- Email: [email protected]
Probing the coherent transport through coupled quantum dots
Superlattices and Microstructures, Vol. 23, No. 6, 1998 Article No. sm960145
Probing the coherent transport through coupled quantum dots R. H. B LICK† , D. W. VAN DER W EIDE‡ , R. J. H AUG§ , K. VON K LITZING , K. E BERL Max-Planck-Institut f¨ur Festk¨orperforschung, Heisenbergstr. 1, D-70569 Stuttgart, Germany
(Received 20 May 1996) We investigate the photoconductance through a double quantum dot or ‘artificial molecule’ induced by a broadband millimeter wave source. This source functions as a heterodyne interferometer, consisting of two nonlinear transmission lines generating harmonic output in the range of 2–400 GHz, and, being coherent, allows tracking of the induced current of the sample in both magnitude and phase. This enables us to monitor effects related to coherent electronic transport through these ‘artificial molecules’. c 1998 Academic Press Limited
Key words: tunneling, quantum dots, microwave sources.
1. Introduction Photon-assisted tunneling (PAT) through single quantum dots has recently been reported, and is applied now as a tool for millimeter wave spectroscopy on such devices [1, 2]. Quantum dots contain approximately 50 electrons and are called ‘artificial atoms’. Transport through these devices is governed by Coulomb blockade (CB) [3]. This results in an oscillating behavior of the conductance through the dot as a function of gate voltages, which vary the electrostatic potentials. The measurements under the influence of microwave radiation performed so far give evidence that the well-known effect of CB can be overcome. Furthermore, such timedependent measurements should give more information about coherent electronic modes in quantum dots. Similar approaches were taken by Karadi et al. [4] for investigations on the dynamic response of a quantum point contact and by Dahl et al. [5] in measurements on metallic discs. Other measurements on superconductor tunnel junctions have shown the importance of electronic phase-effects (quantum susceptance) on electronic transport [6].
2. Motivation In quantum-dot systems to date, only continuous wave (cw) frequency sources have been applied for spectroscopy in the range from some MHz [7] up to 200 GHz [2]. Furthermore, the common method of † Present address: California Institute of Technology, Condensed Matter Physics 114-36, Pasadena, CA 91125, U.S.A.
E-mail: [email protected] ‡ Present address: University of Delaware, Dept. of Electrical Engineering, 140 Evans Hall, Newark, DE 19716, U.S.A. § Present address: Universit¨at Hannover, Institut f¨ur Festk¨orperphysik, Applestr. 2, 30167 Hannover, Germany.
0749–6036/98/061265 + 07
$25.00/0
c 1998 Academic Press Limited
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Superlattices and Microstructures, Vol. 23, No. 6, 1998
detection gives only scalar information, i.e. shows signals in the magnitude of the induced current without information on the phase. Recent work of ours demonstrated the feasibility of integrated broadband antennas to couple effectively cw millimeter and submillimeter radiation to nanostructures, such as quantum dots and double quantum dots, showing the expected PAT resonances in transconductance. The detection mechanism is based upon absorption of photons by the electrons in the leads and in the dot, leading to PAT. The energy acquired by the electrons allows them to tunnel through states of higher energy above the Fermi energy, which are otherwise not accessible. Here we report on transport experiments, performed on a coupled quantum dot structure or ‘artificial molecule’ under irradiation by a new type of broadband millimeter wave interferometer. This source generates picosecond pulses of radiation, corresponding to harmonics in a frequency range of 2–400 GHz. Because these pulses are coherent we can obtain magnitude and phase information on the signal induced by the pulse radiation. This is of major interest, since the double quantum dot we investigate contains only a few electrons and should thus evolve molecular modes, i.e. an electron’s wave function can be spread out across the whole double dot. The influence of the molecular modes can now be probed by the interferometer. The organization of the article is as follows: in the first part we will discuss the quantum dot device used in our experiment and in the second part we introduce the setup of the broadband source. Finally, we present measurements of the photoconductance induced by the broadband source and show the amplitude and phase dependencies.
