Scholarly article on topic 'Mutual phase-locking of planar nano-oscillators'

Mutual phase-locking of planar nano-oscillators Academic research paper on "Physical sciences"

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Academic research paper on topic "Mutual phase-locking of planar nano-oscillators"

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Mutual phase-locking of planar nano-oscillators

K. Y. Xu, J. Li, J. W. Xiong, and G. Wang

Citation: AIP Advances 4, 067108 (2014); doi: 10.1063/1.4881879 View online: http://dx.doi.org/10.1063/1.4881879

View Table of Contents: http://scitation.aip.org/content/aip/journal/adva/4/6?ver=pdfcov Published by the AIP Publishing

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Mutual phase-locking of planar nano-oscillators

K. Y. Xu,1a J. Li,1 J. W. Xiong,1 and G. Wang2

1 Laboratory of Quantum Engineering and Quantum Materials, School of Physics and Telecommunication Engineering, South China Normal University, Guangzhou 510631, China

2 State key Laboratory of Optoelectronic Materials and Technologies, Sun Yat-sen University, Guangzhou 510275, China

(Received 17 April 2014; accepted 25 May 2014; published online 3 June 2014)

Characteristics of phase-locking between Gunn effect-based planar nano-oscillators are studied using an ensemble Monte Carlo (EMC) method. Directly connecting two oscillators in close proximity, e.g. with a channel distance of 200 nm, only results in incoherent oscillations. In order to achieve in-phase oscillations, additional considerations must be taken into account. Two coupling paths are shown to exist between oscillators. One coupling path results in synchronization and the other results in anti-phase locking. The coupling strength through these two paths can be adjusted by changing the connections between oscillators. When two identical oscillators are in the anti-phase locking regime, fundamental components of oscillations are cancelled. The resulting output consists of purely second harmonic oscillations with a frequency of about 0.66 THz. This type of second harmonic generation is desired for higher frequency applications since no additional filter system is required. This transient phase-locking process is further analyzed using Adler's theory. The locking range is extracted, and a criterion for the channel length difference required for realizing phased arrays is obtained. This work should aid in designing nano-oscillator arrays for high power applications and developing directional transmitters for wireless communications. © 2014 Author(s). All article content, except where otherwise noted, is licensed under a Creative Commons Attribution 3.0 Unported License. [http://dx.doi.org/10.1063/L4881879]

Terahertz electromagnetic waves have attracted wide attention due to their numerous applications in areas such as non-destructive imaging and spectroscopy of biological materials, remote detection of hidden objects and explosives, and the manipulations of quantum states in semiconductors.1 However, the development of THz technology has been hampered by the lack of reliable, solid state sources operating at room-temperature.2

One possible way of developing THz devices is to adapt well-known mechanisms previously utilized in the microwave field for higher frequency applications.3 In particular, the Gunn effect is considered to be one of the most promising candidates for developing THz sources due to the possibility of room temperature operation and the compact size of devices based on this effect.4 In order to overcome heat dissipation problems and enable integrated devices for on-chip applications such as high frequency radar and secure imaging,1 planar structures normally replace conventional vertical device architectures. Gunn oscillators based on GaAs HEMT-like layouts have been shown to efficiently operate with a fundamental frequency of up to 0.165 THz.5 Through the use of second harmonic oscillations, a working frequency of 0.218 THz can be achieved.6 The working frequency can be further increased by replacing GaAs with GaN.7

In these HEMT-like oscillators, cathode and anode contacts are directly connected to the Gunn channel in which electric fields are extremely high. Consequently, the careful design of contacts is required to mitigate the effects of high charge concentrations and high electric fields at the anode

aAuthor to whom correspondence should be addressed. Electronic mail: xuky@scnu.edu.cn.

2158-3226/2014/4(6)7067108/8 4, 067108-1 © Author(s) 2014 H M ■

0.45|m

Connecting area \

Left { terminal

Etched area $ 30nm J 50nm i lipj

4 ^ 30nm

0.95|m

Right terminal

J 50nmùipj Etched area $ 30nm

Air 1|m

AlGaN : 30nm

У GaN 1|m J

FIG. 1. Schematic top view (a) and side view (b) of the simulated nano-oscillators (not to scale). The grey areas and the white area in the top view represent insulating trenches and the 2DEG, respectively. A GaN/AlGaN interface is 30 nm below the device surface at which a 2DEG forms. In the simulations, all the insulating trenches are assumed to have vertical sidewalls and pass through the entire GaN/AlGaN heterostructure.

