Josephson currents in two dimensional mesoscopic ballistic conductors Heida, Jan Peter

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1 University of Groningen Josephson currents in two dimensional mesoscopic ballistic conductors Heida, Jan Peter IMPORTANT NOTE: You are advised to consult the publisher's version (publisher's PDF) if you wish to cite from it. Please check the document version below. Document Version Publisher's PDF, also known as Version of record Publication date: 1998 Link to publication in University of Groningen/UMCG research database Citation for published version (APA): Heida, J. P. (1998). Josephson currents in two dimensional mesoscopic ballistic conductors Groningen: s.n. Copyright Other than for strictly personal use, it is not permitted to download or to forward/distribute the text or part of it without the consent of the author(s) and/or copyright holder(s), unless the work is under an open content license (like Creative Commons). Take-down policy If you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediately and investigate your claim. Downloaded from the University of Groningen/UMCG research database (Pure): For technical reasons the number of authors shown on this cover page is limited to 10 maximum. Download date:

2 Chapter 9 Superconducting phase sensitive transport through a two dimensional electron gas channel with superconducting boundaries Abstract We have investigated transport through a channel with superconducting boundaries where electrons are conned byandreev reection. Using an idealized model calculation we show that the decay length of the particle wave function in the channel depends on the superconducting phase dierence between the superconducting boundaries. Theoretically the transfer resistance R t, which is proportional to the current transmitted through the channel has maxima at =. The transfer resistance R t and the resistance R i between the injector and the channel have been measured as a function of. We have not been able to detect an oscillation in R t due to the transmitted current modulation within an accuracy of 0.05 % in this experiment. 9.1 Introduction 1 The superconductor phase plays an important role in supercurrent ow in bulk superconductors as well as in Josephson junctions. Recently much attention has been devoted to superconducting phase sen- 1 This work was done in close collaboration with Dr. J. Nitta of NTT Laboratories Japan during his visit to the Laboratory of Applied Physics of the University of Groningen. 123

3 124 Chapter 9 sitive normal transport. An interference eect on the resistance which is inuenced by the superconducting phase dierence has been pointed out by Spivak et al. and Al'tshuler et al. [1, 2]. Through the Andreevreection (AR) process at the interface between normal metal (N) and superconductor (S), the phase of quasiparticles is shifted by the phase of the superconducting order parameter. The basic idea is that, if the superconducting phase dierence is externally controlled, the resistance of the normal layer can be aected by constructive and destructive interference of quasiparticles. Several types of interferometers have been proposed [3, 4, 5, 6, 7, 8, 9, 10, 11], and such phase sensitive transport has been experimentally observed [12, 13, 14, 15, 16]. In this chapter, wehave study both theoretically and experimentally the transport through a two-dimensional-electron-gas (2DEG) channel with superconducting boundaries. Figure 9.1: Schematic structure of the channel. A two dimensional electron gas (2DEG) is surrounded by superconducting boundaries S1 and S2 9.2 Theory of electron transport through a channel with superconducting boundaries First we discuss the operation principle of the phase sensitive transport in a channel with superconducting boundaries in an idealized case where the 2DEG channel is ballistic, and electrons are conned by AR. The channel is shown schematically in gure 9.1. In such an SNS

