Topological charge pump by surface acoustic waves Yi Zheng(郑一), Shi-Ping Feng(冯世平), Shi-Jie Yang(杨师杰) Citation:Chin. Phys. B . 2016, 25(6): 067301. doi: 10.1088/1674-1056/25/6/067301
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Chin. Phys. B Vol. 25, No. 6 (2016) 067301
Topological charge pump by surface acoustic waves∗ Yi Zheng(郑一), Shi-Ping Feng(冯世平), and Shi-Jie Yang(杨师杰)† Department of Physics, Beijing Normal University, Beijing 100875, China (Received 22 September 2015; revised manuscript received 3 March 2016; published online 20 April 2016)
Quantized electron pumping by the surface acoustic wave across barriers created by a sequence of split metal gates is interpreted from the viewpoint of topology. The surface acoustic wave serves as a one-dimensional periodical potential whose energy spectrum possesses the Bloch band structure. The time-dependent phase plays the role of an adiabatic parameter of the Hamiltonian which induces a geometrical phase. The pumping currents are related to the Chern numbers of the filled bands below the Fermi energy. Based on this understanding, we predict a novel effect of quantized but nonmonotonous current plateaus simultaneously pumped by two homodromous surface acoustic waves.
Keywords: quantized pumping, surface acoustic wave, topological band structure PACS: 73.20.At, 73.21.Hb, 72.50.+b
DOI: 10.1088/1674-1056/25/6/067301
1. Introduction The surface acoustic wave (SAW) is frequently used as an experimental tool to transport electrons in one-dimensional (1D) quantum lines as well as in two-dimensional electron gases (2DEGs). [1,2] The SAW experiments in the 1D channel on GaAs-Alx Ga1−x As heterojunctions reveal quantized plateaus of acoustic-electric current which are proportional to the frequency of the acoustic wave, I = n · e f (n is an integer). [3] It means that an integer number of electrons are pumped through the channel in one oscillating cycle of the SAW. A lot of theoretical works have attempted to interpret this quantum phenomenon. [4–7] It is intuitively considered that the Coulomb-blockade effect may play a key role. The explicit mechanism, however, remains controversial. In Ref. [8], the authors discussed the precision of the plateaus and impurity of the potentials. Other works of modeling the quantized pump are based on the tight-binding model which consists of discrete quantum dots and two leads connecting the electron reservoirs on each side. The non-interacting electron transportation is expressed by reduced nearest-neighbor hopping amplitudes. [9,10] This model is analogous to the discrete 1D Harper model which exhibits a non-trivial topological phase. [11] Topology as a mathematical conception has been serving the condensed matter physics for 30 years since the explanation of the quantized Hall effects. It has become a useful means in understanding the topological insulator, [12] superconductivity, [13–15] and the quantized charge or spin pump. [16,17] It has been implemented in optical lattices [18,19] and in graphene-like materials as well. [20,21] In this paper, we propose a setup with a sequence of split metal gates in the 1D channel, creating a sequence of barrier potentials. We
then explore the topology implications of the quantized charge pumping by a SAW within the band theory. In the quantized charge pumping process, the electrons follow an adiabatically varying potential. [22] The surface acoustic wave is taken as a spatially periodical potential whose energy spectrum possesses the Bloch band structure, whereas the time-dependent phase plays the role of an adiabatic parameter of the Hamiltonian which induces a geometrical phase. The staircase-like acoustic currents are characterized by topological Chern numbers. We investigate the dependence of the number of pumped electrons on physical parameters such as the height of the barrier, the amplitude of the SAW and the Fermi levels and reveal a series of topological transitions. Based on this interpretation, we further predict a novel effect of quantized but nonmonotonous current plateaus simultaneously pumped by two homodromous SAWs.