3. Experimental setup and millimeter wave interferometer The quantum dots used in this experiment were realized in an Alx Ga1−x As–GaAs heterostructure grown by molecular beam epitaxy. The heterostructure has a carrier density of 2.11×1015 m−2 and an electron mobility of 80 m2 V−1 s−1 at liquid helium temperatures. Contacts of gold were evaporated serving as gate-contacts as well as an antenna structure. A number of split gates were then written by electron beam lithography in the center of the gate structure. In Fig. 1A the ‘bow-tie’ antenna is shown; in its center the quantum dots are defined, which can be seen in the SEM micrograph. The measurements were performed in a top-loading 3 He/4 He dilution refrigerator with a sample holder allowing quasi-optical transmission [2]. The method of optical-coupling is useful at high frequencies, since the quasi-optical waveguide allows transmission up to THz-frequencies. The temperature in the measurements presented is around 250 mK, although the electron temperature is slightly above this value, due to the irradiation. The performance of the antenna was verified in detailed numerical analysis with a commercially-available program [8]. The double quantum dot system (see right part of Fig. 1A) is obtained by applying negative voltages to the gate fingers, thus depleting the electrons below the gates. The voltages we used are typically in the range of −0.4 to −0.7 V. In the measurements we found CB energies of E CA = e2 /C6 ≈ 3.0 meV for the small dot A and for the large dot B E CB ∼ = (1.17 ± 0.1) meV, (C6 represents the total capacitance of the quantum dot). The charging energy of the small quantum dot varies, depending on the number of electrons in the dot. The complete characterization of such a double quantum dot is performed within the so-called charging diagram, which is discussed in detail elsewhere [9]. The effect of cw millimeter-wave radiation on a double dot is described by Blick et al. [10]. The principal setup of the photoconductance experiment with the broadband interferometer is shown in Fig. 1. The interferometer consists of two nonlinear transmission lines (NLTLs) with integrated antennas [11], which can generate and radiate high-frequency signals up into the THz regime. The interferometer superposes the two broadband spectra from the NLTLs with an intermediate offset frequency δν. In the data shown, this offset is δν = 23 Hz, while the driving frequency is ν = 7 GHz. These spectra are optically coupled into the waveguide and transduced by the antenna of the artificial molecule. The high frequency radiation then induces a photoconductance through the microstructure, which can easily be measured in both magnitude and phase because of the coherence of the interferometer. This technique thus allows direct probing without
Superlattices and Microstructures, Vol. 23, No. 6, 1998
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3 mm
A
1 µm
B
t (psec)
Synthesizer 7 GHz
Source 'A'
phase-locked Synthesizer 7 GHz + δν
Quantum dot sample
Source 'B'
Vds
Fig. 1. A, The broadband antenna of the device is shown: the antenna arms serve as gate contacts for the quantum point contacts defining the two quantum dots of different size (see right-hand side). B, Schematically shown is the setup of the millimeter-wave interferometer: the radiation of two nonlinear transmission lines (‘A’ and ‘B’) is superimposed, radiated onto the sample, and induces a drain-source photo-conductance.
application of the Kramers–Kronig relation. A more detailed description of the interferometer was given elsewhere [12]. The detection of photoconductivity is performed within the usual conductance measurement setup, which is applied also to characterize the quantum dots. A finite ac excitation voltage of Vexc = 5 µV k B T /e induces a current at νexc = 14 Hz, which is then amplified and detected by a lock-in. The radiation applied induces a current signal, being caused by either a non-resonant bolometric or PAT tunneling. This photocurrent is also preamplified and read out by a spectrum analyzer and lock-in. In locking onto one of the appropriate harmonics, which are separated by nδν, the corresponding high frequency conductance signal is obtained. In the measurements shown the data was taken in amplitude/phase-mode of the lock-in. It should be mentioned that in the current setup the non-resonant bolometric effects remain constant, since all different frequency components of the radiation are applied simultaneously. With phase-sensitive detection the frequency of interest can be selected.