edge.8 Nevertheless, these effects are naturally alleviated in oscillators with intentionally-tailored active layers, since the contacts are separated from the Gunn channel by electron reservoirs.9-11 Self-switching device (SSD)-based nano-oscillators have attracted particular attention due to their numerous applications in the area of logic circuits, rectifiers, memory devices and also THz detectors.12-15 Connecting nano-oscillators to form a phased array not only results in an improved power output, but also enables directional transmitters for wireless communications.16 However, recent studies of SSD-based nano-oscillators have mainly focused on improving the power efficiency of devices.3,17 Little attention has been paid to the study of the phase relationship between SSD-based nano-oscillators. In this work, we focus on analyzing the phase-locking characteristics of SSD-based nano-oscillators based on a combined 2D-3D EMC model.18,19

Figs. 1(a) and 1(b) schematically show the top view and cross section of SSD-based nano-oscillators. The device is based on a GaN/AlGaN heterostructure, in which a 2DEG is formed at the GaN/AlGaN interface with a carrier concentration of 8.0 x 1012 cm-2.3 Each oscillator is defined by two L-shaped insulating trenches (dark grey areas). The trenches are created by etching through the 2DEG layer. Therefore, electrons have to pass through the nanometer channel between the two trenches in order to conduct a current from the left terminal to the right terminal. Consequently, the device shows diode-like characteristics, as demonstrated by Song et al.12 Such a device is known as an SSD, which not only has a strong electric-field effect but also operates at frequencies up to 1.5 THz at room temperature and up to 2.5 THz at 150 K.14 Moreover, Gunn oscillations have been theoretically demonstrated both in GaAs and GaN based SSDs.3,9,11,17

By directly connecting two identical SSD-based oscillators in parallel, one obtains a C-shaped trench with two L-shaped trenches on each side, i.e. the dark grey areas shown in Fig. 1(a). The lighter grey area with a width of W and a length of L in the middle of the two oscillators represents an additional insulating trench, which will be used for adjusting the coupling strength of the two oscillators. The two oscillators are both designed with a channel width of Wc = 50 nm and a trench width of Wt = 30 nm. The distance between the two channels is 200 nm. Other geometric parameters are defined in Fig. 1.

A combined 2D-3D EMC model based on a semi-classical 2D EMC method self-consistently coupled with the 3D Poisson equation is used in this work. This combined model is developed from our entirely 2D model, which has been successfully applied in earlier studies.9,15,20 It is noteworthy that a fully 3D EMC model has been developed to study three terminal T-branch junctions (TBJs) with a top gate terminal modeled as a side gate in purely 2D models.21 Despite minimizing the need for parameter fitting and including the effect of electron transfer from the channel to other layers, the 3D model leads to nearly the same results as those obtained from an entirely 2D model for GaAs-based devices.22

As in an entirely 2D model, all the electrons in the combined model are assumed to be confined within the 2DEG layer, ignoring the effect of electron transfer from the channel to other layers. As a result, the 2D EMC method is sufficient to describe electron transport in the devices. However, the main advantage of the combined model lies in augmenting the 2D Poisson solver a fully 3D Poisson solver. This allows one to describe electric field couplings in a more accurate manner, i.e. not only those coupling within the 2DEG layer are included.18,19 In order to fully include 3D electric-field couplings, the Poisson equation is solved in a sufficiently large domain, beyond the realistic device structure, as shown in Fig. 1(b). Moreover, to avoid trench-depth effects, all the insulating trenches are assumed to vertically pass through the entire GaN/AlGaN heterostructures with constant cross sections.18 The dielectric constant used in the simulations for Air, AlGaN, and GaN is 1, 8.5, and 8.9, respectively. In order to increase the speed of the computation, the effect of surface states at the semiconductor-air interface is included by a simple constant charge model (SCC) but not an advanced self-consistent charge model.23 A negative charge density, Ns = -0.8 x 1012 cm-2, is applied in the SCC model.19 All simulations were performed at room temperature with the left terminal grounded. Further information about the model can be found in our recent

Simulations were performed for a device obtained by directly connecting two identical SSDs, i.e. the lighter grey area shown in Fig. 1(a) is excluded. In the simulations, the right terminal is suddenly biased with a voltage of 21 V, and the time-dependent output current is recorded for a sufficiently long time, up to 2500 ps. Results for the device with a channel separation of 200 nm are shown in Fig. 2(a). The amplitude of current oscillations clearly changes with time randomly. Moreover, from the magnified view of the current oscillations shown in the insets of Fig. 1(a), one can find that the waveform of the oscillations also changes with time. These results imply that the interaction between Gunn oscillations is too weak for the two oscillators to operate in a common state.

