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Interaction of microwaves with backscattering orbits in open quantum dots
Available online at www.sciencedirect.com
Physica E 22 (2004) 514 – 517 www.elsevier.com/locate/physe
Interaction of microwaves with backscattering orbits in open quantum dots R. Brunnera;∗ , R. Meiselsa , F. Kuchara , M. ElHassanb , J. Birdb , K. Ishibashic a Department
of Physics, University of Leoben, Austria of Electrical Engineering, Arizona State University, USA c Semiconducter Laboratory, RIKEN, Saitama, Japan
b Department
Abstract We report on the investigation of ballistic magneto-transport in open AlGaAs/GaAs dots and its response to microwaves in the frequency range around 50 GHz. The results allow to distinguish between backscattering peaks and Shubnikov-de-Haas oscillations in the transition range from classical to quantum-mechanical behaviour. ? 2003 Elsevier B.V. All rights reserved. PACS: 73.20.D Keywords: Open quantum dot; Microwaves
1. Introduction A particularly interesting aspect of the physics of quantum dots regards the crossover from the classical to the quantum-mechanical regime. In this paper, we focus on the behaviour exhibited by open quantum dots in low and medium magnetic ;elds. Open quantum dots are strongly coupled to the 2DEG reservoirs via short constrictions. They support one or more propagating modes so that the Coulomb blockade is suppressed. At high magnetic ;elds quantisation of the electron motion leads to the formation of well separated Landau bands and the Shubnikov-de-Haas (SdH) e>ect. At low ;elds
∗ Corresponding author. Tel.: +43-3842-402-267; fax: +433842-402-762. E-mail address: [email protected] (R. Brunner).
the Landau bands overlap and the peaked structure in the density-of-states is lost. Then, the transport can be treated on a classical basis, where trajectories at the Fermi energy under the inEuence of the Lorentz force are considered. In ballistic open dots it has been shown that certain trajectories exit the dot at the entrance constriction leading to the so-called backscattering peaks [1,2]. The purpose of this work is to investigate the magnetoresistance and its response to microwaves (MW) in the frequency range around 50 GHz at low temperatures. The frequencies are by about a factor of 2 higher than those corresponding to the measuring temperature of 1:2 K. Dot level spacings can be of the same order allowing for transitions at the MW frequencies [3]. The results of the experiments are used to distinguish between backscattering peaks and SdH oscillations in the transition range from classical to quantum-mechanical behaviour.
1386-9477/$ - see front matter ? 2003 Elsevier B.V. All rights reserved. doi:10.1016/j.physe.2003.12.058
R. Brunner et al. / Physica E 22 (2004) 514 – 517
515
2. Experimental set-up
3. Results Magnetoresistance traces, R(B), without MW radiation for di>erent gate voltages are shown in Fig. 1. Additionally, the MW-induced change of the resistance, JR, is shown for a MW frequency of 51 GHz. In R(B) at the higher ;elds SdH oscillations are observed with positions independent of the gate voltage. At medium ;elds the SdH amplitude decreases with decreasing ;eld. e.g. the peak between the minima with ;lling factors (n) 18 and 20 (peak at B = 0:61 T) is much smaller than that one between n = 16 and 18 (B = 0:7 T). However, a stronger peak appears at lower ;eld (B = 0:55 T). A very strong peak occurs at B = 0:22 T which has been attributed to backscattering in previous work [3,4]. This peak shows a dependence on Vg with the position shifted to higher ;elds with increasing Vg . The JR signal depends on the square root of the MW power and saturates at high power levels. At low power the dependence is linear. The power for the JR curve in Fig. 1 is in the linear regime. At low and medium magnetic ;elds the MW response is strongest
0 4
∆R (kΩ )
0.00 0 0 3
R (k Ω )
The experiments were carried out in an array of three dots which were de;ned by ;nger gate structures on top of a two dimensional electron system in AlGaAs/GaAs using electron beam lithography. The connections to neighbouring dots and to the 2DEG reservoirs are made by constrictions (lithographic width 70 nm). The electronic dot diameter was determined to be 185 nm (see below). The carrier density and mobility were about 3 × 1011 cm−2 and 1 × 106 cm2 =V s, respectively, at liquid helium temperature. The measurements were performed in a top-loading 3 He cryostat with a sample holder allowing microwave transmission through a stainless-steel wave guide. The microwave sources employed are Gunn oscillators covering the frequency ranges 45 –55 GHz. The detection of the magnetoresistance R and of the MW-induced signal JR is performed by a double modulation method. The current was modulated at a frequency of 410 Hz, the microwave power at 12 Hz.