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dI/dV (µS)
2
–70 α
β
1
γ 0 0.0
–80
–0.2
–0.4
–0.6
A (–dBm)
VTG (V)
–0.8
ζ2
ζ1 –90
–100
0
100
200 ν (GHz)
300
400
Fig. 2. Response of the quantum dot system to the interferometer radiation up to 400 GHz. The data points with the largest response values are marked by α, β, and γ corresponding to frequencies of 28, 42, and 56 GHz. The resonances labeled by ζ1 , ζ2 represent excitations of the small quantum dot A. The inset shows the Coulomb blockade oscillations of the double quantum dot under variation of the top gate voltage (VT G ). The arrow indicates the point of measurement
4. Results and discussion In the inset of Fig. 2 the conductance resonances of the double quantum dot in dependence of the top gate voltage (VT G ) are shown. One clearly recognizes the two different periodicities corresponding to the small quantum dot A (large period) and the large dot B (smaller period). Although the temperature in the measurements presented is around 250 mK, the CB resonances reveal a broadening due to the constant nonresonant heating of the electrons in the leads by the radiation applied. The double dot was chosen for these measurements, since electron-transport investigations of such devices have shown that the wavefunctions in the two dots effectively couple in a molecular-like fashion [9]. In our case the interdot coupling strength is found in the transport measurements to be on the order of t ≈ 0.15E CB . In Fig. 2 we present the response of the double dot system to broadband radiation with the gate voltages fixed (at the position of the arrow in the inset). The amplitude of the signal induced by the radiation is plotted versus frequency. Each square point of the discrete spectrum corresponds to a harmonic generated by the interferometer. The best response is obtained in the lower frequency regime. At frequencies below the point marked α the cutoff frequency of the waveguide damps the millimeter-wave signal. We note that the usual output power of the interferometer decreases towards higher frequencies as ν −1 . However, the overall sensitivity of the ‘artificial molecule’ is comparatively high, since well-defined signals were measured up to 350 GHz. The spectrum shown can be divided into two parts: the low energy excitations of the large quantum dot (α, β and γ ) and the higher energy excitations of the small quantum dot (ζ1 and ζ2 ). The different energies of the excitations result from the strong variation of the different confinement potentials of the two dots, i.e. the small quantum dot possesses a stronger confinement potential compared to the larger dot. In extension of
Superlattices and Microstructures, Vol. 23, No. 6, 1998 β: 42 GHz π
0.3 0.2 0.1
π –0.2
–0.4
–0.6
0.3 0.2 0.1 0.0
π –0.2
γ: 56 GHz
–0.4
–0.6
π Phase
0.015 σ (µS)
Phase
0.4 Phase σ (µS)
σ (µS)
0.5
α: 28 GHz π
0.4
0.0
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0.010 0.005 π 0.000
–0.2
–0.4 VTG (V)
–0.6
Fig. 3. Induced photo-conductance in R/2-mode for different frequencies α, β, and γ as noted in Fig. 2 versus top gate voltage (VT G ). The solid lines represent the signal amplitude, while the dotted curves give the signal’s phase.