4 Superconductor phase sensitive transport junction bound states are formed in the y-direction, at energies which depend on the superconductor-phase dierence. A supercurrent ows in the y-direction, its magnitude is related to the phase sensitivity of the energy spectrum (Chapter 3 [17, 18, 19, 20, 21]. Here we consider electron transport through the channel in the x-direction, and focus on the properties of the electron wave function at the Fermi energy. There is only a bound state at the Fermi energy when the Andreev reection probability is unity and = ; in all other cases the electron wave functions are evanescent in the channel, and the wave vector k x at the Fermi energy has both a real and an imaginary part. The decay length of the electron wave function in the channel is: Re(k =Im(k x ),1 x ) =, Re(k y ) Im(k y ) (9.1) Where k x and k y are the complex Fermi wave vectors in the x and y direction respectively. The second equality arises from the fact that the total energy in the 2DEG has to be real, and satisfy E F =h 2 (kx 2 + ky)=2m 2. This implies that a decay of the wave function amplitude in the x-direction is compensated by an increase in the wave function amplitude in the perpendicular direction. The electron motion in the y-direction forms a bound state, and the wave functions are given by the solutions to the Bogoliubov-de Gennes equation. They can be described by electron and hole wave functions in the 2DEG (0 <y<l), (y) =[e ikyy + e,ikyy ] 1 0! +[e ikyy + e,ikyy ] 0 1! (9.2) and by quasiparticle wave functions in the superconductor (y < 0;y >L). (y) = 8 >< >: ae ip F y ce ip F (y,l) v u!! + be,ip F y u! v ue i + de v,ip F (y,l) ve i u! (y <0) (y >L) (9.3) Here p F is the real Fermi wave vector in the superconductor which is much larger than the Fermi wave vector k F in the 2DEG when Nb is used as a superconductor. We assume the barrier strength Z at the SN boundaries is zero [22]. The wave functions and their derivatives have to be matched at the boundaries y = 0 and y = L. By eliminating the coecients ; ; and, four equations for a,b,c and d are found.

5 126 Chapter 9 The solvability condition for these four equations gives a phase dependent excitation energy spectrum of quasiparticles in the 2DEG normal channel. We concentrate on the transport at the Fermi energy. We nd that the complex wave vector k y is given by the following equation: k 4 y, 2(m ) 2 p 2 F [1 + cos(2k y L)+2cos] k 2 y +(m ) 4 p 4 F =0 (9.4) cos(2k y L), 1 Here m is the eective mass value of the 2DEG relative to the free electron mass. Equation 9.4 can also be obtained from the equations found in Furusaki et al. [20] by setting E = E F. We can get the following solution for the imaginary part of k y assuming that Re(k y ) m=l < k F where m is the mode index (m= 1,2,..) and Re(k y ) Im(k x ). [23] Im(k y )=, 1 L p 2m p F m q 1 + cos (9.5) (m ) 2 p 2 F, (m=l) 2 L The above assumption Re(k y ) > Im(k y ) holds under the condition m p F k F > m=l which is satised in Nb/InAs/Nb junctions. In equation 9.5, Im(k y ) becomes zero at = (2n + 1) because then the bound state is formed at the Fermi energy. By using the above expression and E F =h 2 (kx+k 2 y)=2m 2, the decay length of the incoming electrons can be obtained by q m = Im(k x ),1 = Lm p F kf 2, (m=l) 2 p p (9.6) 2(m=L) cos The decay length has minima at = 2n and is innite at = (2n+1). It also has a propagation mode dependence. The transmitted L current which is expected to be proportional to exp 2 m shows minima at = 2n, and maxima at = (2n +1). Therefore, we expect that the current transmitted through the channel can be modulated by controlling. 9.3 Samples The schematic structure of the sample is shown in gure 9.2. A channel was made by surrounding an InAs 2DEG with two parallel superconducting Nb electrodes (E 0 ). The channel width W and the length L are 0.5 m and 0.2 m respectively. Electrons were injected from the 0.15 m wide Nb injector electrode (E 1 ) to the channel. The transmitted electrons were detected by a Nb receiver electrode (E 2 ) which was