0 2
-0.2 -2505 Vg =- 0. 44V Vg =- 0. 42V Vg =- 0. 40V Vg =- 0. 38V
0 1
22
20
18
16
14
12
10
51GH z, Vg=- 0.44 V
0.0
0.2
0.4
-0.5 -5000 0.6
0.8
1.0
1.2
1.4
B (T)
Fig. 1. Magnetoresistance at 1:2 K for di>erent gate voltages. The number show the positions of the minima and correspond to ;lling factors of the Landaulevels. The thick line shows the microwave induced change JR for a frequency of 51 GHz and Vg =−0:44 V. The symbols above represent the position of the backscattering peaks obtained from the soft-wall calculation, empty symbols mark the positions of the weaker peaks.
for the 0:22 T peak. At medium ;elds the JR curve is signi;cantly di>erent from R(B) indicating that it does not simply reEect a damping of the SdH oscillations due to electron heating by the MW radiation. Broad structures appear with hardly visible remnants of the SdH oscillations superimposed. Only the SdH remnants are considered as being due to a damping of the oscillation caused by electron heating. In the discussion we will show that the broad structures have the same origin as the strong 0:22 T peak, viz. backscattering. At high ;elds (n 6 10)JR is dominated by the SdH structure. In the following, we present data regarding the positions of the SdH minima and maxima and of the strong low-;eld peaks. Fig. 2 shows the positions of the minima in the magnetic ;eld range 0.35 –4 T for a series of MW frequencies. In the higher ;eld range the positions are independent of frequency whereas a strong frequency dependence develops when lowering the ;eld. For comparison the positions of the minima from the R(B) measurements are given as dashed lines. When numbering the minima in the usual way in terms of Landau level ;lling factor, the period on a 1=B scale is constant at high ;elds and increases at lower ;elds (see also Fig. 3). As mentioned above the oscillations in R(B) get very small around 0:6 T
516
R. Brunner et al. / Physica E 22 (2004) 514 – 517
3
0.315 2.85
n=18 n=16 n=14 n=12 n=10 n=8 n=6 n=4
1
B (T)
-1
1/ B (T )
2
0.270
B (T)
n=22
0.225
0.180 2.80 44
0 46
48
50
52
54
46
48
50
52
54
56
f (GHz)
56
f (GHz)
Fig. 2. Positions of minima of JR are plotted versus the MW-frequency f. The dashed lines show the positions of the minima evaluated from the DC data.
Fig. 4. Magnetic ;eld positions in JR versus MW-frequency. The square symbols show the positions of the maxima at 2:8 T, the round symbols show the maxima at 0:22 T. The dashed lines mark the positions of the maxima obtained from the DC data.
4. Discussion 3 46GHz 47GHz 49GHz 53GHz 54GHz 2 -1
1/ B (T )
DC n0d~k 2/3 (ω 02 +(eB/m*)2 )-1/2 d=370nm
1
0 0
2
4
6
8
10
12
14
16
18
20
22
24
n
Fig. 3. Positions of minima of JR versus ;lling factor. The solid line represent the calculated results assuming hybrid magneto electric subbands.
resulting in the disappearance of the n = 20 minimum. The occurrence of the peak at 0:55 T produces a minimum at about 0:45 T which is preliminarily attributed to n = 22. Its position shows the strongest frequency dependence. In Fig. 4 we compare the positions of the SdH maximum at 2.83 and of the peak at 0:22 T. In contrast to the high-;eld SdH maximum the low-;eld peak shows a strong frequency dependence.