the measurements on quantum-dot lattices with FIR-spectroscopy [13], we now have the ability to perform spectroscopy on single and coupled quantum dots. To focus on the different excitations we select the appropriate harmonic and change the charging state of the artificial molecule by variation of the gate voltage. In Fig. 3 the high-frequency conductance of the dots is given in amplitude/phase-mode of the lock-in; the bias was fixed in the low bias-regime at Vds = −60 µV thus enhancing the resolution. The left axis gives the magnitude and the right axis the relative phase of the conductance signal. From top to bottom the frequency increases, as is marked. The individual curves correspond to the points α, β, and γ in Fig. 3 as indicated, where the response of the dot system was strongest. In general the conductance signal can be understood as a phasor. The coherent signal from the millimeter-wave interferometer enables the detection of conductance (σ ) and relative phase (φ) in the high frequency domain. As seen the conductance amplitude resembles the two periodicities of the conductance resonances found in the common conductance measurement. Towards the highest frequency of 56 GHz shown the amplitude decreases, which is in accordance with the amplitude we found in the broadband response. For comparison we performed measurements with single mode frequency sources (backward wave oscillators) at identical frequencies and power levels. The power levels were calibrated by determining the broadening of the CB conductance resonances. The signals obtained agree with the corresponding amplitudes of the conductance in Fig. 3. In measuring the current with the single sources at very low power levels we also found PAT for the double quantum dot [10]. The current resolution of the interferometer did not enable us to identify additional structure induced by PAT because of resonance broadening: the contributions of all frequencies applied by the interferometer are superimposed. This obviously prevents the observation in direct current. Our main interest here, however, lies in the determination of the phase dependence of the induced current in the high frequency regime. The phase-sensitive data shown in Fig. 3 is strongly influenced when the frequency is varied from the lowest value of α: 28 GHz to the higher values (β: 42 GHz, γ : 56 GHz). Interestingly the variation of the phase in the case α exactly reflects the charging process of the large dot (B), seen as a modulation, which corresponds to the periodicity of the charging process (see Fig. 2). This can be understood as a probing of the admittance of this particular dot and the change from capacitive to inductive behavior and vice versa [14, 15]. Increasing
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the frequency the charging process of the small dot (A) becomes visible, since the change in phase is large and dominated by the CB oscillation period of this dot. Finally, at the highest frequency the phase simply switches as soon as the small dot is to be charged, while the modulation by charging the large dot is not visible any more. Varying the bias across the sample also influences the relative phase signal. Accurate phase signals at lower frequencies were hard to access, since the waveguide cutoff reduced the signal amplitude strongly (see Fig. 2). The question arising is, how the strong variation of the phase signal towards higher frequencies is related to the electronic transport through the artificial molecule. One might state that the high frequency radiation influences a ‘sloshing mode’ [16] of the electronic wave function, i.e. the spread out wave function is localized by the external disturbance of the system. More precisely, this phenomenon can be described as Rabi-oscillations in the double quantum dot, which are probed by the radiation applied, as it was shown theoretically [17, 18]. In measurements of linear transport we found in a detailed resonance shape analysis the tunnel splitting of the double quantum dot to be on the order of δ = 2 − 1 ∼ = 110–140 µeV [19]. This energy corresponds to a Rabi-frequency of ν R = δ/ h ∼ = 27–34 GHz, which is on the order of the transition frequency observed. Hence, our externally applied disturbance with frequencies above 25 GHz tends to destroy these Rabi-oscillations. This can be regarded as direct probing of the electron’s wave function.
5. Summary In summary, we applied a coherent millimeter wave interferometer to perform coherent spectroscopy on coupled quantum dots. The interferometer is tunable over the whole millimeter wave regime and allows both magnitude- and phase-sensitive detection. In the measurements shown we have detected this high-frequency conductance through the double quantum dot. We find a broadband response, i.e. excitations of the small and the large quantum dot. The full strength of the new spectrometer is proven by the phase sensitive detection, where we observe strong variations in the phase-signals when the frequency of the radiation is larger than the Rabi-frequency. Acknowledgements—We thank D. Pfannkuche and J. Smet for stimulating discussions. We also acknowledge the excellent technical help of M. Riek and F. Schartner in sample preparation. This work was supported by the Bundesministerium f¨ur Bildung, Wissenschaft, Forschung und Technologie (BMBF).
References [1] }L. Kouwenhoven, Phys. Rev. Lett. 73, 3443 (1994). [2] }R. Blick, Appl. Phys. Lett. 67, 3924 (1995). [3] }Single Charge Tunneling, edited by H. Grabert and M. Devoret (Plenum, New York, 1992). Vol. 294 of NATO ASI. [4] }C. Karadi, J. Opt. Soc. Am. 11, 2566 (1994). [5] }C. Dahl, Phys. Rev. B 46, 15590 (1992). [6] }Q. Hu, C. Mears, P. Richards, and F. Lloyd, Phys. Rev. Lett. 64, 2945 (1990). [7] }L. Kouwenhoven, Phys. Rev. Lett. 67, 1626 (1991). [8] }Sonnet Software, Liverpool, NY. [9] }R. Blick, Phys. Rev. B 53, 7899 (1996). [10] }R. Blick, Surf. Sci. (in press). [11] }C. Madden, Appl. Phys. Lett. 54, 1019 (1989); D. van der Weide, Appl. Phys. Lett. 62, 22 (1993); D. van der Weide, Appl. Phys. Lett. 65, 881 (1995). [12] }D. van der Weide, R. Blick, F. Keilmann, and R. Haug, Quantum Optoelectronics (OSA) 14, 103 (1995). [13] }B. Meurer, D. Heitmann and K. Ploog, Phys. Rev. Lett. 68, 1371 (1992).