In order to obtain a common operational state, an additional insulating trench, as shown in Fig. 1(a), was introduced in the center of the device. Simulations reveal that the coupling strength between the two oscillators can be enhanced by the additional trench. When the trench has a width of 130 nm and length of 320 nm, the two oscillators operate in a common state over the whole simulation period, as shown in Fig. 2(b). Since the output oscillation is double that obtained from a single oscillator, this common state is likely an in-phase locking or synchronization state.

From Fig. 2(c), when the additional trench is increased to be longer than the channel, i.e. L = 670 nm, the in-phase locking state becomes unstable and quickly transitions to another common state. The later state is distinct from the former one due to the lower oscillation amplitude and the higher oscillation frequency (see Figs. 2(b) and 2(c), and their insets for details). Further simulations (results not shown here) reveal that the time for the above transition into a non-in-phase locking state relates to the width of the additional trench. A minimal transition time has been obtained when W = 130 nm, implying that the coupling strength of the two oscillators has reached its maximum. To uncover the properties of the non-in-phase locking state, the components of the current oscillations I(t) are extracted by fitting the EMC data with a combined sinusoidal function:

work.18,19

0 500 1000 1500 Time (ps)

2000 2500

FIG. 2. Time-dependent output current of (a) directly connected oscillators, (b) oscillators connected through an additional trench with 320 nm length (i.e. the lighter grey area shown in Figs. 1(a)), and 1(c) oscillators connected through an additional trench with 670 nm length. The insets show a magnified view of output currents over a time interval of 12 ps.

where t is time, m = 2.1 (corresponding to a frequency of about 0.33 THz) is the angular frequency, and p is the phase and the subscripts represent different frequency components. Fitting results are shown in Fig. 3(a), with two typical magnified currents in the insets. From these results, one can infer that the EMC data (red curve) obey Eq. (1) (green curve) well. The amplitudes of the oscillation components versus time are also provided in Fig. 3(b). The third-order (pink curve) and the fourth-order (powder blue curve) components are almost zero over the whole simulation period. While the zero-order component (red curve) shows only a slight reduction during the transition process

0.9 0.8 0.7 0.6 0.5 0.4 0.3

100 120 140 1988 1994 2000

Zooming

MC Fitting

0.4 0.06-

0.3 ■ 0.04-

0.2 - 0.01 -

0 Ol.....

\ EMC _ -

\ Adler — -

DC 1st 2nd 3rd 4th

0 500 1000 1500 2000 2500 Time (ps)

FIG. 3. (a) Time-dependent output currents obtained using the EMC technique (red curve) and fitting Equation (1) (green curve). The insets show magnified views of output currents over a time interval of 12 ps. (b) Frequency components of current oscillations versus time. The inset shows the transient evolution of the fundamental component of current oscillations extracted from EMC data (green curve) and obtained by Alder's equation (red curve).

and remains unchanged within the following time, the inter-modulation between the oscillators should be weak. In contrast, the fundamental (green curve) and the second-order (navy blue curve) components show a pronounced change during the transition process. Thus, these components are important for understanding the mutual influence of the oscillators. From Fig. 3(b), one can find that the fundamental component reduces to zero monotonically when the stable state is reached. This fact strongly suggests that the two oscillators experience a de-phasing process, and may reach an anti-phase state in the end. More importantly, the change of the second-order component is non-monotonic. This component reduces to zero with a speed higher than the fundamental but raises to a value almost the same as the initial value. This is to say that the second-order component is almost unchanged after experiencing the transition process. The above behavior of the second-order component is the same as that expected to occur in the anti-phase locking process. According to the above analysis, the two oscillators should experience an anti-phase locking process determined by fundamental-component couplings.