2. Model and band topology The 1D narrow channel was experimentally realized by a split metal gate on the surface of a GaAs-Alx Ga1−x As heterojunction. The time-dependent SAW potential induced by the piezoelectric effects is written as V˜ (x,t) = VSAW cos2 (qπx/a − ωt)
(1)
with VSAW the amplitude. To study the topological properties of the quantum pumping process, we suppose the 1D channel contains a sequence of split metal gates, with a the length of a gate and q the number of spatial periods of the SAW in a single gate. Each split gate is modeled typically by a Gaussian-like barrier potential, [23]
∗ Project
Vg (x) =
V0 , cosh2 [(x − a/2)/σ ]
(2)
supported by the National Natural Science Foundation of China (Grant No. 11374036) and the National Basic Research Program of China (Grant No. 2012CB821403). † Corresponding author. E-mail:
[email protected] © 2016 Chinese Physical Society and IOP Publishing Ltd http://iopscience.iop.org/cpb http://cpb.iphy.ac.cn
067301-1
Chin. Phys. B Vol. 25, No. 6 (2016) 067301 where the parameter σ specifies the width of a barrier which is set to σ = a/4 in our model. Experimentally, Vg is controlled by the gate voltage applied to the 1D channel. We treat the static barrier potential in the q-period of the SAW (0 < x < a) by making use of the periodical condition. The time-dependent Hamiltonian reads: h¯ 2 d 2 ˆ H(x,t) =− +Vg (x) + V˜ (x,t) 2m dx2
(3)
where n is the band index. The lower row of Fig. 1 shows the topographical map of the Berry curvature versus k and t for (c) n = 2 and (d) n = 3 energy bands in panel (a). The topology of a single band is characterized by a Chern number which is the integration of the Berry curvature over the BZ, [24]
ˆ + a,t) = H(x,t). ˆ with H(x 70
70
(a)
(b)
50
Cn =
E/ER
E/ER
50
0.5 t/T
0 -
0
0.5 t/T
1.0
(c)
t/T -
k/(π/a)
Z BZ
n Ωk,t dkdt.
(5)
Table 1. Chern numbers Cn of the six lowest energy bands for various values of V0 . The amplitude of the SAW is fixed at VSAW = 30ER . The slashes indicate that the Chern numbers are not well-defined due to closing of the gap.
(d)
t/T
10
1.0
1 2π
The Chern number of each band can be computed as long as they are well-separated. For example, the integrations of Berry curvature distributions in Figs. 1(c) and 1(d) give a Chern number of 0 and 1, respectively.
30
30
10
shown in Figs. 1(a) and 1(b) which are plotted in the periodic scheme, we note that as V0 increases, the energy gaps may close and reopen and level crossings take place between the adjacent bands. The Berry curvature of a given band in the torus-shape BZ is defined as ∂ un ∂ un n Ωk,t =i | − c.c. , (4) ∂ k ∂t
-
k/(π/a)
Fig. 1. (color online) Energy band structure of E versus time t for (a) V0 = 20ER and (b) V0 = 30ER . The amplitude of the SAW is fixed as VSAW = 30ER and q = 2. Panel (b) shows that a band crossing occurs between the 2nd and 3rd bands as V0 increases. Panels (c) and (d) are the topographical maps of the Berry curvature distributions for the 3rd and 4th bands at V0 = 20ER . The integrations over the BZ give rise to the Chern number of 0 and 1, respectively.
For the slowly oscillating SAW, the system can be treated as a quasi-static problem. The time-dependent phase δ = ωt is taken as a variable parameter that yields a Berry phase as δ changes to complete a cycle from 0 to π. The Bloch wave vector k is approximately a good quantum number at a given δ . The periodicity both in the k-space and the δ -space defines a 2D Brillouin zone (BZ). In the upper row of Fig. 1 we display the energy band structures versus the time or δ for two values of barrier height (a) V0 = 20ER and (b) V0 = 30ER at a fixed SAW amplitude VSAW = 30ER . Here ER = h¯ 2 π 2 /2ma2 is taken as the unit of energy. The dependence of E on the Bloch momentum k leads to broadening of the energy bands and the gaps vary with time. By comparing band structures
V0 (ER )
10
20
30
40
50
60
70
C1 C2
0 1
0 1
0 0
0 0
0 0
0 0
0 0
C3 C4
0 1
0 1
1 0
1 0
0 1
0 1
0 0
C5 C6
/ /
0 /
1 0
1 0
0 1
0 1
1 0
Table 1 lists the Chern numbers of the six lowest bands versus V0 for the model Hamiltonian (3) with q = 2. The amplitude of the SAW is fixed at VSAW = 30ER . The Chern numbers for V0 = 0 are not shown, where the eigen equation is simply the Harper equation and the model is a sliding lattice. Each pair of adjacent bands corresponds to a single band of the V0 = 0 model, e.g., C1 + C2 instead of C1 or C2 separately is well-defined. The change of V0 opens or closes a gap, leading to jumps of the Chern number or topological transitions of the bands.
