Controlling the charging energy of arrays of tunnel junctions

Physica E 5 (2000) 274–279

www.elsevier.nl/locate/physe

Controlling the charging energy of arrays of tunnel junctions C.  Kurdaka; b; ∗;1 , A.J. Rimberga; b; 2 , T.R. Hoa; b; 3 , John Clarkea; b , J.D. Walkerc; 4 b Materials

a Department

of Physics, University of California, Berkeley, CA 94720, USA Sciences Division, Lawrence Berkeley National Laboratory, Berkeley, CA 94720, USA c Lawrence Livermore National Laboratory, Livermore, CA 94550, USA Received 18 March 1999; accepted 22 September 1999

Abstract We describe a new technique to control in situ charging energy of systems of coupled metallic or superconducting islands. To illustrate the technique, we have fabricated two-dimensional arrays of Al islands on GaAs=AlAs heterostructures. Each island is coupled to its nearest-neighbor by a submicron Al=AlOx =Al tunnel junction and to the three-dimensional electron gas (3DEG) located below the surface of the heterostructure by a capacitance Cg . We vary Cg , which dominates the charging energy of the array, by depleting the electrons in the 3DEG by means of a negative voltage applied to the array. With the array driven normal by a magnetic eld, a decrease in Cg increases in both the oset voltage and the period of the Coulomb blockade oscillations. ? 2000 Elsevier Science B.V. All rights reserved. PACS: 73.40.Gk; 73.40.Rw; 73.50.Yg Keywords: Josephson junction array; Charging energy; Three-dimensional electron gas

1. Introduction There have been many studies on the properties of structures consisting of submicron tunnel junctions, ∗ Correspondence address: Department of Physics, University of Michigan, Ann Arbor, MI 48109, USA. Tel.: +1-734-647-4650. E-mail address: [email protected] (C. Kurdak) 1 Current address: Department of Physics, University of Michigan, Ann Arbor, MI 48109, USA. 2 Current address: Department of Physics and Center for Nanoscale Science and Technology, Rice University, Houston, TX 77251, USA. 3 Current address: Plumtree Software, 615 Battery St., 2nd Floor, San Francisco, CA 94111, USA. 4 Current address: Genoa Corporation, 2342 Shattuck Av. # 524, Berkeley, CA 94704, USA.

separated by small metallic islands [1]. Such systems are unique in that their Hamiltonians depend only on quantum macroscopic variables. Consequently, all the important parameters of their Hamiltonians can be engineered via fabrication to address speci c physical phenomena. Topics of interest include multijunction devices such as the single-electron transistor (SET) [2,3], the turnstile [4], and the electron pump [5] and, in arrays of junctions, the superconductor– insulator-phase transition [6 –10], charge solitons [11–14], transport in disordered systems [15,16], and the properties of vortices [17–19]. Such structures are almost invariably fabricated from thin lms of Al, because of the very high-quality Al=AlOx =Al tunnel junctions that can readily be achieved. Experiments

1386-9477/00/$ - see front matter ? 2000 Elsevier Science B.V. All rights reserved. PII: S 1 3 8 6 - 9 4 7 7 ( 9 9 ) 0 0 2 5 8 - 1

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are performed either with the Al in the superconducting state or with the Al driven normal by means of a magnetic eld. In some studies, such structures are deposited over, but insulated from, a conducting ground plane. For a particular subset of this general class of systems, speci cally one- or two-dimensional arrays over a ground plane, the array characteristics are determined by the resistance R and capacitance C of each tunnel junction and by the capacitance Cg of each island to the ground plane. The so-called charging energy of each junction — the energy required to impose a charge of one electron — is e2 =2C , where C is a function of C and Cg . For a normal-state array, one observes the Coulomb blockade provided the inequalities e2 =2C /kB T and R/RQ = h=e2 are satis ed, and so thermal uctuations and quantum uctuations, respectively, are unimportant. When the array is superconducting, Josephson tunneling may occur; the phase dierence  across a junction is canonically conjugate to its charge Q, and their uctuations satisfy the uncertainty relation
Q
¿2e. An ongoing issue in the study of these arrays and other multijunction structures has been the fact that most of the parameters are xed at the time of fabrication. Thus, for normal arrays, the values of R, C and (where appropriate) Cg cannot be varied, and so one is required to make a series of arrays to explore a range of parameters. In the case of superconducting arrays, the situation is somewhat less restrictive in that one can vary the critical current I0 of the junctions by means of a magnetic eld, thus varying the Josephson coupling energy. Recently, however, Rimberg et al. [20] demonstrated a technique that allows one to vary the dissipation experienced by an array in situ. They fabricated arrays on GaAs substrates containing a two-dimensional electron gas (2DEG) just below the surface. The 2DEG could be depleted by applying a negative voltage to a back-gate on the reverse side of the substrate, thereby increasing the resistance of the 2DEG and decreasing the dissipation to which the array is subjected. Thus, one can vary the dissipation independently of all the other parameters of the system. This technique enabled the authors to demonstrate a dissipation-driven superconductor– insulator quantum-phase transition. In this paper, we describe a method that enables one to vary the island-to-ground-plane capacitance