6 Superconductor phase sensitive transport Figure 9.2: Schematics of the fabricated structure. A channel is formed by surrounding an InAs 2DEG with two parallel superconducting Nb electrodes (E 0 ). Both ends are connected, and form a superconducting ring geometry to control the phase dierence. The current I 1 is injected from the Nb injector electrode (E 1 ). The transmitted current I t is detected by the Nb receiver electrode (E 2 ). Si is used as an insulating layer between the superconducting ring and the lead for the receiver electrode. placed inside the superconducting ring. Because the receiver electrode has to be close to the channel, all electrodes were made of Nb at the same time for alignment accuracy. Therefore, Josephson coupling between the injector electrode and the Nb ring (E 1 -E 0 ), and between the receiver electrode and the Nb ring (E 2 -E 0 ) is expected. The separations between E 1 and E 0, and between E 2 and E 0 were 0.1 m, and 0.3 m, respectively. Si with 100 nm thickness was used as an insulating layer between the superconducting ring and the lead for the receiver electrode. To control the phase dierence, a superconducting ring geometry was formed. The area S of the ring was about 60 m 2. The 2DEG is formed in a 20 nm InAs layer of InAs/GaSb heterostructures [24]. The electron carrier concentration and the electron mobility of the 2DEG InAs were N s = 1: m,2 and = 3:0 m 2 =V s. The carrier concentration gives a Fermi wavelength of 18 nm which provides 20 propagating modes in the channel. From these values, the electron mean free path is estimated to be 0.65 m. The superconducting electrodes were fabricated by standard electron beam lithography and lift-o techniques. Prior to deposition of 60 nm Nb, the free surface of InAs was Ar ion-cleaned. The InAs layer underneath the Nb pattern was not completely removed by the Ar ion-cleaning process.

7 128 Chapter 9 Figure 9.3: Equivalent circuit of the channel. R i and R r present the resistance between the injector and the Nb ring, and the resistance between the receiver and the ring. 9.4 Experimental results Equivalent circuit The equivalent circuit of the sample is as shown in gure 9.3. V 1 is measured between probes E 1 and E 0 and V 2 between probes E 2 and E 0. The currents I 1 and I 2 are sent to probe E 0 from E 1 and E 2 respectively. Because the eective channel resistance R c is much larger than the resistance from the injector to the ring R i and the resistance from the receiver to the ring R r,dv 1 =di 1 = R i (R r + R c )=(R i + R r + R c ) R i, and dv 2 =di 2 = R r (R i + R c )=(R i + R c + R r ) R r. These resistances were measured; dv 1 =di 1 =16,and dv 2 =di 2 = When I 1 is supplied from E 1 to E 0, V 2 is a measure of the current transmitted through the channel between the superconductors. We call R t =dv 2 =di 1 the transfer resistance, it is 1.6. From the equivalent circuit the transfer resistance is given by R i R r =(R i + R r + R c ) R i R r =R c. From these measurements, R c 350. This value is smaller than the Sharvin resistance which is estimated to be 610. A possible reason is that hole conduction in the GaSb layer is not negligible, or, as recently pointed out by Magnee et al. [25], the increase of the electron carrier concentration during the fabrication process. The current transmitted through the channel I t is given by I t (R i =R c ) I 1. From this expression it is deduced that about 4.8 % of the injected electrons can reach the receiver electrode E 2 which is about 0.3 m away from the channel. In other samples, the transmitted currents were reduced to 0.8 % and 0.15 % by putting the receiver electrode

8 Superconductor phase sensitive transport m and 1.0 m away from the channel. This result suggests that electrons transmitted through the channel do not reach the receiver electrode ballistically. The dv=di,i characteristics were measured to estimate the Josephson coupling. A sharp dip structure was observed in dv 1 =di 1 - I 1 characteristics below I 1 =1A, but dv 1 =di 1 did not reach zero. This sharp dip is attributed to a supercurrent I c of about 0.5 A. The dip structure was not observed in dv 2 =di 2 - I 2 from which we conclude that the Josephson coupling between the Nb ring and the receiver electrode was negligible. R t = dv 2 /di (f) 1.74 (e) (d) 1.72 (c) (b) Oscillation through the channel 1.6 (a) T=50mK B (Gauss) Figure 9.4: Transfer resistance R t =dv 2 =di 1 as a function of magnetic eld for dierent DC biases I dc. (a) I dc =0A, (b) I dc =3A, (c) I dc =10A, (d) I dc =15A, (e) I dc =20A, (f) I dc =40A Magnetic eld dependence of the transfer resistance The transfer resistance R t =dv 2 =di 1 which is proportional to the transmitted current was measured at T = 50 mk, as a function of the magnetic eld B, for dierent DC current biases I dc. The phase dierence can be controlled by an applied magnetic eld B which yields a phase dierence = 2(BS)= 0, where 0 is a ux quantum ( 0 = h 2e ).

