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SECTION I

## INTRODUCTION AND BACKGROUND

SUPERCONDUCTING mixers are playing an important role in radio astronomy at far-IR to millimetre wavelengths, where receivers with large bandwidth and low noise are needed [1]. At frequencies above 1.2 THz superconducting hot-electron bolometer (HEB) [2] mixers have become the device of choice for heterodyne receivers due to the superior sensitivity and low Local Oscillator (LO) power requirement (${<}{\hbox{1}}\ \mu$ W) [3]. Radio astronomical instruments like the Herschel Space Observatory (HIFI/1.4–1.9 THz bands) [4], TELIS, APEX telescope [5], [6] etc., have been equipped with HEB mixers made of ultrathin NbN and NbTiN films. However, some important issues still have to be resolved. One of the issues is the limited gain bandwidth (GBW) of HEB mixers [7], [8].

The GBW of so called phonon-cooled HEB mixers is determined by the electron–phonon interaction time and the phonon escape time into the substrate. At the HEB mixer operation condition, the electron–phonon interaction time $\tau_{{\rm e} \hbox{-} {\rm ph}}$ is inversely dependent on the critical temperature, $T_{c}$, of the superconducting film whereas the phonon-escape time $\tau_{\rm esc}=4d/(\alpha \cdot u)$ depends on the thickness of the film $d$, the speed of the sound $u$ and the film/substrate acoustic phonon transmission coefficient $\alpha$ [9]. In thin NbN films $\tau_{{\rm e} \hbox{-} {\rm ph}}$ has been measured experimentally to be 12 ps (at 10 K, [10]) whereas the $\tau_{\rm esc}$ was 40 ps in 3–4 nm NbN film [11]. Currently, NbN HEB mixers demonstrate the largest GBW among the phonon cooled HEB mixers (3–4 GHz on Si), resulting in a noise bandwidth of about 5 GHz [11]. For this figure to be increased, thinner NbN films, an improved film/substrate acoustic matching, or materials with faster response are required. Further reduction of the NbN film thickness (thinner than 3 nm) was reported to lead to a drastic reduction of the critical temperature which weakens the electron–phonon interaction. Replacing NbN with a material with a faster optical response could be an option. As an alternative to phonon-cooled HEB mixers, HEB mixers which utilize electron cooling via diffusion into the contact area were proposed [12]. With Nb and Al thin films a low noise THz mixing was demonstrated with a gain bandwidth of 9 and 3 GHz, respectively [13], [14]. However, the result for the Nb mixer was not associated with a stable low noise performances. Recently, Tretyakov et al. [15] demonstrated a GBW (6.5 GHz) with a low noise (600 K at 2.5 THz) in NbN HEBs. This was explained by the authors by an extra electron cooling path via the out-diffusion of the electrons into the contact pads. Such mixers required to be extremely short (100 nm), as well as a special treatment of the HEB's contacts.

Since the discovery of superconductivity in magnesium diboride (MgB2) by Akimitsu's group in 2001 [16], great interest has been generated in the fabrication of electronic devices based on it. A high critical temperature (39 K in bulk), caused by strong electron–phonon coupling [17], makes it very attractive to replace NbN with MgB2, aiming for a better HEB mixer performance. Furthermore, using time domain spectroscopy, the electron–phonon interaction time has been measured to be 3 ps [18] in MgB2 films on Si substrate which is shorter compared to superconducting NbN films. In principle, HEBs based on MgB2 film are expected to operate faster than their NbN counterparts. Recently, superconducting MgB2 films as thin as 5 nm have been demonstrated [19].

In [17], MgB2 HEBs operating at THz frequency have been reported, made of 20 nm films on silicon substrates. The GBW was 2 GHz and the noise temperature $T_{r}$ was 11000 K. Recently, we demonstrated a $T_{r}$ of 600 K measured at 2 K and 0.6 THz using devices made of 10 nm MgB2 films [20]. Preliminary GBW data for HEBs made on different MgB2 films thickness were reported [21] showing a clear dependence of the GBW on the films thickness.

In this paper, we report results of the noise and the gain bandwidth investigation which have been done using HEB mixers made of 10 nm MgB2 film with a critical temperature as high as 15 K. Furthermore, analyzing experimental data using the two-temperature model, the electron–phonon time, the phonon-escape time and the electron and phonon specific heats have been inferred as a function of the films thickness and the critical temperature. These results indicate a step forward for a new category of terahertz detectors with a low noise, and a wide GBW.