275

Fig. 1. Schematic diagram of an array of coupled Al islands on a GaAs=AlAs heterostructure. The array and the 3DEG below the surface of the heterostructure have separate electrical contacts. The island-to-island capacitance, C, and island-to-3DEG capacitance, Cg , are shown in the expanded view of the active region. The electrons under the Al islands can be depleted by applying a voltage Vg between the 3DEG and the array, thus varying Cg . The layers of the heterostructure are shown in the bottom left corner.

Cg in situ, independently of any other parameter. The technique involves a GaAs substrate containing a three-dimensional electron gas (3DEG) just below the surface on which the array is grown. By applying a small positive voltage to the 3DEG relative to the array, one increases the width of the depletion layer, thus decreasing the value of Cg . We illustrate the method by showing that both the oset voltage and period of the Coulomb blockade oscillations of a normal-state array increase as Cg is reduced. 2. Sample structure A schematic diagram of an array with a controllable Cg is shown in Fig. 1. The array is fabricated on a GaAs=AlAs heterostructure with a three-dimensional electron gas (3DEG) below the surface, electrically isolated from the array. The electrons in the 3DEG can be depleted controllably by biasing it positively relative to a gate on the upper surface. For a macroscopic gate, the depletion width can be calculated provided the electron density and the separation of the top surface and the 3DEG layer are known. However, such calculations are much more dicult when the islands

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of a junction array are used as the top gate. First, the dimensions of an island are typically comparable to the separation of the island and the ground plane, and so there is a signi cant contribution to Cg from fringing elds. Second, applying a voltage Vg between the 3DEG and array does not deplete the electrons in the heterostructure uniformly. To study exactly how Cg varies with Vg , we fabricated arrays with high resistance tunnel junctions for which the charging energy dominates electrical transport at low temperatures. We can determine Cg very precisely from the period of the Coulomb blockade oscillations with the arrays in the normal state. Furthermore, we expect the oset voltage produced by the Coulomb blockade to increase as Cg is reduced. To investigate these eects, we measured the current– voltage (I –V ) characteristics of the arrays as a function of Vg . The two arrays, A and B, used in these measurements are 40 × 40 square arrays of Al islands linked by Al=AlOx =Al tunnel junctions and fabricated by electron-beam lithography and shadow evaporation techniques [21]. The two arrays were fabricated on two dierent GaAs=AlAs heterostructures, both of which were grown by molecular beam epitaxy on undoped GaAs substrates. As shown in Fig. 1, the heterostructure used for array A consists of a 1:5 m thick undoped GaAs buer layer, a 250 nm thick Si-doped conducting GaAs layer with a dopant density of 3:3 × 1017 cm−3 , a 100 nm thick undoped AlAs layer, and a 7 nm thick undoped GaAs cap layer. The heterostructure used for array B consists of layers identical to those in the rst sample except that the Si-doped conducting GaAs layer, which constitutes the 3DEG, is 1 m thick and has a dopant density of 7:7 × 1016 cm−3 . We determined the dopant densities at T = 4:2 K from the period of the Shubnikov–de Haas magnetoresistance oscillations and from Hall resistance measurements. Furthermore, we measured the capacitance between a macroscopic top gate and the 3DEG as function of gate voltage for both heterostructures. The dependence of the measured depletion width on gate voltage is found to be consistent with the heterostructure parameters. The charging energy of an array with a ground plane depends on both Cg and the tunnel junction capacitance C. To estimate C, we fabricated a test array along with each measured sample. We inspected these