9 130 Chapter 9 The results are shown in gure 9.4. For I dc = 0, oscillations in the transfer-resistance R t were observed with a period of about 0.35 G. This magnetic eld corresponds to a ux quantum 0 through the Nb ring area. The oscillations exhibit resistance minima at = 2n. This is because the Josephson coupling between E 1 and the ring leads to a DC-SQUID oscillation [12] in R i, and thus in R t R i R r =R c. This DC-SQUID oscillation was used as a reference signal to determine the phase dierence = 2n. A DC bias current higher than the critical current was added between the injector electrode and the Nb ring in order to suppress the Josephson coupling. Under low DC bias conditions, a phase shift in the oscillations and a rapid reduction in amplitude were observed. The transfer-resistance for I dc = 3 A has maxima at = 2n. This is because the Josephson coupling still dominated the experiment. By increasing I dc, the resistance with I dc =10A has minima again at =2n, and a gradual phase shift in the oscillations was observed. Under a high DC bias condition, I dc > 30 A, the transferresistance has maxima at = 2n again. This can not be explained by the idealized model calculation because the transmitted current theoretically has minima at = 2n. Therefore, the transfer-resistance should have minima at =2n. If the DC bias current is not equally supplied from the injector to both superconducting boundaries E 0, this asymmetric DC bias current gives rise to an additional magnetic ux by the geometrical inductance L eff of the Nb ring which is estimated to be 60 ph. This also changes the phase dierence by = L eff ji dc1, I dc2 j= 0. An asymmetric DC bias of ji dc1, I dc2 j = 30 A causes a phase dierence of. This asymmetry seems to be too large for the sample conguration, and can not explain the additional phase shift. The resistance dv 1 =di 1 R i between the injector and the ring has also been measured in comparison to the transfer resistance R t proportional to the transmitted current through the channel, as shown in gure 9.5. This is the same measurement conguration as used for the quasiparticle interferometer [12]. The DC bias dependence was almost the same as in gure 9.4. The resistance maxima at = 2n were also observed at I dc > 3 A. The oscillation amplitude of R t through the channel seems to scale with the oscillation amplitude of R i between the injector and the Nb ring. This is because R t is given by R i R r =R c, and depends on R i. The eect of R c is smaller than the eect of R i in this experiment. The dierence from the resistance oscillation of R t is that a small increase in oscillation amplitude was observed under high DC bias conditions.

10 Superconductor phase sensitive transport R i = dv 1 /di (b) (f) (e) (d) (c) 15.8 T= 50 mk (a) B (Gauss) Figure 9.5: Dierential resistance R i =dv 1 =di 1 as a function of magnetic eld for dierent DC biases I dc. (a) I dc = 0 A, (b) I dc = 3 A, (c) I dc =10A, (d) I dc =15A, (e) I dc =20A, (f)i dc =40A. 9.5 Discussion Figure 9.6 shows a comparison in oscillation amplitude as a function of DC bias current. In this plot, the phase of the oscillation was not taken into account. The oscillation amplitudes q by the Josephson coupling were plotted using the relation V = R (Idc 2, Ic 2 ) with I c = 0.5 A.[26] The rapid reduction in oscillation amplitude under a low DC bias condition can be well explained by the DC-SQUID action of the device. Under high bias conditions, the amplitude of the oscillations is higher than that expected for a DC-SQUID. This fact suggests a quasiparticle interference eect. However, the resistance maxima at = 2n can not be explained by interferometer operation. In the previous quasiparticle interferometer experiment [12], resistance minima at = 2n were observed under high bias conditions. Therefore, the transition from the DC-SQUID oscillations to the interferometer type oscillations is not yet clear. If present, the oscillation amplitude of R t due to the decay length modulation is less than 0.05 % in this experiment. In the idealized model calculation, we assumed a ballistic channel and well-dened superconducting boundaries with Z = 0. After the