According to Adler's theory, the phase difference of two coupled oscillators can be written as,24

1 VK2 + 1 B(t - t0)

a(t) = n - 2tan [--1--tan-

Vk 2 +1],

where t is time, B is the locking range and K = Aco/B. The quantity Am is the angular frequency difference of the oscillators. By combining two similar sinusoidal signals together, one obtains the

(a) random

o • ^

.2 1 1

0.9 0.8 0.7 0.6 0.5 0.4

(b) in-phase

(c) anti-phase

0.8 0.6 0.4 0.2 0 -0.2 -0.4 -0.6 -0.8 -1

h-» •

h-» •

23456789 10 Time (ps)

FIG. 4. Electric potential fluctuations along the center of the connecting area for oscillators with (a) a random-phase relationship, (b) an in-phase relationship and (c) an anti-phase relationship. The positions corresponding to the entrance and the exit of the channels (see Fig. 1(a) for more information) are indicated by white lines.

amplitude of the total output as,

I(t) = 2Is cos ^[«(i) + kvt].

Here, the amplitude of both signals is assumed to be Is. Substituting Eq. (2) into Eq. (3), one obtains,

. 1 VK2 + 1 B(t - t0) /—r,- KBt

~ L- + tan \ °V K2 + 1] + -—}. (4)

I(t) = 2Is sin{tan-

When Is = 36.5 nA, K = 9.5 x 10-5, B = 0.076 THz, and 0 = 0 ps, Eq. (4) well describes the behavior of the fundamental component extracted from EMC data during the transient process, as shown in the inset of Fig. 3(b). Of course, this EMC data is noisy, and possesses slight deviations from Eq. (4). The deviations can be attributed to the influence of a small inter-modulation between the oscillators. From the fitting results, one obtains Am/m & 3.4 x 10-6. This quantity is small but reasonable, since the two oscillators used in the simulations have the same geometry. It is known that the frequency of Gunn oscillators is typically determined by the domain velocity divided by the length of the channel. In this study, to fulfill the locking range requirement, the channel length difference of the oscillators should be shorter than ALmax & L0B/m0 & 17 nm (where L0 = 450 nm is the length of the Gunn channel and m0 & 0.33 x 2n THz is the fundamental angular frequency of the oscillator).

To further understand the emergence of above three phase relationships, i.e. random, in-phase locking and anti-phase locking, we will make an insight into the variation of electric potential within the connecting area. In Fig. 4, electric potential fluctuations along the center of the connecting area are shown. One can find that when the phase relationship of the two oscillators is random, potential fluctuations along the centre of the connecting area are both random and small, as shown in Fig. 4(a). When the phase relationship is in-phase or exhibits anti-phase locking, potential fluctuations exhibit dominant oscillations with a frequency of about 0.33 THz, as shown in Figs. 4(b) and 4(c). Since the fundamental frequency of the oscillator is also about 0.33 THz, this further confirms that phase-locking phenomena occur due to fundamental-component couplings. Moreover, the dominant potential oscillations under different phase-locking states do not appear in the same area, indicating the existing of two coupling paths. One path occurs between Gunn channels, as shown in Fig. 4(b). The other path occurs between the output lips of the oscillators, as shown in Fig. 4(c). Coupling through the former path could lead to an in-phase locking and that through the latter path results in an anti-phase locking.

In conclusion, based on a combined 2D-3D EMC technique, we have demonstrated that for Gunn effect-based oscillators, there are three phase relationships: random, in-phase locking and anti-phase locking. The different phase locking phenomena are mainly determined by the coupling of the fundamental component of the oscillations. We also show that each special phase relationship can be realized by adding insulating trenches with the proper geometry at the connecting area of the oscillators. When the insulating trench is shorter than the Gunn channel, a dominant coupling occurs between the channels, leading to an in-phase locking. However, when the insulating trench is longer than the Gunn channel, the dominant coupling occurs between the output lips of the oscillators, resulting in an anti-phase locking. It is interesting to see that for two identical oscillators operating with anti-phase locking, the outputs consist of only purely second harmonic oscillations, since the fundamental components are cancelled. This effect is desirable for higher frequency applications, since no additional filter system is needed. Moreover, by applying Adler's theory to analyze the transient phase-locking process, we show that the locking range can be obtained. Further analysis also reveals that according to the locking range requirement, the channel length difference of the oscillatorsstudied in this work should be shorter than 17 nm.

This work was supported by Natural Science Foundation of Guangdong Province, China (No. S2013010012711); NSFC (Nos. 11374185, 61072029); Science and Technology Program of Guangzhou, China (No. 2014J4100049) and FOK YING TONG Education Foundation (No. 122004). The authors also thank the high-performance computing platform of South China Normal University for technical support.

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