Slanted coupling of one-dimensional arrays of small tunnel junctions

Transcription

1 JOURNAL OF APPLIED PHYSICS VOLUME 84, NUMBER 1 15 DECEMBER 1998 Slanted coupling of one-diensional arrays of sall tunnel junctions G. Y. Hu Departent of Physics and Astronoy, Louisiana State University, Baton Rouge, Louisiana and Departent of Physics, Cheju National University, Cheju , Korea R. F. O Connell a) Departent of Physics and Astronoy, Louisiana State University, Baton Rouge, Louisiana Jai Yon Ryu Departent of Physics, Cheju National University, Cheju , Korea Received 19 January 1998; accepted for publication 14 Septeber 1998 We have studied the electrostatic proble of the slanted coupling of two one-diensional 1D arrays with equal junction capacitances C, equal stray capacitances C o, equal coupling capacitances C c, and with both arrays biased. In the weak coupling liit (C c /C1), we obtain an analytic solution for the potential profile and the corresponding Gibbs free energy, and we derive threshold voltages for various charge transport odes. Our results show that C 0, C c, and the bias voltage V 1 all play iportant roles in deterining the threshold voltage of the syste. In the sall stray capacitance liit (C 0 /C1), the threshold voltage is proportional to 1/C, while in the large stray capacitance liit (C 0 /C1), the threshold voltage becoes independent of C. Also, in the sall C c /C liit, single electron tunneling always has a lower threshold voltage than that of the electron-hole and ixed tunneling. In addition, we find that V 1 has a ore draatic effect on the electron-hole tunneling threshold voltage than on that of the single electron tunneling, i.e., at soe favored value of C c /C a sall change in V 1 can switch the transport of the syste fro single electron to electron-hole transport Aerican Institute of Physics. S I. INTRODUCTION a Electronic ail: rfoc@rouge.phys.lsu.edu In the study of Coulob blockade of single electron tunneling, capacitively coupled one-diensional 1D arrays of sall tunnel junctions have attracted a lot of attention. 1 7 These systes can be used to study both tie correlated and space correlated single electron tunneling, and holds proise for being the basis of a new etrological standard. 1 Nevertheless, as shown in recent experients, 7 it is exceedingly difficult to keep a single charge state propagating in a 1D array, because of the extree sensitivity to stray charges. Thus uch effort has been ade to develop a ore sophisticated syste of 1D arrays. 5 One proising experiental syste consists of two parallel 1D arrays see Fig. 1, where each electrode of one array is coupled to the electrodes of the other array with one straight, Fig. 1a or two slanted, Fig. 1b capacitors C c. When a single electron is placed on an electrode in the first array, the total energy will be lower if a hole is added an electron is reoved fro a neighboring electrode in the other array. As a result, the electron and the hole e-h tend to ove together in the arrays resulting in two identical currents of opposite signs in the two arrays. 1, Recently, Delsing et al. 3 have reported that they have observed indications of correlated electron-hole transport. However, in the straight coupling case, the low energy state can be conserved only if the e-h tunnels siultaneously, which is a second-order cotunneling process and a less likely event. It was noted by Delsing et al. 3 that the slanted coupling is a better choice where the low energy e-h state can be preserved even for sequential tunneling so that we are still dealing with a lowest order process, and it is very iportant to understand the syste theoretically. Nevertheless, to date theoretical investigations have dealt ainly with the case of straight coupling and zero bias on one of the arrays. Here, we study the electrostatic proble of the slanted coupled two 1D arrays, and investigate the charge profiles and threshold voltages of various charge solitons. In particular, we show that the electrostatic proble can be solved in the weak coupling liit, and we give a quantitative analysis for the case when both arrays are biased. In Sec. II, we present the general foralis underlying our ethod and investigate the charge profiles