3. Topological transition and quantized pumping We examine the relation of the band topology with the system parameters. Figure 2 plots the six lowest gaps versus V0 at different times ωt = 0, π/8, π/4, 3π/8, and π/2, respectively. The amplitude of the SAW potential is VSAW = 30ER . From Fig. 2(b) we observe that the second energy gap is closed and reopened at V0 /ER ' 24, which corresponds to the jump of the Chern number in C2 and C3 as displayed in Table 1. It
067301-2
Chin. Phys. B Vol. 25, No. 6 (2016) 067301
(a)
in the fifth gap at V0A1 /ER ' 46 and V0A2 /ER ' 78 (Fig. 2(e)), respectively. The topological transition points are intimately related to the closing points of the energy gaps.
ωt=π/2 ωt=π/4 ωt=3π/8
ωt=π/8 ωt/
∆2/ER
∆1/ER
implies that a topological transition occurs. The same transitions occur in the third gap at V0 /ER ' 43 (Fig. 2(c)), in the fourth gap at V0A1 /ER ' 24 and V0A2 /ER ' 64 (Fig. 2(d)), and
(b)
V0/ER
(c)
∆4/ER
∆3/ER
V0/ER (d)
A2
A1
V0/ER
(e)
A2
A1
∆6/ER
∆5/ER
V0/ER (f)
(g)
B1
V0/ER
B2
∆6/ER
∆5/ER
V0/ER (h) B1
B3
B2
V0/ER
V0/ER
Fig. 2. (color online) Energy gaps ∆n versus the amplitudes V0 of the barrier potential at different time points ωt = 0 (blue), π/8 (red), π/4 (yellow), 3π/8 (purple), π/2 (green). Panels (a)–(f) correspond to n = 1–6 gaps with VSAW = 30ER . The zeros of the curves indicate that the gaps vanish. As a comparison, panels (g) and (h) respectively illustrate ∆5 and ∆6 for a larger SAW amplitude VSAW = 60ER where the 6th gap is reopened because of nonzero ∆6 . It means that the amplitude of the SAW modifies the topology of the energy bands.
On the other hand, topological transitions can also be implemented by adjusting the SAW amplitude VSAW . As VSAW increase from 30ER (Fig. 2(f)) to 60ER (Fig. 2(h)), gap-closing points appear at B1 , B2 , and B3 . To give a complete description of the relations of the band topology with the system parameters, we show in Fig. 3 the topographical map of the minimum gaps in the time period versus V0 and VSAW . The horizontal lines A and B in Figs. 3(e) and 3(f) mark VSAW = 30ER and VSAW = 60ER , respectively. The crossing points A1 , A2 , and B1 and B2 in Fig. 3(e) correspond to the transition points in Figs. 2(e) and 2(g). Accordingly, the points B1 , B2 , and B3 in Fig. 3(f) correspond to the transition points in Fig. 2(h). For the particle pumping, we adopt the adiabatic approximation analysis of the geometric phase [25–27] by considering that the velocity of the acoustic wave is much smaller than the Fermi velocity, [28,29] Z i t 0 0 iγn |ψt i = e exp − dt En (t ) h¯ t0 # " |un0 ihun0 |∂t |un i × |un i − i h¯ ∑ . (6) En − En0 n6=n0 It satisfies the time-dependent Schr¨odinger equation ˆ i h¯ ∂t |ψt i = H(t)|ψ t i. We have used the instantaneous eigenˆ states (the Bloch states) |un i of the Hamiltonian H(t) and R γn = i tt0 dt 0 hun |∂t 0 |un i. The system has a complete temporal
dependence on the SAW potential, which is periodically varying with the period T = π/ω. The expectation of the velocity operator vˆn = ∂ H(k,t)/∂ (¯hk) is hvn (k)i = hψt |vˆn |ψt i ∂ En hun |∂ H/∂ k|un0 ihun0 |∂t |un i = −i× ∑ h¯ ∂ k En − En0 n6=n0 =
∂ En (k) n − Ωk,t . h¯ ∂ k
(7)
The last equivalence is obtained by making use of the relation hun |∂ H/∂ k|un0 i = (En − En0 )h∂ un /∂ k|un0 i. Here insulating states rather than conducting states contribute to the induced current, thus only initially full-filled bands are of concern. The pumping current is the integration of hvn (k)i over the BZ, in which the zeroth-order term vanishes. The total number of particles pumped in an oscillating cycle is Z T
Qn = i
Z
dt 0
BZ
dk [h∂t un |∂k un i − h∂k un |∂t un i] . 2π
(8)
This is exactly the Chern number calculated from formulism (5). It demonstrates that the adiabatic pumping is quantized and the total number of pumped particles is equal to the summation of the Chern numbers of all completely filled bands. Thus the Fermi level should also be considered to determine the amount of bands that are of concern.