SECTION II

## DEVICE FABRICATION AND DC RESULTS

MgB2 films, 30, 15, and 10 nm thick, were grown on c-sapphire substrates via molecular-beam epitaxy (MBE). Mg and B were evaporated using e-guns and the growth temperature measured at the backside of the substrate holder was 300 °C [22], [23]. The films were covered by a 20 nm in situ gold layer, that prevents the film degradation due to the interaction with the air, as well as improves the MgB2/Au contact resistance. The bolometers were patterned as microbridges at the feed point of a planar antenna. The critical temperatures, $T_{c}$ were 25, 23 and 19 K as measured in the continuous 30, 15 and 10 nm films, respectively. The first step was to define the MgB2 bridges using the UV-lithography and the lift off process. Another 350 nm gold layer was deposited all over the wafer and lifted up from the bridges. For the next step, the remaining 20 nm Au layer over the bolometer was etched through the window via Ar-ion milling. The last step was to define the spiral antenna by the UV-lithography, followed by the Ar-ion milling of the thick gold layer (350 nm) and the MgB2 film (voltage of 400 V and current density of 0.2 mA/cm2). The etching rate of MgB2 was measured to be 2 nm/min at the used conditions. The bolometer area was in the range of 100–500 $\mu$ m2 for samples #1, #2 and #3, whereas for samples #4 and #5 the bolometer area was smaller (3– ${\hbox{42}}\ \mu$ m2).

Fig. 1. SEM image of MgB2 HEB integrated with a spiral antenna (grey) on a sapphire substrate (black).
Fig. 2. Resistance versus temperature curves of HEBs. Sample numbers correspond to Table I. The curves are normalized to the resistance at 40 K.

Fig. 1 shows a scanning electron microscope (SEM) image of an HEB bolometer integrated with a spiral antenna (similar to samples #4 and #5). The real impedance of the antenna is simulated to be approximately 90 $\Omega$ in the frequency range of 0.5– 3 THz, providing a reasonable impedance match to the bolometers. In fact, because the RF impedance of HEB mixers is real [17], they can be integrated with many planar antennas as well as in waveguide receivers.

Fig. 2 shows resistance versus temperature curves of HEBs patterned in MgB2 films with different thicknesses. For thinner films the critical temperature is reduced. Devices of batches C and D are made of 10 nm films. However, due to differences in the initial quality of the films, the $T_{c}$ of the devices from these two batches is 15 and 8.5 K, respectively. Devices #3 and #5 exhibit the same critical temperature, transition width and residual resistance, whereas the transition width is larger in device #4. The resistivity is comparable in devices #3, #4, and #5. A summary of the Fig. 2 is also presented in Table I.

$I{\hbox{--}}V$ characteristics of MgB2 HEBs #4 and #5 (of 1 $\mu$ m×3 $\mu$ m and 6 $\mu$ m×7 $\mu$ m, respectively), with and without local oscillator (LO) power applied, are shown in Fig. 3. The critical currents, measured at 4.2 K, were 160 $\mu$ A and 650 $\mu$ A for samples #4 and #5, respectively. The resulting critical current densities were 0.55 MA/cm2 and 0.93 MA/cm2.

TABLE I MgB2 THICKNESS (d), CRITICAL TEMPERATURE $({\rm T}_{\rm C})$, TRANSITION WIDTH $(\Delta {\rm T}_{\rm C})$, RESISTIVITY $(\rho_{300})$ AND RESISTANCE AT 300 K $({\rm R}_{300})$
Fig. 3. (a) I-V curves of 1 $\mu$ m×3 $\mu$ m (device #4). (b) I-V curves of 6 $\mu$ m×7 $\mu$ m (device #5). HEBs at 4.2 K bath temperature, with LO power and without LO power applied.

Both a high resistivity (see Table I) and a low critical current density of the MgB2 micro bolometers suggest a low crystal quality of the films. However, within a single batch, the dc parameters of the devices are quite consistent.

Upon application of the LO power the critical current in the HEBs is suppressed. Due to the lower $T_{c}$ and the smaller bolometer size, the required LO power for device #4 was much smaller compared to device #5.