test samples with scanning electron microscopy, and determined the junction areas to be 0:005 m2 . For the speci c capacitance of similarly sized junctions, various groups have quoted values ranging from 75 to 110 fF=m2 from measurements on two-dimensional (2-D) arrays [6,10,14] and a value of 45 fF=m2 from measurements on one-dimensional (1-D) arrays [22]. These speci c capacitances were generally determined from the I –V characteristics of arrays without a ground plane by using the expression Vo = eN=2C for the oset voltage, where e is the electron charge and N is the number of islands over the length of the array [23]. However, Bakhvalov et al. [13] have pointed out that this formula is not valid for 2-D arrays in which the energetics of electron tunneling depend not only on the capacitance of the tunnel junction across which the electron tunnels but also on all the other capacitances of the array (global rule). Consequently, the important capacitance is the equivalent capacitance between nearest-neighbor islands, Cnn , which for a 2-D square array with no ground plane can be shown to be exactly Cnn = 2C. Accordingly, one should use the expression Vo = eN=4C to determine the junction capacitance in a 2-D square array [13,14]; using this formula for the 2-D arrays of Refs. [6,10,14], we obtain speci c capacitance values ranging from 37.5 to 55 fF=m2 . These lower speci c capacitance values are in quite good agreement with those obtained from 1-D arrays [22]. In our current work, we assume a speci c capacitance of 50 fF=m2 , yielding C = 0:25 fF. 3. Results and discussions We made the electrical measurements in a dilution refrigerator at 20 mK and in an applied magnetic eld of 0.3 T which was sucient to make the arrays normal. The sample leads were ltered by - lters at room temperature, low-pass RC and microwave lters at 4.2 K, and a second set of microwave lters at the temperature of the mixing chamber [24]. Battery-operated, low-noise voltage and current ampli ers in the screened room surrounding the refrigerator were used to make four-terminal measurements. For high Vg (≈ 1 V), there was a small leakage current between the array and the 3DEG. Consequently, we limited our measurements to lower values of Vg

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Table 1 Values of Vo (Vg = 0) and dVo =dVg measured (m) on arrays A and B and predicted (p) from estimated values of C and values of Cg determined from Coulomb blockade oscillations (m)

(p)

Sample

Vo (Vg = 0) (mV)

Vo (Vg = 0) (mV)

(dVo =dVg )(m) (10−6 )

(dVo =dVg )(p) (10−6 )

A B

4.05 3.78

2:56 ± 0:18 2:61 ± 0:18

34 230

45 ± 14 190 ± 20

Fig. 2. I –V characteristics of arrays A and B at 20 mK and 0.3 T. Solid lines are for A and B at Vg = 0 V, and the dashed line is for B at Vg = 0:7 V.

for which the leakage current was less than 1 pA. The I –V characteristics of arrays A and B are shown in Fig. 2. At low voltages, the characteristics are very nonlinear and obey a scaling law I ˙ (V − Vth ) , where Vth is the threshold voltage and  is the scaling exponent [15,16]. Scaling behavior in both the I –V characteristics and the Coulomb blockade oscillations is discussed in detail in [25]. In this paper, we focus on the dependence of the transport properties on Vg . As shown by the dashed line in Fig. 2, the I –V characteristic of array B changed with Vg . A much smaller change was also observed for array A but is not evident on the scale of the gure. To quantify the changes, we use data at higher voltages (V ≈ 10 mV, not shown) at which the I –V characteristics can be tted to the linear form I = (V − Vo )=R, where R is the average junction resistance. For Vg = 0 V, we nd R = 810 k and Vo = 4:05 mV for array A and R = 530 k and Vo = 3:78 mV for array B (Table 1). Furthermore, R is found to be independent of Vg whereas Vo increases with increasing Vg for both samples. This is consistent with our expectation that Vg should change only Cg .

Fig. 3. (a) and (b) Current through array B at a constant bias voltage of 1.6 mV versus Vg for two widely separated ranges; (c) Current spectral density, SI , versus 1=Vg calculated by taking the Fourier transform of two I versus Vg traces, one around Vg = 0 V and the other around Vg = 0:68 V. Each trace contains about 100 oscillations.

To determine Cg , we measured the current through the array as a function of Vg at constant bias voltage. Two typical measurements of I versus Vg are shown in Fig. 3(a) and (b). The current exhibits Coulomb blockade oscillations as a function of Vg . The period of these oscillations is e=Cg , and is found to be independent of bias voltage but dependent on Vg . The change in the period can be seen by carefully counting the number of oscillations in Figs. 3(a) and (b): the former contains 25 periods and the latter 22.5 periods. To determine Cg accurately, we Fourier transform traces of I versus Vg , traces similar to those of

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Cg /C, we nd Cnn = (Cg =2)[1 + 5(C=Cg ) − 3(C=Cg )2 + · · · ]:

(1)

Using the rst three terms, our estimated value of C = 0:25 fF, and our measured values of Cg , we calculate the values of Vo (Vg = 0) listed in Table 1. On average, the predicted values are about 23 of the measured values; the origin of this discrepancy is not clear. We now turn to the shift in oset voltage,
Vo = Vo (Vg ) − Vo (0), as we vary Vg and hence Cg . In Fig. 4(b) we plot
Vo versus Vg for both samples, and list the inferred values of dVo =dVg in Table 1. We calculate this slope by dierentiating Eq. (1) with respect to Vg to nd     eN [1 + 3(C=Cg )2 ] dCg dVo =− : dVg Cg2 [1 + 5(C=Cg ) − 3(C=Cg )2 ]2 dVg (2) Fig. 4. (a) Cg obtained from current power spectrum, and (b) the shift in oset voltage obtained from I –V measurements versus Vg for both arrays. Solid lines are linear ts to the data.