067301-3
Chin. Phys. B Vol. 25, No. 6 (2016) 067301 For example, if the lowest four bands are occupied, there are two transition points A1 and A2 in Fig. 2(d) as V0 increases. We notice that V0A1 is also the transition point of the second gap, as shown in Fig. 2(b). Consequently, there exist two quantized current plateaus. In the same way, the two transition points A1 and A2 in Fig. 2(e) indicate three current plateaus of I = 2e f for 0 ≤ V0 < V0A1 , I = 1e f for V0A1 < V0 < V0A2 , and I = 0 for V0A2 < V0 . Here f = ω/π is the frequency of
the SAW. In order to obtain higher current plateaus, one just needs to implement a larger SAW amplitude in which more electrons are classically trapped in a spatial period. In our frame of analysis, we need more transition points in the wellseparated gaps, as the transition points B1 , B2 , and B3 in the fifth (Fig. 2(g)) and sixth gaps (Fig. 2(h)) for a larger SAW amplitude VSAW = 60ER .
(a) ∆1/ER VSAW/ER
(b) ∆2/ER
V0/ER
V0/ER
(d) ∆4/ER
VSAW/ER
(c) ∆3/ER
V0/ER
V0/ER
VSAW/ER
(e) ∆5/ER A1
B2
B1
B1
A2
B2
B3 B
B A
A
(f) ∆ 6/ER
Fig. 3. (color online) Topographical map of the minimum energy gaps ∆n (n = 1–6) versus V0 and VSAW . The map is divided into several regimes by valleys where band gaps are closed. The rightmost region corresponds to the lowest plateaus of a pumping current of I = 0e f . The plateaus rise up from 0e f to ne f (with an integer n) while we assign parameters from lower right areas to upper left ones. 110 ef
ef
ef
Based on the above analysis, in Fig. 4 we summarize the
ef
current plateau diagram versus V0 and VSAW . Each region
90 C VSAW/ER
is labeled by the summation of Chern numbers of the wellef
ef
70
ef B
50 ef
ef
closing point of the gaps. Deeper insights into the transitional
ef
ef
ef
I/ef
A 30
separated bands. The ‘phase’ boundary is determined by the
ef
ef
ef
issues at these boundaries are beyond the capability of topo-
ef
ef ef
logical interpretation. Experimentally, the boundary may be broadened due to thermal fluctuations. The gap must be large
Ef/ER
enough so that the insulating condition is satisfied. We argue
10 100
200
300
that a partially-filled band has no contributions to the current
V0/ER
Fig. 4. (color online) Phase diagram of current plateaus versus V0 and VSAW . Each region corresponds to the summation of Chern numbers of the filled bands. Inset: schematic plot of staircase-like pumping current versus the Fermi energy for VSAW = V0 = 100ER .
induced by adiabatic pumping. Our theoretical treatment of non-interacting electrons qualitatively agrees with the experimental observations. [8,30]
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Chin. Phys. B Vol. 25, No. 6 (2016) 067301 4. Non-monotonous pumping plateaus by two SAWs
with different spatial period but the same temporal frequency. The two SAWs have the same acoustic velocity. It can be realized by implementing an additional SAW propagating at an angle θ to the direction of the 1D channel. Explicitly, we consider the following SAW potential,
The performance of SAW in split gates without gate voltage is simply a sliding potential, which gives rise to the standard of particle pumping. An extended research has confirmed the equivalence between topological properties and quantized charge pumping of a continuous Rice–Mele pump. [19] In this section, we study the pumping effect of two sinusoidal SAWs
V˜SAW (x,t) = V1 cos2 (πx/a − ωt) +V3 cos2 (3πx/a − ωt). (9) The propagating angle of the first SAW is cos θ = 1/3. t/T
4
6 6
0.03
Pumped particles
0.02 j1
4 0.01 2
(a)
0
4
2
0 (b)
0
-0.02
3
4
5
I/ef
ef
ef V1/V3/ER
ef
ef 2 (d2) V1/ER, V3/ER ef ef 1 0 0
-6 6
ef
ef
(d1)
1 0
I/ef
-4
Pumped particles
2
-0.01
2
6 6
0.01
(c)
1
4
0.02
0
-2 j3
2
0
0
0
0
j2
2
Pumped particles
t/T 0
ef
ef 10
t/T
20 Ef/ER
30
40
Fig. 5. (color online) Pumping currents (blue line) and particle number (red line) by the SAW potential (9) in panel (a) the first band, (b) the second band, and (c) the third band, respectively. Note that the pumping current is reversed in the third band, which is related to the Chern number C3 = −1. (d) Non-monotonous current plateaus versus the Fermi energy for V1 = 10ER . Upper panel (d1): V3 = 10ER . Lower panel (d2): V3 = 20ER .