Generally, in the magneto-transport of 2D electron systems the SdH maxima occur at the maxima of the density-of-states of the Landau levels. A backscattering peak occurs when an electron enters and exits the dot through the same constriction. In a hard-wall approximation the condition for backscattering is r = rc =tan( =(2k + 1)). r is the dot radius, rc = h(2 ns )1=2 =eB is the classical cyclotron radius, k = 1; 2; 3; : : : [1]. k = 1 corresponds to a quasi-triangular trajectory with two reEections in the dot. The electron density ns determines the Fermi velocity vf via vf = h(2 ns )1=2 =m. Therefore, in contrast to the SdH maxima the backscattering peaks are not related to structure in the density-of-states but to certain vf values ;tting the conditions given above. We could show that a soft-wall calculation yields much better agreement also for higher n values. For any symmetric smooth continuous varying potential having minimum centre at the dot second-order parabolic potential is a good approximation. This potential has the bene;t that the movement of the electron can be calculated analytically. In general the electrons enter the dot at di>erent angles w.r.t. to the direction of the centre. While the trajectories for the backscattering peak k = 1 vary strongly with in the hard-wall (abrupt con;ning potential) the trajectories return almost the same point (after two reEections) for di>erent in the soft wall model. As
R. Brunner et al. / Physica E 22 (2004) 514 – 517
a result the resistance peaks in the hard-wall model are completely smoothed out [4]. Therefore, in the following discussion we will use the soft-wall results only. In order to distinguish backscattering peaks and SdH oscillations we mainly use details of our MW-JR results: (1) Backscattering peaks show a stronger response to the microwave radiation as clearly demonstrated by the strong JR signal at 0:22 T (k = 1). This proves the usefulness of the microwave experiment. Also the microwave-induced broad peaks at medium ;elds are at least of the same strength as the remnants of the SdH peaks. This is a ;rst indication that they can be attributed to backscattering with higher k values. This is supported by the soft-wall calculation (positions given in Fig. 1). The calculation yields backscattering peaks partly in close vicinity to SdH maxima, however, not to all of them (n = 15 and 11). (2) The peak in R at 0:55 T might be attributed to the ;lling factor n = 21 assuming hybrid magneto-electric subbands as shown by the solid curve in Fig. 3. This curve is calculated according to n ˙ 1=(!02 + !c2 )1=2 . !0 = h(2 ns )1=2 =mr is the con;ning-potential frequency, !c = eB=m the cyclotron frequency. The value of r (185 nm) is obtained from ;tting the position of the 0:22 T peak by the soft-wall calculation. However, the JR-peak at 0:55 T is stronger than expected from the decay of the SdH oscillation amplitude. Moreover, this JR-peak shows a dependence of its position on the gate voltage and MW frequency as the 0:22 T backscattering peak does. Therefore, it is also considered as a backscattering peak possibly with the SdH maximum of n = 21 superimposed. The appearance of the peak at 0:4 T in the soft-wall results
517
and in the JR experiments only again demonstrates that the microwave response is particularly sensitive to the backscattering peaks. (3) The calculation shows that the backscattering peaks are broader than the SdH period [4]. With dominating backscattering peaks in JR the peak structures should be broad as observed experimentally. (4) Considering the points above and the correspondences indicated in Fig. 1 we conclude that beside the strong peak at 0:22 T backscattering peaks occur also at 0.4, 0.55, 0.7, and 1 T. They are resolved in the MW-induced JR but not in R. 5. Conclusion The measurement of the microwave response of the magnetoresistance enabled us to identify backscattering peaks in the cross-over regime from classical to quantum-mechanical behaviour of the transport in open quantum dots. Acknowledgements This work was supported by the “Fonds zur FMorderung der wissenschaftlichen Forschung”, Austria (P15513). References [1] [2] [3] [4]
L.H. Lin, et al., Physica E 7 (2000) 750. H. Linke, et al., Phys. Rev. B 56 (1997) 1440. R. Brunner, et al., Phys. Stat. Sol. (c) (2003) 1321. R. Brunner, et al., Physica E (2004), in press.