Superlattices and Microstructures, Vol. 23, No. 6, 1998
[14] [15] [16] [17] [18] [19]
}Y. Fu and S. Dudley, Phys. Rev. Lett. 70, 65 (1993). }T. Ivanov, V. Valtchinov, and L. Wille, Phys. Rev. B 50, 4917 (1994). }J. Tucker, IEEE J. Quantum. Electron. QE 15, 1234 (1979). }N. Tsukuda, M. Gotoda, and M. Nunoshita, Phys. Rev. B 50, 5764 (1994). }Ch. Stafford and N. S. Wingreen, Phys. Rev. Lett. 76, 1916 (1996). }R. Blick, to be published (1996).
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Probing the coherent transport through coupled quantum dots R. H. B LICK† , D. W. VAN DER W EIDE‡ , R. J. H AUG§ , K. VON K LITZING , K. E BERL Max-Planck-Institut f¨ur Festk¨orperforschung, Heisenbergstr. 1, D-70569 Stuttgart, Germany
(Received 20 May 1996) We investigate the photoconductance through a double quantum dot or ‘artificial molecule’ induced by a broadband millimeter wave source. This source functions as a heterodyne interferometer, consisting of two nonlinear transmission lines generating harmonic output in the range of 2–400 GHz, and, being coherent, allows tracking of the induced current of the sample in both magnitude and phase. This enables us to monitor effects related to coherent electronic transport through these ‘artificial molecules’. c 1998 Academic Press Limited
Key words: tunneling, quantum dots, microwave sources.
1. Introduction Photon-assisted tunneling (PAT) through single quantum dots has recently been reported, and is applied now as a tool for millimeter wave spectroscopy on such devices [1, 2]. Quantum dots contain approximately 50 electrons and are called ‘artificial atoms’. Transport through these devices is governed by Coulomb blockade (CB) [3]. This results in an oscillating behavior of the conductance through the dot as a function of gate voltages, which vary the electrostatic potentials. The measurements under the influence of microwave radiation performed so far give evidence that the well-known effect of CB can be overcome. Furthermore, such timedependent measurements should give more information about coherent electronic modes in quantum dots. Similar approaches were taken by Karadi et al. [4] for investigations on the dynamic response of a quantum point contact and by Dahl et al. [5] in measurements on metallic discs. Other measurements on superconductor tunnel junctions have shown the importance of electronic phase-effects (quantum susceptance) on electronic transport [6].
2. Motivation In quantum-dot systems to date, only continuous wave (cw) frequency sources have been applied for spectroscopy in the range from some MHz [7] up to 200 GHz [2]. Furthermore, the common method of † Present address: California Institute of Technology, Condensed Matter Physics 114-36, Pasadena, CA 91125, U.S.A.
E-mail: [email protected] ‡ Present address: University of Delaware, Dept. of Electrical Engineering, 140 Evans Hall, Newark, DE 19716, U.S.A. § Present address: Universit¨at Hannover, Institut f¨ur Festk¨orperphysik, Applestr. 2, 30167 Hannover, Germany.