The receiver noise temperature $T_{r}$ was measured for devices #4 and #5, whereas for other devices $T_{r}$ was not measured because of the lack of the LO power. For MgB2 films with $T_{c} > 15$ K, bolometers of submicrometer sizes are needed to reduce the LO power requirement to the practical values.

SECTION III

## EXPERIMENTAL TECHNIQUE

Achievement of the GBW [as well as of the noise bandwidth (NBW)] superior to that of NbN HEB mixers is the main motivation of this work. Therefore, our primary measurements were the mixer gain and the noise temperature across a wide IF band, at various bias conditions and LO power levels.

For the RF measurements the MgB2 mixer chips were glued on the backside of a 12 mm elliptical silicon lens, without an AR-coating, defining a quasi-optical setup. The mixer block was placed on the cold plate of a LHe cryostat (4.2 K bath temperature).

The gain bandwidth of the devices #1–3 was measured using two backward wave oscillators (BWOs) at 600 GHz. The frequency of the LO BWO was kept constant, while the frequency of the signal BWO was tuned. At each frequency point, the amplitude of the signal was modulated by a mechanical chopper at 18 Hz. The direct detection signal was read out via the mixer bias line with a lock-in amplifier, which was later used for the calibration of the signal input power. The IF signal was amplified with two 0.1–12 GHz room temperature amplifiers and measured by a microwave spectrum analyzer. Both the LO and the signal beams were collimated by Teflon lenses, spatially mixed by a thin film (Mylar™) beam splitter, and led into the cryostat through a high density polyethylene pressure window. Three sheets of Zitex G108 IR filters were installed at the 77 K and 4.2 K shields. Devices #1–3 were heated to a temperature a few Kelvin below $T_{c}$. At such conditions the energy gap of the MgB2 film $2\Delta$ is suppressed so that is becomes smaller than the photon energy of the 600 GHz signal/LO sources. Experimental data, obtained under such conditions, are also easier to interpret (see Section V), because the electron and the phonon heating (by both the LO and dc power) above the substrate is small.

The noise temperature for devices #4 and #5 was measured using the Y-factor technique with a 290 K and a 77 K (liquid nitrogen) black body sources (Eccosorb sheets2). The 600 GHz BWO local oscillator was used, as in the previous measurements, with a 50 $\mu$ m thick (Mylar™) beam splitter. The intermediate frequency signal from the mixer was amplified using a set of cold and room temperature low noise amplifiers. Cold low noise IF amplifiers covered frequency ranges of 1–4 GHz and 3–8 GHz. The input noise temperature of both IF amplifiers was approximately 3–10 K (higher at the ends of the bands). The IF passband was set by a tunable (1–9 GHz) 50 MHz band pass filter. The gain of the device #4 was extrapolated from the Y-factor measurements using an expression $P_{\rm if} = (P_{300} -P_{77})\times G_{m}$, where $P_{300}$ and $P_{77}$ are the single mode Planck power in the IF bandwidth, and $G_{m}$ is the mixer gain.

The 3 dB roll-off GBW, $f_{0}$ was determined by performing the least square fitting of the measured mixer (relative) gain $G(f_{\rm IF})$ (after the calibration for the IF amplifier transfer function) with a single-pole Lorentzian (1), where $G(0)$ is the mixer gain at zero IF frequency: TeX Source $$G({f_{\rm IF}}) = G(0)[1 + (f_{\rm IF}/ f_{0})^{2}]^{- 1}.\eqno{\hbox{(1)}}$$

The effective mixer time, $\tau_{m}$ constant is obtained as $\tau_{m} = 1/(2\pi f_{0})$. The GBW depends on the HEB mixer bias point because the electrothermal feedback modifies the mixer time constant as $\tau_{m}=\tau_{\theta}/(1-C_{0}(R_{L}-R_{0})/(R_{L}+R_{0}))$, where $\tau_{\theta}$ is the time constant in the limit of a zero bias, $C_{0} = (R_{d} -R_{0})/(R_{d} +R_{0})$ is the self-heating parameter, $R_{0}={\rm V}/{\rm I}$ is the dc resistance at the mixer bias point, $R_{d}$ is the differential resistance $(dV/dI)$, and $R_{L}$ is the IF load resistance (50 $\Omega$) [2], [24].

SECTION IV

## RESULTS

The relative mixer gain as a function of the intermediate frequency of the mixers fabricated from 30, 15, and 10 nm ($T_{c}= 15$ K) MgB2 films is given in Fig. 4. Fig. 5 shows the response of HEB fabricated from a 10 nm film (${T}_{c} = 8.5$ K). The GBW was obtained by the least square fitting of the measured data with the (1).