Figs. 3(a) and (b), over a range of Vg that contains about 100 oscillations. The power spectra calculated in this way, one from oscillations around Vg = 0 V and the other from oscillations around Vg = 0:68 V, are shown in Fig. 3(c). From the positions of the sharp peaks we determine Cg with an accuracy of ±10 aF. The values of Cg obtained from the power spectra of a series of I versus Vg traces for both samples are shown in Fig. 4(a). As we expect, for both samples Cg decreases with increasing Vg . This decrease is much more pronounced in array B, since array A has an electron density that is 4.5 times higher and would require a 4.5 times higher voltage to produce the same degree of depletion as B. We note that for a given value of Vg , the change in Cg for the arrays is only 40% of the capacitance change we obtain from measurements on test samples with macroscopic gates. We attribute this dierence to a nonuniform depletion of electrons in the heterostructure under the array. Given the measured values of Cg at Vg = 0, to predict the oset voltage, Vo = eN=2Cnn , we require Cnn in terms of C and Cg . For a 2-D square array with a conducting ground plane, we can calculate Cnn perturbatively using a capacitance network; in the limit

Again assuming C = 0:25 fF, we insert the measured values of Cg and dCg =dVg into Eq. (2) to nd the predicted values of dVo =dVg listed in Table 1. Given the errors in the predicted values of dVo =dVg , the agreement with the measured values is reasonable. 4. Concluding remarks We have fabricated 2-D arrays of metallic islands interconnected with tunnel junctions on GaAs=AlAs heterostructures with a 3DEG located below the surface. We characterized the arrays with I –V measurements and Coulomb blockade oscillations. In the array with the lower density 3DEG, we were able to change Cg in situ by as much as 10% by depleting the electrons electrostatically. The ability to vary Cg and hence the charging energy in situ can be exploited in a variety of ways. For example, one could hope to observe a superconductor–insulator quantum-phase transition driven by changes in the charging energy. Another example might be a single-electron transistor (SET), which consists of a metallic island that is coupled to leads by two tunnel junctions and to a gate by a capacitance Cg . The gain of a SET is determined by the ratio Cg =C [26,27], and so one could control the gain by varying the gate capacitance. This control could be particularly useful in a dierential ampli er con guration requiring two SETs with identical gain:

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by matching the gains one might hope to increase the common mode rejection ratio substantially. Acknowledgements We thank M. Fuhrer for his help with the high magnetic eld characterization of the heterostructures. The work at Berkeley was supported by the Oce of Naval Research, Order No. N00014-95-F-0099 through the US Department of Energy under Contract No. DE-AC03-76SF00098, and the work at Livermore was performed under the auspices of the U.S. Department of Energy by Lawrence Livermore National Laboratory under contract No. W-7405-Eng-48. References [1] H. Grabert, M. Deverot (Eds.), Single Charge Tunneling, Coulomb Blockade Phenomena in Nanostructures, NATO ASI, Vol. 294, Plenum Press, New York, 1992. [2] D.V. Averin, K.K. Likharev, J. Low Temp. Phys. 62 (1996) 345. [3] T.A. Fulton, G.J. Dolan, Phys. Rev. Lett. 59 (1987) 109. [4] L.J. Geerligs, V.F. Anderegg, P. Holweg, J.E. Mooij, H. Pothier, D. Esteve, C. Urbina, M.H. Deverot, Phys. Rev. Lett. 64 (1990) 2691. [5] H. Pothier, P. Lafarge, C. Urbina, D. Esteve, M.H. Deverot, Europhys. Lett. 17 (1992) 249. [6] L.J. Geerligs, M. Peters, L.E.M. de Groot, A. Verbruggen, J.E. Mooij, Phys. Rev. Lett. 63 (1989) 326. [7] H.S.J. van der Zant, F.C. Firtschy, W.J. Elion, L.J. Geerligs, J.E. Mooij, Phys. Rev. Lett. 69 (1992) 2971.

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