0749–6036/98/061265 + 07
$25.00/0
c 1998 Academic Press Limited
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Superlattices and Microstructures, Vol. 23, No. 6, 1998
detection gives only scalar information, i.e. shows signals in the magnitude of the induced current without information on the phase. Recent work of ours demonstrated the feasibility of integrated broadband antennas to couple effectively cw millimeter and submillimeter radiation to nanostructures, such as quantum dots and double quantum dots, showing the expected PAT resonances in transconductance. The detection mechanism is based upon absorption of photons by the electrons in the leads and in the dot, leading to PAT. The energy acquired by the electrons allows them to tunnel through states of higher energy above the Fermi energy, which are otherwise not accessible. Here we report on transport experiments, performed on a coupled quantum dot structure or ‘artificial molecule’ under irradiation by a new type of broadband millimeter wave interferometer. This source generates picosecond pulses of radiation, corresponding to harmonics in a frequency range of 2–400 GHz. Because these pulses are coherent we can obtain magnitude and phase information on the signal induced by the pulse radiation. This is of major interest, since the double quantum dot we investigate contains only a few electrons and should thus evolve molecular modes, i.e. an electron’s wave function can be spread out across the whole double dot. The influence of the molecular modes can now be probed by the interferometer. The organization of the article is as follows: in the first part we will discuss the quantum dot device used in our experiment and in the second part we introduce the setup of the broadband source. Finally, we present measurements of the photoconductance induced by the broadband source and show the amplitude and phase dependencies.
3. Experimental setup and millimeter wave interferometer The quantum dots used in this experiment were realized in an Alx Ga1−x As–GaAs heterostructure grown by molecular beam epitaxy. The heterostructure has a carrier density of 2.11×1015 m−2 and an electron mobility of 80 m2 V−1 s−1 at liquid helium temperatures. Contacts of gold were evaporated serving as gate-contacts as well as an antenna structure. A number of split gates were then written by electron beam lithography in the center of the gate structure. In Fig. 1A the ‘bow-tie’ antenna is shown; in its center the quantum dots are defined, which can be seen in the SEM micrograph. The measurements were performed in a top-loading 3 He/4 He dilution refrigerator with a sample holder allowing quasi-optical transmission [2]. The method of optical-coupling is useful at high frequencies, since the quasi-optical waveguide allows transmission up to THz-frequencies. The temperature in the measurements presented is around 250 mK, although the electron temperature is slightly above this value, due to the irradiation. The performance of the antenna was verified in detailed numerical analysis with a commercially-available program [8]. The double quantum dot system (see right part of Fig. 1A) is obtained by applying negative voltages to the gate fingers, thus depleting the electrons below the gates. The voltages we used are typically in the range of −0.4 to −0.7 V. In the measurements we found CB energies of E CA = e2 /C6 ≈ 3.0 meV for the small dot A and for the large dot B E CB ∼ = (1.17 ± 0.1) meV, (C6 represents the total capacitance of the quantum dot). The charging energy of the small quantum dot varies, depending on the number of electrons in the dot. The complete characterization of such a double quantum dot is performed within the so-called charging diagram, which is discussed in detail elsewhere [9]. The effect of cw millimeter-wave radiation on a double dot is described by Blick et al. [10]. The principal setup of the photoconductance experiment with the broadband interferometer is shown in Fig. 1. The interferometer consists of two nonlinear transmission lines (NLTLs) with integrated antennas [11], which can generate and radiate high-frequency signals up into the THz regime. The interferometer superposes the two broadband spectra from the NLTLs with an intermediate offset frequency δν. In the data shown, this offset is δν = 23 Hz, while the driving frequency is ν = 7 GHz. These spectra are optically coupled into the waveguide and transduced by the antenna of the artificial molecule. The high frequency radiation then induces a photoconductance through the microstructure, which can easily be measured in both magnitude and phase because of the coherence of the interferometer. This technique thus allows direct probing without
Superlattices and Microstructures, Vol. 23, No. 6, 1998
1267
3 mm
A
1 µm
B
t (psec)
Synthesizer 7 GHz
Source 'A'
phase-locked Synthesizer 7 GHz + δν
Quantum dot sample
Source 'B'
Vds
Fig. 1. A, The broadband antenna of the device is shown: the antenna arms serve as gate contacts for the quantum point contacts defining the two quantum dots of different size (see right-hand side). B, Schematically shown is the setup of the millimeter-wave interferometer: the radiation of two nonlinear transmission lines (‘A’ and ‘B’) is superimposed, radiated onto the sample, and induces a drain-source photo-conductance.
