1 cm1 at zero gate bias indicates that the InAs nanowire is already highly n-doped. The high carrier concentration is also reflected in the Seebeck coefficient. Fig. 2(b) shows S as a function of gatevoltage VG for the same temperatures as in Fig. 2(a). The Seebeck coefficient of the contact metal, Ni, has already been substracted using reference data for the thermopower 2 of pure Nickel;4 SNi ð1:89 19:17 300T K þ 1:953 300T K Þl V=K. The Seebeck coefficient of around 140 lV/K at room temperature and zero gate bias is consistent with previous measurements5,6 and indicates that the Fermi-level is located well above the InAs conduction band edge. By applying a positive gate bias, the magnitude of the Seebeck coefficient is further reduced, indicating charge carrier accumulation. To analyze the data, a sufficiently accurate description of the Seebeck coefficient is required. To this end, the Fermi distribution fo and a constant v ¼ 3:406 1021 ðeVÞ3=2 cm3 ,

FIG. 2. (a) Electrical conductivity of an InAs nanowire versus gate voltage for four different temperatures; (b) Seebeck coefficient versus gate voltage for four different temperatures.

defined in terms of the reduced Planck constant h and the electron mass mo, are given in Eqs. (1) and (2). It is known that the C-valley conduction band of InAs exhibits a pronounced non-parabolicity, with a dispersion relation that can be h2 k 2 described as 2m m ¼ ð1 þ aÞ, with the dimensionless effeco * tive mass m and the electron mass mo. Here, a is the non-parabolicity parameter, taken to be a ¼ 1.4 eV1 according to Ref. 7. The C-valley density of states D() then takes the form given in (3). One can see that pin ffiffi the limit a ! 0 the density of states acquires its common dependence. We use the symbol here to indicate that the energy is measured with respect to the conduction band edge; likewise the Fermi level g is given with respect to the conduction band edge. The constant g is the valley degeneracy, which is 2 in case of the C-valley (because of spin) fo ¼

1 ; g 1 þ exp kB T

(1)

3=2 1 2mo ; v¼ 2 4p h2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi DðÞ ¼ g v m3=2 ð1 þ aÞ ð1 þ 2aÞ :

(2) (3)

We used formulae and parameters of Vurgaftman et al.8 to calculate the temperature-dependent effective mass m* as m ¼

EP ðEC þ 23 Dso Þ ð1 þ 2FÞ þ EC þ Dso

!1 ;

(4)


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EC ¼ ECo

aC T 2 ; T þ bC

(5)

with ECo ¼ 0.417 eV, aC ¼ 2:76 104 eV=K, bC ¼ 93 K, F ¼ 2.9, EP ¼ 21.5 eV, and Dso ¼ 0.39 eV. These parameters cause the effective mass m* to decrease from 0.0236 at 200 K to 0.0204 at 400 K. With these prerequisites the density of states is fully defined and the electron concentration n can be calculated ð1 fo DðÞd ; (6) n¼ 0

2 e so ; 3 m mo ð 1 b @fo ð1 þ aÞ r ¼ ec DðÞd ; ð1 þ 2aÞ2 0 @ o c¼

ð 1 b @fo ð1 þ aÞ ð gÞ Sr ¼ c DðÞd : ð1 þ 2aÞ2 T 0 @ o

(7)

(8)

(9)

The expression for the electrical conductivity r, given in (8), contains six implicit assumptions: (i) that quantization effects can be neglected, (ii) that the conductivity can be treated as being isotropic (well met for the C-valley); (iii) that deviations of the true distribution function from the equilibrium distribution function fo are small enough to justify the use of the Boltzmann transport equation in its linearized form; (iv) that it is sufficient to only consider the Cvalley, i.e., that the material is sufficiently n-doped to neglect both valence band and other conduction band contributions (in InAs X- and L-valley are located well above the C-valley minimum and can be neglected); (v) that we can make use of the relaxation time approximation; and (vi) that the relaxation time s can be approximated as s ¼ so ð=o Þb with a relaxation time constant so and a normalization constant o ¼ kB 300 K necessary for dimensionality reasons. The coefficient b describes the energy dependence of the relaxation time which, via the temperature dependence of the Fermi function fo, implicitly causes a temperature dependence of s of approximately ðT=300 KÞb . This energy dependence is often neglected and instead relaxation time is made explicitly temperature-dependent. While this is sufficient to describe the conductivity behaviour, it falls short if also the Seebeck coefficient needs to be modelled. In the latter case, customarily, a power law energy dependence of relaxation time is assumed. A similar expression as for the conductivity can be found for the product of Seebeck coefficient S and conductivity r, see (9). From (8) and (9) the Seebeck coefficient is trivially obtained S ¼ Sr r . The mobility l is defined as l ¼ ern. It should be understood that, in general, electrical conductivity, Seebeck coefficient, and mobility depend both on the parameter b and on the Fermi level g. That is, unless a ¼ 0 and b ¼ 0, mobility aquires a dependence on g. Although this dependence is negligible in the Boltzmann limit, it cannot be neglected if a degenerately doped semiconductor is considered. In the latter case, the g-dependence of the mobility is pronounced, and a single, g-independent value of the mobility can not be determined.