Fig. 4. Intermediate frequency response of MgB2 mixers made of 15 nm (device #2), 30 nm (device #1) and 10 nm (device #3) MgB2 films. Solid lines are fits to the experimental data. Dashed lines are results of the two-temperature model.
Fig. 5. Intermediate frequency response of MgB2 mixers made on 10 nm (device #4) film. The solid and the dotted lines are the fit to the experimental data and the result of the two-temperature model.

The 3 dB gain roll-off frequency was 1.3 GHz and 2.3 GHz for devices 1 and 2 fabricated from 30 nm and 15 nm films. A GBW of 3.4 GHz was observed for the mixers (#3) made of a 10 nm film with a $T_{c}$ of 15 K. Much smaller GBW, 1.5 GHz, was measured for a mixer (#4) fabricated from the films with the same thickness (10 nm) but with a $T_{c}$ of 8.5 K.

A summary of the GBW obtained from the least square fitting for all devices, along with the fitting errors (for the GBW) is given in Table II.

TABLE II MgB2 THICKNESS (d), CRITICAL TEMPERATURE $({\rm T}_{\rm C})$, ELECTRON-PHONON INTERACTION TIME $(\tau_{{\rm e} \hbox{-} {\rm ph}})$, PHONON ESCAPE TIME $(\tau_{\rm esc})$, SPECIFIC HEAT RATIO $({\rm c}_{\rm e}/{\rm c}_{\rm ph})$, MIXER TIME CONSTANT $(\tau_{\rm m})$, THE GAIN BANDWIDTH (GBW) AND ITS ERROR*

The noise temperature spectra for devices #4 and #5 (10 nm, $T_{c}$ of 8.5 K and 15 K) are given in Fig. 6. The optimum bias points, in which the noise temperature has a minimum, are: ${\rm V}=1$ mV, ${I}=60\ \mu$ A and ${V}=30$ mV, ${\rm I}=300\ \mu$ A for HEB #4 and #5, respectively. Defining the NBW as an IF corresponding to the $T_{r}$ rise by a factor of 2, the NBW appears to be 3 GHz and 6.5 GHz for mixers #4 and #5, respectively. For comparison reasons in the same figure we plot the noise temperature spectrum of a HEB mixer made of a 3 nm thick NbN film [3].

Fig. 6. Receiver noise temperature versus the intermediate frequency. Blue stars: NbN HEB mixer at 1.9 THz [3]. Black squares: MgB2 HEB mixer #4. Red circles: MgB2 HEB mixer #5. All data correspond to the bath temperature of 4.2 K.

The IF response calibration included the IF amplifier, the bias-T, and the coaxial cables. There was no isolator used during the gain bandwidth measurements. Due to the HEB/IF chain impedance mismatch and other parasitics in the mixer unit, the measured IF response was disturbed by a standing wave in the long cable in the cryostat (scattering of the experimental points in Fig. 4). This effect is device specific, and especially strong at higher IFs, i.e., at and above the 3 dB roll-off frequency. For the noise temperature measurements (Fig. 6) the IF amplifier was mounted much closer to the mixer (namely, next to it). Therefore, in Fig. 5 (the IF response obtained from the hot/cold load measurements) this problem does not appear, except for the frequencies below 2 GHz, where the amplifier input impedance matching is very poor.

Previously, the noise temperature of NbN HEB mixers was reported to be sensitive to the bath temperature [3], [25], increasing almost immediately as the bath temperature rises. Similar behavior has been observed for a MgB2 mixer with a low $T_{c}$ [20]. Mixer #5, discussed just above, has a $T_{c}$ of 15 K. Therefore, it was interesting to verify how sensitive the noise temperature of such devices is to the bath temperature. As it can be observed from Fig. 7, the $T_{r}$ of this device (measured at IF of 3 GHz) remains constant from 4.2 K up to 10.5 K, and rises only at higher temperatures.

Fig. 7. Receiver noise temperature at ${\rm IF} = 3$ GHz (with 3–8 GHz LNA) versus the bath temperature. MgB2 HEB mixer #5.
SECTION V

## DISCUSSION

As discussed above, the bolometer response depends on several parameters, such as the critical temperature, $T_{c}$, the film thickness, $d$, and the electron and phonon specific heats. Study of devices made of 30 nm, 15 nm and 10 nm MgB2 films provides a good comparison of the physical parameters that determine the mixer GBW.