application of the Kramers–Kronig relation. A more detailed description of the interferometer was given elsewhere [12]. The detection of photoconductivity is performed within the usual conductance measurement setup, which is applied also to characterize the quantum dots. A finite ac excitation voltage of Vexc = 5 µV k B T /e induces a current at νexc = 14 Hz, which is then amplified and detected by a lock-in. The radiation applied induces a current signal, being caused by either a non-resonant bolometric or PAT tunneling. This photocurrent is also preamplified and read out by a spectrum analyzer and lock-in. In locking onto one of the appropriate harmonics, which are separated by nδν, the corresponding high frequency conductance signal is obtained. In the measurements shown the data was taken in amplitude/phase-mode of the lock-in. It should be mentioned that in the current setup the non-resonant bolometric effects remain constant, since all different frequency components of the radiation are applied simultaneously. With phase-sensitive detection the frequency of interest can be selected.
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dI/dV (µS)
2
–70 α
β
1
γ 0 0.0
–80
–0.2
–0.4
–0.6
A (–dBm)
VTG (V)
–0.8
ζ2
ζ1 –90
–100
0
100
200 ν (GHz)
300
400
Fig. 2. Response of the quantum dot system to the interferometer radiation up to 400 GHz. The data points with the largest response values are marked by α, β, and γ corresponding to frequencies of 28, 42, and 56 GHz. The resonances labeled by ζ1 , ζ2 represent excitations of the small quantum dot A. The inset shows the Coulomb blockade oscillations of the double quantum dot under variation of the top gate voltage (VT G ). The arrow indicates the point of measurement
4. Results and discussion In the inset of Fig. 2 the conductance resonances of the double quantum dot in dependence of the top gate voltage (VT G ) are shown. One clearly recognizes the two different periodicities corresponding to the small quantum dot A (large period) and the large dot B (smaller period). Although the temperature in the measurements presented is around 250 mK, the CB resonances reveal a broadening due to the constant nonresonant heating of the electrons in the leads by the radiation applied. The double dot was chosen for these measurements, since electron-transport investigations of such devices have shown that the wavefunctions in the two dots effectively couple in a molecular-like fashion [9]. In our case the interdot coupling strength is found in the transport measurements to be on the order of t ≈ 0.15E CB . In Fig. 2 we present the response of the double dot system to broadband radiation with the gate voltages fixed (at the position of the arrow in the inset). The amplitude of the signal induced by the radiation is plotted versus frequency. Each square point of the discrete spectrum corresponds to a harmonic generated by the interferometer. The best response is obtained in the lower frequency regime. At frequencies below the point marked α the cutoff frequency of the waveguide damps the millimeter-wave signal. We note that the usual output power of the interferometer decreases towards higher frequencies as ν −1 . However, the overall sensitivity of the ‘artificial molecule’ is comparatively high, since well-defined signals were measured up to 350 GHz. The spectrum shown can be divided into two parts: the low energy excitations of the large quantum dot (α, β and γ ) and the higher energy excitations of the small quantum dot (ζ1 and ζ2 ). The different energies of the excitations result from the strong variation of the different confinement potentials of the two dots, i.e. the small quantum dot possesses a stronger confinement potential compared to the larger dot. In extension of
Superlattices and Microstructures, Vol. 23, No. 6, 1998 β: 42 GHz π
0.3 0.2 0.1
π –0.2
–0.4
–0.6
0.3 0.2 0.1 0.0
π –0.2
γ: 56 GHz
–0.4
–0.6
π Phase
0.015 σ (µS)
Phase
0.4 Phase σ (µS)
σ (µS)
0.5
α: 28 GHz π
0.4
0.0
1269
0.010 0.005 π 0.000
–0.2
–0.4 VTG (V)
–0.6
Fig. 3. Induced photo-conductance in R/2-mode for different frequencies α, β, and γ as noted in Fig. 2 versus top gate voltage (VT G ). The solid lines represent the signal amplitude, while the dotted curves give the signal’s phase.