In order to analyse the measurement data concerning Seebeck coefficient and conductivity of the InAs nanowire, the first step is to find a sufficiently accurate estimate for the parameter b from functional compliance of the computed Seebeck coefficient S with experimental bulk InAs reference data. As shown in Fig. 3(a), where the room temperature Seebeck coefficient of InAs calculated for three different values of b is shown together with the InAs data of Rode,9 excellent agreement can be obtained for b ¼ 0.78 (black curve), which will therefore be assumed throughout. The black curve in Fig. 3(a), computed using Eqs. (6)–(9) can be regarded as the calibration curve by means of which to relate n to S. In order to give an impression how sensitive the calculated Seebeck coefficient is with respect to the choice of b, the two dashed curves in Fig. 3(a) indicate the functional relation between S and n for b ¼ 0.58 and b ¼ 0.98. Using this model the Fermi level g, can be deduced from the measured Seebeck coefficient. In Fig. 3(b) g is given as a function of gate-voltage for a temperature T of 278 K. As already indicated by the low electrical resistivity and the low Seebeck coefficient, the Fermi level at zero gate bias is located well within the conduction band, about 0.1 eV above the edge, and by applying a positive gate bias g moves by 0.1 eV further up the conduction band due to charge carrier accumulation. After using S to infer the Fermi level g one can in a second step use g to calculate the electron concentration n. In a third step, this value of n can be combined with the corresponding value of the electrical conductivity r, measured at the same conditions regarding gate voltage and temperature, to obtain the mobility l. This was done for the complete data set. The obtained mobility for three different temperatures

FIG. 3. (a) Seebeck coefficient versus electron concentration of bulk InAs. Black dots: data taken from Rode;9 black curve computed for b ¼ 0.78. (b) Fermi level g (left) and electron concentration (right) as a function of gate voltage for 278 K.


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mobility is indirectly fully covered by the choice of b. We believe that Eqs. (1)–(9), together with the determined values for b and so, convincingly and consistently describe the experimental observations. To summarize, gate-voltage- and temperaturedependence data on the Seebeck coefficient and the electrical conductivity of a degenerately n-type doped InAs nanowire were analyzed by first deducing Fermi level and charge carrier concentration from the Seebeck coefficient. Taking the ratio of electrical conductivity and charge carrier concentration the mobility, showing a dependence on the Fermi level, could be inferred, leading to a relaxation time Þ0:78 s. s 9 1015 ðkB 300K We wish to acknowledge funding received from the EU 7th Framework Programs Steeper (257267) and the Marie Curie Fellowships WISE (276595) and TETTRA (303254). We also wish to thank U. Drechsler and M. Tschudy for technical support and M. Borg, P. Khomyakov, and W. Riess for fruitful discussions.

1

C.-H. Lee, G.-C. Yi, Y. M. Zuev, and P. Kim, Appl. Phys. Lett. 94, 022106 (2009). P. M. Wu, J. Gooth, X. Zianni, S. F. Svensson, J. G. Gluschke, K. A. Dick, C. Thelander, K. Nielsch, and H. Linke, Nano Lett. 13, 4080 (2013). 3 N. Preissler, O. Bierwagen, A. T. Ramu, and J. S. Speck, Phys. Rev. B 88, 085305 (2013). 4 C. L. Foiles, “Thermopower at low temperatures,” in Springer Materials The Landolt-B€ ornstein Database edited by K.-H. Hellwege and J. L. Olsons (Springer, Berlin, Heidelberg, 1985), Vol. 15b, Chap. 3.4, p. 60. 5 S. Karg, P. Mensch, B. Gotsmann, H. Schmid, P. D. Kanungo, H. Ghoneim, V. Schmidt, M. T. Bj€ ork, V. Troncale, and H. Riel, J. Electron. Mater. 42, 2409–2414 (2013). 6 P. Mensch, S. Karg, B. Gotsmann, P. D. Kanungo, V. Schmidt, V. Troncale, H. Schmid, and H. Riel, in Proceedings of ESSDERC Conference, Bucharest, Romania (2013), pp. 252–254. 7 K. Brennan and K. Hess, Solid-State Electron. 27, 347 (1984). 8 I. Vurgaftman, J. R. Meyer, and L. R. Ram-Mohan, J. Appl. Phys. 89, 5815 (2001). 9 D. L. Rode, Phys. Rev. B 3, 3287 (1971). 10 H. Ghoneim, P. Mensch, H. Schmid, C. D. Bessire, R. Rhyner, A. Schenk, C. Rettner, S. Karg, K. E. Moselund, H. Riel, and M. T. Bj€ ork, Nanotechnology 23, 505708 (2012). 11 A. Lin, J. N. Shapiro, A. C. Scofield, B. L. Lang, and D. L. Huffaker, Appl. Phys. Lett 102, 053115 (2013). 12 T. Bryllert, L.-E. Wernerson, L. E. Fr€ oberg, and L. Samuelson, IEEE Electron Device Lett. 27, 323–325 (2006). 13 S. A. Dayeh, D. P. R. Aplin, X. Zhou, P. K. L. Yu, E. T. Yu, and D. Wang, Small 3, 326–332 (2007). 2

FIG. 4. InAs nanowire (a) Electron mobility versus Fermi level; (b) electron relaxation time so as a function of temperature.

are shown in Fig. 4(a) as a function of the Fermi level g. The value of around 2000 cm2/Vs at room temperature is greater than the field effect mobilities of Ghoneim et al.10 and Lin et al.11 but smaller than the value of Bryllert et al.12 and Dayeh et al.13 The solid lines in Fig. 4(a) are computed using so as fitting parameter. Note that so determines both magnitude and slope of the fit curves, so that the fact that the g dependence of l agrees well with the mobility data, speaks in favor of the validity of the model. Fig. 4(b) displays the resulting values for so as a function of temperature and it becomes apparent that so can be approximated by a constant value of about so ¼ 9 1015 s. As mentioned in the beginning, the energy dependence of the relaxation time also indirectly causes a temperature dependence. The fact so does not show any temperature dependence means that the temperature dependence of the