and threshold voltages of various charge solitons. Our results are suarized in Sec. III. II. FORMULATION The syste with which we are concerned is illustrated in Fig. 1b, where two 1D arrays each with sall tunnel junctions, with equal junction capacitances C and equal stray capacitances C o, are doubly coupled to each other by the coupling capacitance C c. Specifically, the ith island of the first array is coupled through C c to both the (i1)th and ith island of the second array. The bias potentials of the two edges of the upper first array are, respectively, 0 and, while that of the lower second array are 0 and. Also, the tunnel resistance R of each junction is assued to be the sae and Rh/e, which ensures that the wave function of an excess electron on an island is localized there. We denote /98/84(1)/6713/5/$ Aerican Institute of Physics

2 6714 J. Appl. Phys., Vol. 84, No. 1, 15 Deceber 1998 Hu, O Connell, and Ryu E 1 C 1 o j C j, E C 1 o j C j, E 1 C 1 c j j j, FIG. 1. Capacitively coupled two 1D arrays each with sall tunnel junctions, with equal junction capacitances C, equal stray capacitances C 0, and equal coupling capacitances C c : a straight coupling; b slanted coupling. the potential on each of the individual 1 islands of the first and the second array by the colun vectors 1,,..., 1 T and 1,,..., 1 T, respectively. In addition, the nuber of excess electrons on each of the individual 1 islands of the first and the second array is denoted by the colun vectors N N 1,N,...,N 1 T and n n 1,n,...,n 1 T, respectively. In general, the Gibbs free energy for the biased doublecoupled two 1D arrays contains two ters, the electrostatic energy E S and the work done due to the charge redistribution associated with the change of the charge profile N,n on the islands, which can be written as FE s W, 1 where the work done due to the charge redistribution associated with the change of the charge profile N,n is given by 1 W i1 i1 i Q g i Q c ia i q g i Q c ia i Q c ib i1 Q c ib V i Q i i q i, where (Q s i,q c i,q i ) and (q s i,q c i,q i ) are the charges on the stray capacitor, the coupling capacitor, and the tunneling junction, respectively. In particular, one has Q c ia C c ( i i ), and Q c ib C c ( i i1 ). Also, the local voltage for the upper array is defined as V i i1 i, with 0 V, and that of the lower array is v i i1 i, with 0 U. In addition, the electrostatic energy in Eq. 1 is defined as with E s E 1 E E 1 e N i i n i i, i0 3 representing the charging energy for the first array, the second array, and the coupling between the two arrays, respectively. We note that the use of Eq. 1 has at least two advantages. First, the iportant process of inner junction charge transfer, i.e., charge transfer between two neighboring islands without any other transfer on the other islands, can now be studied in detail. The other advantage is that it is now possible to obtain the electrostatic equations for the syste directly fro Eq. 1, which we discuss in the following. Our goal here is to find out an explicit for of Eq. 1 at the fixed potential profile, and changeable island charge nuber N,n. For this purpose, we take the derivative of Eq. 1 with respect to the potentials i and i (i 1,,...,1), after which we get (1) equilibriu conditions. These equilibriu conditions for (1) linear equations for the potential profile, and the island charge nuber N,n, and they can be conveniently put into a siple atrix for see Appendix M Y Y M e C N n N n, 7 where the coupling constant C c /C, and we have used a double bar to denote the 1 by1 tridiagonal atrices M and Ȳ. Specifically, M is a syetrical tridiagonal atrix having the sae diagonal eleents D C 0 /C, and the sae off-diagonal eleents 1/, while Y is an antisyetrical tridiagonal atrix having the sae diagonal eleents 0, and the sae off-diagonal eleents 1/. In addition, the first and last eleents of N and n in Eq. 7 are understood to be N 1 C 0 /e and N 1 C /e, n 1 C 0 /e and n 1 C /e, respectively, to accoodate the effects of the bias voltages. Our foralis is quite general. In the case of a syetric bias voltage, we have 0 V/, V/, and 0 0. For the experients of Ref. 4, where both arrays are biased, one has 0 V 1, 0 V, and. We note that Eq. 7 is also known as the electrostatic equation for the syste, the 1D version of which has been obtained by the consideration of the charge conservation law. 4 It is straightforward to show that the off-diagonal block atrix Y in Eq. 7 will contribute to the proble only to order. Thus, in the weak coupling (1) region, it is justified to neglect the off-diagonal block atrix Y and, as a result, Eq. 7 can be solved analytically. By an extension of the techniques used in Ref. 4, for atrix inversion of a syetric tridiagonal atrix, we obtain a solution of Eq. 7 in the weak coupling region as