Physica E 22 (2004) 514 – 517 www.elsevier.com/locate/physe
Interaction of microwaves with backscattering orbits in open quantum dots R. Brunnera;∗ , R. Meiselsa , F. Kuchara , M. ElHassanb , J. Birdb , K. Ishibashic a Department
of Physics, University of Leoben, Austria of Electrical Engineering, Arizona State University, USA c Semiconducter Laboratory, RIKEN, Saitama, Japan
b Department
Abstract We report on the investigation of ballistic magneto-transport in open AlGaAs/GaAs dots and its response to microwaves in the frequency range around 50 GHz. The results allow to distinguish between backscattering peaks and Shubnikov-de-Haas oscillations in the transition range from classical to quantum-mechanical behaviour. ? 2003 Elsevier B.V. All rights reserved. PACS: 73.20.D Keywords: Open quantum dot; Microwaves
1. Introduction A particularly interesting aspect of the physics of quantum dots regards the crossover from the classical to the quantum-mechanical regime. In this paper, we focus on the behaviour exhibited by open quantum dots in low and medium magnetic ;elds. Open quantum dots are strongly coupled to the 2DEG reservoirs via short constrictions. They support one or more propagating modes so that the Coulomb blockade is suppressed. At high magnetic ;elds quantisation of the electron motion leads to the formation of well separated Landau bands and the Shubnikov-de-Haas (SdH) e>ect. At low ;elds
∗ Corresponding author. Tel.: +43-3842-402-267; fax: +433842-402-762. E-mail address: [email protected] (R. Brunner).
the Landau bands overlap and the peaked structure in the density-of-states is lost. Then, the transport can be treated on a classical basis, where trajectories at the Fermi energy under the inEuence of the Lorentz force are considered. In ballistic open dots it has been shown that certain trajectories exit the dot at the entrance constriction leading to the so-called backscattering peaks [1,2]. The purpose of this work is to investigate the magnetoresistance and its response to microwaves (MW) in the frequency range around 50 GHz at low temperatures. The frequencies are by about a factor of 2 higher than those corresponding to the measuring temperature of 1:2 K. Dot level spacings can be of the same order allowing for transitions at the MW frequencies [3]. The results of the experiments are used to distinguish between backscattering peaks and SdH oscillations in the transition range from classical to quantum-mechanical behaviour.
1386-9477/$ - see front matter ? 2003 Elsevier B.V. All rights reserved. doi:10.1016/j.physe.2003.12.058
R. Brunner et al. / Physica E 22 (2004) 514 – 517
515
2. Experimental set-up
3. Results Magnetoresistance traces, R(B), without MW radiation for di>erent gate voltages are shown in Fig. 1. Additionally, the MW-induced change of the resistance, JR, is shown for a MW frequency of 51 GHz. In R(B) at the higher ;elds SdH oscillations are observed with positions independent of the gate voltage. At medium ;elds the SdH amplitude decreases with decreasing ;eld. e.g. the peak between the minima with ;lling factors (n) 18 and 20 (peak at B = 0:61 T) is much smaller than that one between n = 16 and 18 (B = 0:7 T). However, a stronger peak appears at lower ;eld (B = 0:55 T). A very strong peak occurs at B = 0:22 T which has been attributed to backscattering in previous work [3,4]. This peak shows a dependence on Vg with the position shifted to higher ;elds with increasing Vg . The JR signal depends on the square root of the MW power and saturates at high power levels. At low power the dependence is linear. The power for the JR curve in Fig. 1 is in the linear regime. At low and medium magnetic ;elds the MW response is strongest
0 4
∆R (kΩ )
0.00 0 0 3
R (k Ω )
The experiments were carried out in an array of three dots which were de;ned by ;nger gate structures on top of a two dimensional electron system in AlGaAs/GaAs using electron beam lithography. The connections to neighbouring dots and to the 2DEG reservoirs are made by constrictions (lithographic width 70 nm). The electronic dot diameter was determined to be 185 nm (see below). The carrier density and mobility were about 3 × 1011 cm−2 and 1 × 106 cm2 =V s, respectively, at liquid helium temperature. The measurements were performed in a top-loading 3 He cryostat with a sample holder allowing microwave transmission through a stainless-steel wave guide. The microwave sources employed are Gunn oscillators covering the frequency ranges 45 –55 GHz. The detection of the magnetoresistance R and of the MW-induced signal JR is performed by a double modulation method. The current was modulated at a frequency of 410 Hz, the microwave power at 12 Hz.