A very good understanding of the superconductor response on a modulated RF radiation can be obtained using a two temperature approach presented in [26]. The general approach is that at the given mixer bath temperature, the LO power and the bias voltage are optimized for the maximum mixer gain (largest IF signal). Under such circumstances, the electron temperature $T_{e}$ rises close to the $T_{c}$ due to the dissipation of the LO and the dc power. The phonon temperature $T_{\rm ph}$ is ${\sim}0.9 \times T_{e}$, and it can be estimated from the heat balance equations as, e.g., in [28].

The effect of the self-heating electrothermal feedback is taken in account as in [28]. The HEB conversion gain as a function of the intermediate frequency is [expressed in decibels (dB)] TeX Source $$G_{\rm IF} (\omega) \propto 20\lg \left[{\left\vert {{G(0)C_{0}\over\xi (\omega) + {R_{0} - R_{L}\over R_{0} + R_{L}}C_{0}}} \right\vert} \right] \eqno{\hbox{(2)}}$$ where TeX Source \eqalignno{\xi (\omega)& = {(1 + j\omega \tau_{1})(1 + j\omega \tau_{2})\over(1 + j\omega \tau_{3})} &\hbox{(3)} \cr \tau_{3}^{- 1}& = \tau_{\rm esc}^{- 1} + \tau_{\rm eph}^{- 1} {c_{e}\over c_{\rm ph}} &\hbox{(4)} \cr \tau_{1,2}^{- 1}& = {\tau_{3}^{- 1} + \tau_{\rm eph}^{- 1}\over 2} \left[{1\pm \sqrt {1 - 4{\left({\tau_{3}^{- 1} + \tau_{\rm eph}^{- 1}} \right)^{- 2}\over\tau_{\rm esc} \tau_{\rm eph}}}} \right] &\hbox{(5)}} and $\omega = 2\pi f_{\rm IF}$ is the angular intermediate frequency, and the rest of the parameters are as defined in Section III.

The electron specific heat as a function of the electron temperature is TeX Source $$c_{e} ({T_{e}}) = \gamma T_{e}. \eqno{\hbox{(6)}}$$ Here, $\gamma$ is the electron specific heat coefficient. The published value of $\gamma$ is in the range from 3 to 5.5 mJ/mol K2 [29], [30], [27].

The phonon specific heat at an arbitrary phonon temperature $T_{p}$ can be calculated in the Debye approximation [31]: TeX Source $$c_{\rm ph} (T_{\rm ph}) = 9nK\left[{T_{\rm ph}\over T_{D}} \right]^{3} \int_{0}^{T_{D}\over T_{\rm ph}} {e^{x} x^{4}\over(e^{x} - 1)^{2}} dx \eqno{\hbox{(7)}}$$

The atomic density in MgB2 is $n=3.54\times 10^{22}$ cm -3 (from the mass density $\rho =2.7$ g/cm3 [29], [30]) and the molar mass of MgB2 $M=45.925$ g/mol). $K$ is the Boltzmann constant. The Debye temperature, $T_{D}$ in MgB2 was experimentally obtained to be much larger compared to other intermetallic as well as cuprate superconductors: from 700 to 1100 K [29], [30], [27]. Mean values for the electron specific heat coefficient and for the Debye temperature were taken for our simulations: $\gamma = 4.24, T_{D}= 900$ K.

Using the electron–phonon interaction time and the phonon escape time as fitting parameters, we applied the 2T model (2–5) to our results. The dashed curves in the plots of the intermediate frequency response (Figs. 4, 5) represent the fit of the 2T model. Table II shows a summary of the electron–phonon interaction time, the phonon escape time and the specific heat ratio $({\rm c}_{\rm e}/{\rm c}_{\rm ph})$ for different MgB2 film thicknesses given by the 2T model applied to the experimental results. For thinner films the electron–phonon time and the specific heat ratio increase due to the lower critical temperature. The phonon-escape time decreases proportionally to the film thickness.

