Coulomb oscillations in double metal tunnel junctions

PHYSICAD

Physica B 194-196 (1994) 1267-1268 North-Holland

Coulomb

oscillations

in double

metal

tunnel

junctions

L.J. Geerligs , M. M a t t e r s a n d J . E . Mooij Delft Institute for Microelectronics and Submicron Technology P.O. Box 5046, 2600 G A Delft, The Netherlands The Coulomb blockade of electron tunneling through two tunnel junctions in series can be suppressed periodically by applying a gate voltage to the central island. The resulting conductance peaks ("Coulomb oscillations") have been studied in aluminum-oxide- aluminum tunnel junctions. For decreasing temperature the peaks narrow, in agreement with theoretical predictions for a continuous energy spectrum of the island. We observe an increase of the conductance in the tails of the peaks for lower resistance devices, well described by thermally enhanced inelastic electron co-tunneling. 1. I n t r o d u c t i o n The discreteness of the electric charge of a small conducting island, separated by tunneling barriers from the leads, gives rise to Coulomb blockade of electron tunneling. The conductance at low bias shows a significant peak only for a specific value of the Fermilevel of the island, controlled through a capacitively coupled gate voltage. In this situation one electron alternatingly enters and leaves the island. [1]. We present transport measurements of small metallic islands in the linear response regime. These measurements can serve as cross-check for the measurements on (quasi-) zero-dimensional systems. The latter systems reveal information about quantized energy levels and show a cross-over in the character of the resonant tunneling as a function of the ratio of the intrinsic level width (i.e. coupling to the leads) and the thermal energy [2]. In metallic systems, as the coupling to the leads increases, co-tunneling of electrons is known to be important [3]. Co-tunneling is a process where in one event an electron charge crosses both tunnel barriers Here we study this effect as a function of tunnel resistance, isolated from resonant tunneling effects. 2. C o n d u c t a n c e peak lineshapes Kulik and Shekhter [4], and later many others [5], have studied the situation where a continuum of energy levels in an island contributes to the conductance. In this case (kBT >> A E , where A E is the level separation) the conductance of the dot is given by

the island, C~ is the total island capacitance. The peak positions Vo(") are spaced by e/Cg corresponding to changes of e in the net island charge. For resonant tunneling the energy level discreteness needs to be clearly resolved (kBT << A E ) . When the intrinsic level broadening is negligible (kBT >> hr), a very similar lineshape as given by (1) is obtained, the main difference being an apparent temperature T* = 0.8T. For ksT < hF, the intrinsically broadened level yields a (thermally broadened) Lorentzian lineshape. Finally, the thermally enhanced cotunneling conductance for systems with an effectively continuous density of states is given by [3] G

=

amax

sinh(AC/kBT)

(1)

where Gm~x = G1Gz/2(GI q-G2) is the peak conductance in the Coulomb blockade regime (Gi are the tunnel barrier conductances). AU is the minimum energy cost of a tunneling event: C AU = e-~ (Vs - V("3), Cg is the gate capacitance to

(2)

Recently Averin et al. [6] have determined a formula describing the conductance peaks for metallic systems which smoothly connects the two limits (1) and (2) and describes the development of co-tunneling wings in the line shape as a function of coupling to the leads:

G = 8kBTre 2hGG I~ dx

sinh (x~2kBT

1

1

× [ ( E l + x) 2 + 3'7 + (E2 - ~c)2 + ~'~

AU/kBT G :

h (a 3e+

4

2

+

-

((El + x)~ + ~ ) ((E~

where E1 = ec~-~

0921-4526/94/$07.00 © 1994 - Elsevier Science B.V. A l l rights reserved SSDI 0921-4526(93)E1214-7

(3)

x)~ + ~ ) 1

- V o (n) , E~ = ~

(G1 + a2)E, coth G~)E2.

-

and

=

-El,

7a =

h

+

1268

i

I

r

i

|

1

i

!

,

L_i

|

i

i

i

i

.~.~,

~

4

• i::ifi .::iil: iiI! i'. •, ' . . . .

'] I: :

~'~ - 2

"

" .. ~ -

J

°

I'I

•

Vg (mV) Figure 1. Conductance G vs gate voltage Vg for a high-resistance sample (R~ = 620 kfL) taken at three values of the temperature. From top to bottom T = 400, 200 and 6 mK

~._~ - 2 -3

S I

3. E x p e r i m e n t s In Fig.1 we give a set of conductance peaks of a high-resistance metallic double junction, for different temperatures kBT < E c / 1 0 , where Ec = e2/C~. In Fig.2, we give the combined classical~co-tunneling fit (3) to the measurements for both a high and lower resistance metallic double junction. The peaks are scaled to Gm~x, and all capacitance values are independently determined. Then the temperature and one value of Gx/G2 (R~ = R1 +R2 = (Ga +G2)/G1 G2 is known from the large scale I-V) are left as fitparameters. The fitted temperature agrees very well, to within 20mK with the temperature of the mixing chamber to which the sample was anchored. We note that at low temperature the top of the peaks is wider, and/or the bottom narrower than the classical lineshape, probably due to heating and nonequilibrium effects. We have not identified significant deviations which would indicate discrete energy levels, such as an anomalous temperature dependence or modulation of the peak height by a magnetic field. We have also confirmed that the shape of the conductance peaks with co-tunneling wings, fits much better to (3) than to a thermally broadened Lorentzian. This is not unexpected since a significant level separation requires a maximum island volume of order (40am) 3, taking a reasonable estimate of 2 - 101° eV -1 #m -3 for the level density in aluminum. Such a size is one order of magnitude smaller then our island size, however it is not entirely out of reach of present fabrication

--5

'

'

-3

I

-2

I

|

i

-1

I

t

0

|

1

i

i

2

i

i

,

3

Vg (mY) Figure 2. Scaled logarithmic conductance vs gate voltage. Data points and combined classical/cotunneling fits (solid lines) are shown. Top figure (C~ = 0.37 IT, R~ = 620 k~2): T = 300, 200, 100 and 50 inK. Bottom figure (C~ -- 0.50 IT, R~ = 133 kf~): T - 500, 350, 200 and 80 mK (top to bottom). techniques. We would like to thank D.V. Averin for providing theoretical results prior to publication and P.L. MeEuen for stimulating discussions. This work was supported by the Dutch Foundation for Fundamental Research on Matter (FOM). References 1. D.V. Averin and K.K. Likharev, in Mesoscopic Phenomena in Solids, edited by B.L. Altshuler, P. Lee and R. Webb (Elsevier, Amsterdam, 1991). 2. E.B. Foxman et al., preprint. 3. D.V. Averin and Yu.V. Nazarov, Phys. Rev. Lett.

65, 2446 (1990). 4. 5. 6.

I.O. Kulik and R.I. Shekhter, Zh. Eksp. Teor. Fiz. 68, 623 (1975) [Sov. Phys. J E T P 41,308 (1975)]. E.g., C.W.J. Beenakker, Phys. Rev. B 44, 1646 (1991). D.V. Averin et al., these proceedings.