3 J. Appl. Phys., Vol. 84, No. 1, 15 Deceber 1998 Hu, O Connell, and Ryu 6715 e A B C B A N 8 n, where the eleents of the syetric atrix A and B are given by, respectively, A 1 ij R ij R ij, B 1 ij R ij R ij, with defined by ch D/ Also, is the sae as in Eq. 7, DC o /C, and R ij chji chij. 1 sh sh We note that in Eqs. 9 1, one can check that in general and R ij ( )R ij ( )0, so that AB ij 0. ij In the 0 liit, B ij tends to zero ( ) and the two arrays becoes decoupled. Also, Eq. 8 can be explicitly written as i 0 i A i1 i0 0 A i,1 i B i1 0 B i,1, 13 i 0 i A i1 i0 0 A i,1 i B i1 0 B i,1, 14 where the bias independent ters in Eqs. 13 and 14 are defined, respectively, as 1 i 0 e C 1 i 0 e C AN ij j Bn ij j, BN ij j An ij j We note that in writing Eqs. 13 and 14, we have added Kronecker delta ters to ensure that the equations are correct for i0,1,..., recall in Eq. 8 that i1,,..., 1. Equation 8, suppleented by Eqs. 9 11, isakey result of this article. Once a charge profile N,n is known, we can use Eq. 8 to deterine the potential profile,. Before pursuing this subject, let us analyze our result of Eq. 8 for the siplest case, where no net charge exists in any of the island (N i n i 0) and the arrays act like a network of capacitors, and where one can easily check the results. In this case, at the bias voltage 0 V and 0 0, Eq. 8 reduces to, j A j1 j0 V, 17 j B j1 V. 18 Two liiting cases of Eqs. 17 and 18 are of particular interest. First, when C c 0, one finds B 0, ij and j 0. If one also has C o 0 then j V(1j/), and all the junctions have the sae voltage V/. Next, we use Eqs. 7 and 8 to evaluate Eq. 1 exactly, and the result is 1 FE 0 e C i, N i AN ij j n i An ij j n i BN ij j N i Bn ij j 0 Q 0 Q 0 q 0 q, 19 where E 0 is a quantity independent of the charge profile N,n, and Q 0 N 0 ec 0 1, Q N ec 1, 0a q 0 n 0 ec 0 1, q n ec 1. 0b Equation 19 is a general expression for the Gibbs free energy F of double-coupled two 1D arrays with changeable charge en,n on the islands, at the fixed bias voltage 0,, 0, and the fixed potential profile, on the islands. By eans of Eq. 19, one can directly deterine the change of F due to soe charge transfer event. To be definite, here we discuss the case N k,n k,n k,n k to N k,n k,n k,n k, i.e., the charge transfer between two islands k and k, while the charges on the other islands are unchanged. We denote the net transferred charge as Q Q can be electron-hole, single electron, or the cobined electronhole and single electron case, which will be discussed later. Thus, we obtain fro Eq. 19 the F Q (k,k) due to the charge transfer N k,n k,n k,n k to N k,n k,n k,n k, F Q k,kfn,nfn,n E Q k,kw Q k,k, 1 where the detailed for of the change of the charging energy E Q (k,k) and the work done W Q (k,k) in Eq. 1 can directly be worked out fro Eq. 19. One can now deterine the tunneling threshold voltage by eans of Eq. 1. AtT0, the tunneling of charge Q fro the k to the k island restricting our discussion to the case of kk1 is energy favorable when F Q (k,k) of Eq. 1 is less than zero. Thus, the threshold energy V t for the injection of a charge fro the kth island onto the neighboring k island can be obtained by equating F Q (k,k) 0. Here we study three cases of particular interest: i the electron-hole pair e-h case, where an electron is transferred in the upper array fro the k island to the k island, and at the sae tie in the lower array an electron fro the k island to the k island is transferred, i.e., N k N k 1, n k n k 1, N k N k 1, n k n k 1; ii the single charge e case, where an electron is transferred fro the k island to the k island in the upper array, i.e., N k N k 1, n k n k 0, N k N k 1, n k n k 0; iii the cobined eh,e case, where, in addition to i a single electron is transferred fro k to k, with the results N k N k 1, N k N k 0, n k n k 1, N k N k 1, n k n k 1. It turns out that the final for of V t depends on the setup of the bias voltage. In the case of both arrays biased with 0 V 1, 0 V, and 0, we obtain fro Eqs. 19 and 1,