11 132 Chapter R/R (%) Josephson effect R t = dv 2 /di 1 R i = dv 1 /di I dc (µa) Figure 9.6: Comparison of the oscillation amplitude in resistance as a function of I dc. fabrication process, it is usually observed that the carrier concentration of the InAs underneath the Nb electrodes is enhanced, and the electron mobility is much reduced [25]. This gives a mean free path of 10 nm under the Nb electrodes. It is not clear whether the principle of operation will hold under such interface conditions. 9.6 Conclusions In conclusion, we have shown that the decay length of the incoming electrons in a 2DEG channel surrounded by superconducting boundaries is sensitive to the superconducting phase dierence. The oscillation amplitude in the transfer-resistance R t through the channel scales with the resistance R i between the injector and the channel. The oscillation amplitude in R t due to the decay length modulation is less than 0.05 % in this experiment. It might be necessary to measure the sample with normal injector and receiver electrodes to eliminate the DC-SQUID action of the device. References [1] B. Z. Spivak and D. E. Khmel'nitskii, JETP. Lett. 35, 412 (1982). [2] B. L. Al'tshuler and B. Z. Spivak, Sov. Phys. JETP 65, 343 (1987).

12 Superconductor phase sensitive transport [3] H. Nakano and H. Takaynagi, Solid Stat Commun. 80, 997 (80). [4] H. Nakano and H. Takayanagi, Phys. Rev. B 47, 7986 (1993). [5] F. W. J. Hekking and Y. V. Nazarov, Phys. Rev. Lett. 71, 1625 (1992). [6] C. J. Lambert, J. Phys.: Condens. Matter 5, 707 (1993). [7] V. C. Hui and C. J. Lambert, Europhys. Lett. 23, 203 (1993). [8] C. J. Lambert and S. J. Robinson, Physica B B, 1641 (1994). [9] A. V. Zaitsev, Phys. Lett. A194, 315 (1994). [10] A. V. Zaitsev, JETP Lett. 61, 771 (1995). [11] N. K. Allsopp, J. S. Canizares, R. Raimondi, and C. J. Lambert, Jour. Phys. 8, 377 (1996). [12] A. Dimoulas et al., Phys. Rev. Lett. 74, 602 (1995). [13] S. G. den Hartog et al., Phys. Rev. Lett. 76, 4592 (1996). [14] P.G.N.deVegvar, T. A. Fulton, W. H. Mallison, and R. E. Miller, Phys. Rev. Lett. 73, 1416 (1994). [15] H. Pothier, S. Gueron, D. Esteve, and M. H. Devoret, Phys. Rev. Lett. 73, 2488 (1994). [16] V. T. Petrashov, V. N. Antonov, P. Delsing, and T. Claeson, JETP Lett. 60, 606 (1994). [17] I. O. Kulik, Zh. Eksp. Teor. Fiz. 57, 1745 (1969), [Sov. Phys. JETP 30, 944 (1970)]. [18] C. Ishii, Prog. Theor. Phys. 44, 1525 (1970). [19] J. Bardeen and J. L. Johnson, Phys. Rev. B 5, 72 (1972). [20] A. Furusaki and M. Tsukada, Phys. Rev. B 43, (1991). [21] C. W. J. Beenakker, Phys. Rev. Lett. 67, 3836 (1991). [22] G. E. Blonder, M. Tinkham, and T. M. Klapwijk, Phys. Rev. B 25, 4515 (1982).

13 134 Chapter 9 [23] All higher mode indices belong to states that are evanescent in the normal state. These states decay on a length scale comparable to the Fermi wavelength, and their contribution to the electron transport is negligible. [24] Due to the bandstructure of the InAs-GaSb system, a hole gas is present in the GaSb layer. [25] P. H. C. Magnee et al., Appl. Phys. Lett. 67, 3569 (1995). [26] A. Barone and G. Paterno, Physics and Applications of Josephson junctions (John Wiley & Sons, New York, 1982), pp. 405{406.