0 2
-0.2 -2505 Vg =- 0. 44V Vg =- 0. 42V Vg =- 0. 40V Vg =- 0. 38V
0 1
22
20
18
16
14
12
10
51GH z, Vg=- 0.44 V
0.0
0.2
0.4
-0.5 -5000 0.6
0.8
1.0
1.2
1.4
B (T)
Fig. 1. Magnetoresistance at 1:2 K for di>erent gate voltages. The number show the positions of the minima and correspond to ;lling factors of the Landaulevels. The thick line shows the microwave induced change JR for a frequency of 51 GHz and Vg =−0:44 V. The symbols above represent the position of the backscattering peaks obtained from the soft-wall calculation, empty symbols mark the positions of the weaker peaks.
for the 0:22 T peak. At medium ;elds the JR curve is signi;cantly di>erent from R(B) indicating that it does not simply reEect a damping of the SdH oscillations due to electron heating by the MW radiation. Broad structures appear with hardly visible remnants of the SdH oscillations superimposed. Only the SdH remnants are considered as being due to a damping of the oscillation caused by electron heating. In the discussion we will show that the broad structures have the same origin as the strong 0:22 T peak, viz. backscattering. At high ;elds (n 6 10)JR is dominated by the SdH structure. In the following, we present data regarding the positions of the SdH minima and maxima and of the strong low-;eld peaks. Fig. 2 shows the positions of the minima in the magnetic ;eld range 0.35 –4 T for a series of MW frequencies. In the higher ;eld range the positions are independent of frequency whereas a strong frequency dependence develops when lowering the ;eld. For comparison the positions of the minima from the R(B) measurements are given as dashed lines. When numbering the minima in the usual way in terms of Landau level ;lling factor, the period on a 1=B scale is constant at high ;elds and increases at lower ;elds (see also Fig. 3). As mentioned above the oscillations in R(B) get very small around 0:6 T
516
R. Brunner et al. / Physica E 22 (2004) 514 – 517
3
0.315 2.85
n=18 n=16 n=14 n=12 n=10 n=8 n=6 n=4
1
B (T)
-1
1/ B (T )
2
0.270
B (T)
n=22
0.225
0.180 2.80 44
0 46
48
50
52
54
46
48
50
52
54
56
f (GHz)
56
f (GHz)
Fig. 2. Positions of minima of JR are plotted versus the MW-frequency f. The dashed lines show the positions of the minima evaluated from the DC data.
Fig. 4. Magnetic ;eld positions in JR versus MW-frequency. The square symbols show the positions of the maxima at 2:8 T, the round symbols show the maxima at 0:22 T. The dashed lines mark the positions of the maxima obtained from the DC data.
4. Discussion 3 46GHz 47GHz 49GHz 53GHz 54GHz 2 -1
1/ B (T )
DC n0d~k 2/3 (ω 02 +(eB/m*)2 )-1/2 d=370nm
1
0 0
2
4
6
8
10
12
14
16
18
20
22
24
n
Fig. 3. Positions of minima of JR versus ;lling factor. The solid line represent the calculated results assuming hybrid magneto electric subbands.
resulting in the disappearance of the n = 20 minimum. The occurrence of the peak at 0:55 T produces a minimum at about 0:45 T which is preliminarily attributed to n = 22. Its position shows the strongest frequency dependence. In Fig. 4 we compare the positions of the SdH maximum at 2.83 and of the peak at 0:22 T. In contrast to the high-;eld SdH maximum the low-;eld peak shows a strong frequency dependence.