4 6716 J. Appl. Phys., Vol. 84, No. 1, 15 Deceber 1998 Hu, O Connell, and Ryu V eh k,k e C R k R k k,k k,0 R 1,k R 1,k V 1, a V e k,k e C V eh,e k;k,k e C A k A k k,k k,0 A 1,k A 1k V 1 A k k B 1k B 1k, k,0 A 1,k A 1,k b A k R k kk B k k k,0 A 1k B 1k k,0 A 1k B 1k V 1, c k,0 A 1k B 1k where A ij, B ij, and R ij R ij ( ) are defined by Eqs. 8 11, respectively. Equations a c are interesting results. First, by using the one iediately finds that dv t /d0 and dv t /dk0. This iplies two facts: i for fixed values of C, C o, and C c, an array of larger will generally have larger V t ; ii once a charge is injected into the array, it will have no difficulty in traveling through with increasing k the array. This is to say that fro Eq. one can directly check that V Q t (0,1)V Q t (1,) so that the actual threshold voltage value of the syste is V Q t V Q t (0,1). Thus, the threshold energy V t for the injection of a single charge can be deduced fro Eq. as V eh V eh 0,1 e C V e V e 0,1 e C R 1,1 1R 1,1 V 1, A 11 1A 11 V 1 B 11 1A 11, 3a 3b V eh;e V eh;e 0;1, e B 1 A A 11 1A 11 B 1 V C 1A 1 B A 1 B 11. 3c By using Eqs. 3a 3c, we can study the C o and C c effects, the nuber effect, as well as the bias voltage V 1 effects on the threshold voltages for the various tunneling cases. First, we exaine the capacitor s (C o and C c effects. For siplicity, we set V 1 0 in Eq. 3. It is straightforward to deduce fro Eqs. 9 1 that B ij tends to zero, while A ij and R( ij ) both approach the sae value, (C o /C) i for C o /C1 and i( j)/ for C o /C1, respectively. It follows that in the case of C o /C1, 3 reduces to V eh e C 1V 1, V e e C 1, V eh;e e C 35V 1. Also, when C o /C1, Eq. 3 reduces to V eh e C o V 1, 4a 4b 4c 5a FIG.. Threshold voltage V t in units of e/c 0 for injecting a charge into the first island of a slanted coupled two 1D array, each with ten sall junctions and no bias votage for the other array, as a function of C 0 /C, at three different values of C c /C0, 0.1, 0. fro top to botto. Full curves are for electron-hole pair, dotted curves are for single electron, and dashed curves are for cobined electron-hole pair and single electron case. Also, C c, C, and C o, are the coupling capacitances, junction capacitances, and stray capacitances, respectively. V e e C o, V eh;e e C o V 1. 5b 5c Equations 4 and 5 show that the stray capacitors play an iportant role in deterining the threshold voltage V t.in the sall stray capacitance liit see Eq. 4, the threshold voltage is proportional to 1/C, while in the large stray capacitance liit, the threshold voltage becoes independent of C. These features of the threshold voltage V t are further illustrated in Fig., where we plot V t in units of e/c o vs C o /C, for 0, 0.1 and 0., for a double-coupled two 1D arrays. Also seen fro the figure, in the sall liit single electron tunneling always has a lower threshold voltage than that of the e-h and ixed tunneling. Next, we study the bias voltage V 1 effects on the threshold voltages. For this purpose, in Fig. 3 we plot the threshold voltage V in units of e/c c for injecting a charge into the first island of a slanted coupled two 1D array each with ten sall junctions and bias voltage V 1 for the other array, as a function of C c /C, at three different values of CV 1 /e0, 0.5, 0.9 fro top to botto. As can be seen fro the figure, overall the presence of V 1 will down shift the threshold voltage of V for any kind of charge injection. Nevertheless, it has a ore draatic effect on the e-h tunneling threshold voltage than on that of the single electron tunneling threshold voltage. For exaple, in the region of C c /C0.1, V e V e-h at CV 1 /e0, and it changes to V e V e-h at CV 1 /e0.5. In other words, at soe favored value of C c /C the presence of V 1 can switch the transport of the syste fro single electron to electron-hole transport.