The electron–phonon interaction time is a function of the temperature [32]: TeX Source $$\tau_{{\rm e} \hbox{-} {\rm ph}} \propto T^{- \mu} \eqno{\hbox{(8)}}$$

Fig. 8 shows the electron–phonon interaction time at the critical temperature of each film: the open circles are for the $\tau_{{\rm e} \hbox{-} {\rm ph}}$ extrapolated from the 2T model (see Table II), whereas the open triangle is the value of the $\tau_{{\rm e} \hbox{-} {\rm ph}} \approx 3$ ps reported in [18] for a $T \approx 40$ K. The data of Fig. 8 can be fitted using (8) with $\mu\approx 1.08$.

Fig. 8. Electron-phonon interaction time in MgB2 as a function of the temperature, obtained from the gain bandwidth measurements and the 2T-model analysis. Open circles: this work; the triangle: from the literature. The solid line is (8) fit to the data (the exact formula is given in the inset). The dashed lines show the estimated error margins.

Fig. 9 shows the electron and phonon specific heats $({\rm c}_{\rm e}, {\rm c}_{\rm ph})$ and the specific heat ratio $({\rm c}_{\rm e}/{\rm c}_{\rm ph})$ extrapolated from the 2T model (see Table II) as a function of the temperature. The specific heat ratio is an important factor together with the electron–phonon interaction time and the phonon escape time in the system response [26]. Indeed in order to achieve a short relaxation time all of these parameters must be optimized.

Fig. 9. Electron and phonon specific heats and the specific heat ratio for MgB2 as a function of the temperature. The ${\rm c}_{\rm e}/{\rm c}_{\rm ph}$ ratio is plotted for electron temperature equal to the phonon temperature. Filled circles indicate ${\rm c}_{\rm e}/{\rm c}_{\rm ph}$ at critical temperatures of the films used in this work.

Within the 2T model, it has been demonstrated that the phonon cooled HEB mixer requires a film with a high critical temperature $T_{c}$ to minimize the $\tau_{{\rm e} \hbox{-} {\rm ph}}$ and the $c_{e}/c_{\rm ph}$. On the other hand, to ensure fast removal of the energy from the phonons, the film thickness should be small. The larger GBW measured in device #3 compared to device #4 can also be explained in the frame of the 2T model. As it is possible to see from the Table II, the $\tau_{{\rm e} \hbox{-} {\rm ph}}$ is larger in device #4, furthermore the specific heat ratio is also three time higher in device #4 with the consequence of a larger relaxation time and smaller GBW than device #3.

The $T_{c}$ of the MgB2 film as in batch C (devices #3 and #5) is much below its value reported for thick films (${\sim} {\hbox{39}}$ K). Therefore, there is much room for improvement. Recently, it was demonstrated that MgB2 films as thin as 7.5 nm can exhibit a critical temperature as high as 34 K [33]. It suggests that both the GBW and the NBW of MgB2 HEB mixers can be further increased.

SECTION VI

## CONCLUSION

In conclusion, THz mixing in MgB2 HEBs was investigated with respect to the gain and the noise bandwidths. A number of devices were fabricated of 30, 15, and 10 nm MgB2 films with gain bandwidths of 1.3, 2.3, and 3.4 GHz and energy relation times of 130, 70 and 47 ps respectively.

By fitting results with the two temperatures model the electron–phonon interaction time, $\tau_{{\rm e} \hbox{-} {\rm ph}}$ of 7 to 15 ps, the phonon escape time $\tau_{\rm esc}$ of 4.8 to 42 ps, and the specific heat ratio, $c_{e}/c_{\rm ph}$ of 1.35 to 9 were deduced for the given MgB2 thicknesses. The noise bandwidth of a mixer made of a 10 nm film with a $T_{c}$ of 15 K is 6–7 GHz, i.e., close to the one for HEB mixers made of 3–4 nm NbN films. Further work has to be directed to obtaining thinner MgB2 films with higher critical temperatures.

## Footnotes

This work was supported by the Swedish Research Council (VR), by the Swedish National Space Board, and by the European Research Council.

S. Bevilacqua, S. Cherednichenko, V. Drakinskiy, and J. Stake are with the Department of Microtechnology and Nanoscience (MC2), Chalmers University of Technology, SE-41296 Göteborg, Sweden (e-mail: stellab@chalmers.se).

H. Shibata and Y. Tokura are with NTT Basic Research Laboratories, Morinosato, Atsugi, Kanagawa 243-0198, Japan.

Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org.

1CST—Computer Simulation Technology, Microwave Studio. [Online.] Available: http://www.cst.com

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