the measurements on quantum-dot lattices with FIR-spectroscopy [13], we now have the ability to perform spectroscopy on single and coupled quantum dots. To focus on the different excitations we select the appropriate harmonic and change the charging state of the artificial molecule by variation of the gate voltage. In Fig. 3 the high-frequency conductance of the dots is given in amplitude/phase-mode of the lock-in; the bias was fixed in the low bias-regime at Vds = −60 µV thus enhancing the resolution. The left axis gives the magnitude and the right axis the relative phase of the conductance signal. From top to bottom the frequency increases, as is marked. The individual curves correspond to the points α, β, and γ in Fig. 3 as indicated, where the response of the dot system was strongest. In general the conductance signal can be understood as a phasor. The coherent signal from the millimeter-wave interferometer enables the detection of conductance (σ ) and relative phase (φ) in the high frequency domain. As seen the conductance amplitude resembles the two periodicities of the conductance resonances found in the common conductance measurement. Towards the highest frequency of 56 GHz shown the amplitude decreases, which is in accordance with the amplitude we found in the broadband response. For comparison we performed measurements with single mode frequency sources (backward wave oscillators) at identical frequencies and power levels. The power levels were calibrated by determining the broadening of the CB conductance resonances. The signals obtained agree with the corresponding amplitudes of the conductance in Fig. 3. In measuring the current with the single sources at very low power levels we also found PAT for the double quantum dot [10]. The current resolution of the interferometer did not enable us to identify additional structure induced by PAT because of resonance broadening: the contributions of all frequencies applied by the interferometer are superimposed. This obviously prevents the observation in direct current. Our main interest here, however, lies in the determination of the phase dependence of the induced current in the high frequency regime. The phase-sensitive data shown in Fig. 3 is strongly influenced when the frequency is varied from the lowest value of α: 28 GHz to the higher values (β: 42 GHz, γ : 56 GHz). Interestingly the variation of the phase in the case α exactly reflects the charging process of the large dot (B), seen as a modulation, which corresponds to the periodicity of the charging process (see Fig. 2). This can be understood as a probing of the admittance of this particular dot and the change from capacitive to inductive behavior and vice versa [14, 15]. Increasing
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the frequency the charging process of the small dot (A) becomes visible, since the change in phase is large and dominated by the CB oscillation period of this dot. Finally, at the highest frequency the phase simply switches as soon as the small dot is to be charged, while the modulation by charging the large dot is not visible any more. Varying the bias across the sample also influences the relative phase signal. Accurate phase signals at lower frequencies were hard to access, since the waveguide cutoff reduced the signal amplitude strongly (see Fig. 2). The question arising is, how the strong variation of the phase signal towards higher frequencies is related to the electronic transport through the artificial molecule. One might state that the high frequency radiation influences a ‘sloshing mode’ [16] of the electronic wave function, i.e. the spread out wave function is localized by the external disturbance of the system. More precisely, this phenomenon can be described as Rabi-oscillations in the double quantum dot, which are probed by the radiation applied, as it was shown theoretically [17, 18]. In measurements of linear transport we found in a detailed resonance shape analysis the tunnel splitting of the double quantum dot to be on the order of δ = 2 − 1 ∼ = 110–140 µeV [19]. This energy corresponds to a Rabi-frequency of ν R = δ/ h ∼ = 27–34 GHz, which is on the order of the transition frequency observed. Hence, our externally applied disturbance with frequencies above 25 GHz tends to destroy these Rabi-oscillations. This can be regarded as direct probing of the electron’s wave function.
5. Summary In summary, we applied a coherent millimeter wave interferometer to perform coherent spectroscopy on coupled quantum dots. The interferometer is tunable over the whole millimeter wave regime and allows both magnitude- and phase-sensitive detection. In the measurements shown we have detected this high-frequency conductance through the double quantum dot. We find a broadband response, i.e. excitations of the small and the large quantum dot. The full strength of the new spectrometer is proven by the phase sensitive detection, where we observe strong variations in the phase-signals when the frequency of the radiation is larger than the Rabi-frequency. Acknowledgements—We thank D. Pfannkuche and J. Smet for stimulating discussions. We also acknowledge the excellent technical help of M. Riek and F. Schartner in sample preparation. This work was supported by the Bundesministerium f¨ur Bildung, Wissenschaft, Forschung und Technologie (BMBF).
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