5 J. Appl. Phys., Vol. 84, No. 1, 15 Deceber 1998 Hu, O Connell, and Ryu 6717 the U.S. Ary Research Office under Grant No. DAAH04-94-G-0333, and in part by the Korea Science and Engineering Foundation KOSEF. APPENDIX: DERIVATION OF EQ. 7 The (1) equilibriu conditions as deduced fro Eq. 1 are i1 i1 D i i i1 e C N i i1,,...1, A1 i1 i1 D i i i1 FIG. 3. Threshold voltage V in units of e/c c for injecting a charge into the first island of a slanted coupled two 1D array each with ten sall junctions and bias voltage V 1 for the other array, as a function of C c /C, at three different values of CV 1 /e0, 0.5, 0.9 fro top to botto. Full curves are for electron-hole pair, dotted curves are for single electron, and dashed curves are for cobined electron-hole pair and single electron case. Also, C c, C, and C o, are the coupling capacitances, junction capacitances, and gate capacitances, respectively. III. SUMMARY In suary, in this article we have studied the electrostatic proble of the slanted coupled two 1D arrays, and investigated the charge profiles and threshold voltages of various charge solitons. In particular, in the weak coupling liit we have obtained an analytic solution of Eq. 8 for the potential profile and the corresponding Gibbs free energy of Eq. 19. By eans of Eq. 19, we have derived the threshold voltages Eq. for various cases of charge transport in the slanted coupled 1D arrays with both arrays biased. Our results show that the stray capacitor C 0 the coupling capacitor C c, and the bias voltage V 1 all play iportant roles in deterining the threshold voltage of the syste. In the sall stray capacitance liit see Eq. 4, the threshold voltage is proportional to 1/C, while in the large stray capacitance liit, the threshold voltage becoes independent of C. Also, in the sall liit, single electron tunneling always has a lower threshold voltage than that of the e-h and ixed tunneling cases. For the effects of bias voltage V 1, we find that it has a ore draatic effect on the e-h tunneling threshold voltage than that of the single electron tunneling, i.e., at soe favored value of C c /C the presence of V 1 can switch the transport of the syste fro single electron to electronhole transport. ACKNOWLEDGMENTS The authors would like to thank Dr. P. Delsing for stiulating discussions. The work was supported in part by e C n i i1,,...1, A where DCg/C, and the coupling constant C c /C. Next, we introduce a pair of new variables defined by S i i i, A3 T i i i, A4 and use Eqs. A3 and A4 to put Eqs. A1 and A into the following for: S i1 S i1 DS i S i S i1t i1 S i1t i1 e C N in i, A5 T i1 T i1 DT i T i S i1t i1 S i1t i1 e C N in i. A6 It is not difficult to see that the atrix for of Eqs. A5 and A6 is Eq. 7 of the text. 1 P. Delsing, in Single Charge Tunneling, NATO ASI, Series B, Vol. 94: Physics, edited by H. Grabert and M. H. Devoret Plenu, New York, 199, p. 49. D. V. Averin, A. N. Korotkov, and Yu. V. Nazarov, Phys. Rev. Lett. 66, P. Delsing, J. Petterson, P. Wahlgren, D. B. Haviland, N. Rorsan, H. Zirath, P. Davidson, and T. Claeson, Proceedings of the 1st International Syposiu on Advanced Physics Fields Tsukuba, Japan, G. Y. Hu and R. F. O Connell, Phys. Rev. B 54, J. A. Melsen, U. Hanke, H.-O. Muller, and K.-A. Chao, Phys. Rev. B 55, K. A. Matsuoka and K. K. Likharev, Phys. Rev. B 57, M. W. Keller, J. M. Martinis, and R. L. Kautz, Phys. Rev. Lett. 80, , and references therein.