Generally, in the magneto-transport of 2D electron systems the SdH maxima occur at the maxima of the density-of-states of the Landau levels. A backscattering peak occurs when an electron enters and exits the dot through the same constriction. In a hard-wall approximation the condition for backscattering is r = rc =tan( =(2k + 1)). r is the dot radius, rc = h(2 ns )1=2 =eB is the classical cyclotron radius, k = 1; 2; 3; : : : [1]. k = 1 corresponds to a quasi-triangular trajectory with two reEections in the dot. The electron density ns determines the Fermi velocity vf via vf = h(2 ns )1=2 =m. Therefore, in contrast to the SdH maxima the backscattering peaks are not related to structure in the density-of-states but to certain vf values ;tting the conditions given above. We could show that a soft-wall calculation yields much better agreement also for higher n values. For any symmetric smooth continuous varying potential having minimum centre at the dot second-order parabolic potential is a good approximation. This potential has the bene;t that the movement of the electron can be calculated analytically. In general the electrons enter the dot at di>erent angles w.r.t. to the direction of the centre. While the trajectories for the backscattering peak k = 1 vary strongly with in the hard-wall (abrupt con;ning potential) the trajectories return almost the same point (after two reEections) for di>erent in the soft wall model. As
R. Brunner et al. / Physica E 22 (2004) 514 – 517
a result the resistance peaks in the hard-wall model are completely smoothed out [4]. Therefore, in the following discussion we will use the soft-wall results only. In order to distinguish backscattering peaks and SdH oscillations we mainly use details of our MW-JR results: (1) Backscattering peaks show a stronger response to the microwave radiation as clearly demonstrated by the strong JR signal at 0:22 T (k = 1). This proves the usefulness of the microwave experiment. Also the microwave-induced broad peaks at medium ;elds are at least of the same strength as the remnants of the SdH peaks. This is a ;rst indication that they can be attributed to backscattering with higher k values. This is supported by the soft-wall calculation (positions given in Fig. 1). The calculation yields backscattering peaks partly in close vicinity to SdH maxima, however, not to all of them (n = 15 and 11). (2) The peak in R at 0:55 T might be attributed to the ;lling factor n = 21 assuming hybrid magneto-electric subbands as shown by the solid curve in Fig. 3. This curve is calculated according to n ˙ 1=(!02 + !c2 )1=2 . !0 = h(2 ns )1=2 =mr is the con;ning-potential frequency, !c = eB=m the cyclotron frequency. The value of r (185 nm) is obtained from ;tting the position of the 0:22 T peak by the soft-wall calculation. However, the JR-peak at 0:55 T is stronger than expected from the decay of the SdH oscillation amplitude. Moreover, this JR-peak shows a dependence of its position on the gate voltage and MW frequency as the 0:22 T backscattering peak does. Therefore, it is also considered as a backscattering peak possibly with the SdH maximum of n = 21 superimposed. The appearance of the peak at 0:4 T in the soft-wall results
517
and in the JR experiments only again demonstrates that the microwave response is particularly sensitive to the backscattering peaks. (3) The calculation shows that the backscattering peaks are broader than the SdH period [4]. With dominating backscattering peaks in JR the peak structures should be broad as observed experimentally. (4) Considering the points above and the correspondences indicated in Fig. 1 we conclude that beside the strong peak at 0:22 T backscattering peaks occur also at 0.4, 0.55, 0.7, and 1 T. They are resolved in the MW-induced JR but not in R. 5. Conclusion The measurement of the microwave response of the magnetoresistance enabled us to identify backscattering peaks in the cross-over regime from classical to quantum-mechanical behaviour of the transport in open quantum dots. Acknowledgements This work was supported by the “Fonds zur FMorderung der wissenschaftlichen Forschung”, Austria (P15513). References [1] [2] [3] [4]
L.H. Lin, et al., Physica E 7 (2000) 750. H. Linke, et al., Phys. Rev. B 56 (1997) 1440. R. Brunner, et al., Phys. Stat. Sol. (c) (2003) 1321. R. Brunner, et al., Physica E (2004), in press.