The Chern number of each band can be computed in the same way. For V1 = V3 = 10ER , the Chern numbers of the lowest four bands are Cn = 1, 1, −1, 1. It is straightforward to calculate the current and pumped number of particles. Figure 5 shows the pumping currents of the three lowest bands. Each band contributes exactly one particle in an oscillating circle. Figure 5(c) indicates a pump in the negative direction, which corresponds to the Chern number C3 = −1. The current plateaus are depicted according to the band structure and the Chern invariants, as shown in Fig. 5(d1). The pumping currents exhibit a non-monotonous rather than the staircase-like behavior. For V1 = 10ER and V3 = 20ER the Chern numbers of the lowest four bands are Cn = 1, −1, 1, 1. The pumping current is zero if the lowest two band are filled, as shown in Fig. 5(d2). Finally, if we keep V1 = 30ER while increasing the second SAW amplitude from V3 = 30ER to V3 = 40ER , we can observe a jump from the I = 3e f plateau to the I = 1e f plateau. In this case, the Chern numbers of the four lowest bands experience
a transition from Cn = 1, 1, 1, −1 to Cn = 1, 1, −1, 1.
5. Summary In summary, we have investigated the SAW pumping effects in a 1D channel with barrier potentials caused by the gate voltage. The quantized pumping of electrons is interpreted in the viewpoint of topological invariants of the filled bands. The jump between two adjacent current plateaus is related to a topological transition. One of the benefits of topological theory is that the quantized plateaus are robust against perturbations as long as the gap keeps finite. Based on our interpretation, we predicted non-monotonous current plateaus pumped by two homodromous SAWs. Our prediction can be readily verified by current experiment techniques. [31,32]
References [1] Wixforth A, Scriba J, Wassermeier M, Kotthaus J P, Weimann G and Schlapp W 1989 Phys. Rev. B 40 7874 [2] Barnes C, Shilton J and Robinson A 2000 Phys. Rev. B 62 8410
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Chin. Phys. B Vol. 25, No. 6 (2016) 067301 [3] Shilton J, Talyanskii V, Pepper M, Ritchie D, Frost J, Ford C, Smith C and Jones G 1996 J. Phys.: Condens. Matter 8 L531 [4] Maksym P A 2000 Phys. Rev. B 61 4727 [5] Robinson A M and Barnes C H W 2001 Phys Rev. B 63 165418 [6] Chen X S. and Gao J 2010 Solid State Commun. 150 91 [7] Galperin Y M, Entin-Wohlman O and Levinson Y 2001 Phys. Rev. B 63 153309 [8] Ahlers F J, Fletcher N E, Ebbecke J and Janssen T J B M 2004 Curr. Appl. Phys. 4 529 [9] Aharony A and Entin-Wohlman O 2002 Phys. Rev. B 65 241401 [10] Kashcheyevs V, Aharony A and Entin-Wohlman O 2004 Eur. Phys. J. B-Condens. Matter and Complex Systems 39 385 [11] Lang L J, Cai X and Chen S 2012 Phys. Rev. Lett. 108 220401 [12] Hasan M Z and Kane C L 2010 Rev. Mod. Phys. 82 3045 [13] Zheng Y J, Song J T and Li Y X 2016 Chin. Phys. B 25 037301 [14] Liu Y, Zhao J Z Yu L, et al. 2015 Chin. Phys. Lett. 32 067303 [15] Wray L A, Xu S Y, Xia Y, Hor Y S, Qian D, Fedorov A V, Lin H, Bansil A, Cava R J and Hasan M Z 2010 Nat. Phys. 6 855 [16] Kitagawa T, Berg E, Rudner M and Demler E 2010 Phys. Rev. B 82 235114 [17] Meidan D, Micklitz T and Brouwer P W 2010 Phys. Rev. B 82 161303
[18] Wang L, Troyer M and Dai X 2013 Phys. Rev. Lett. 111 026802 [19] Zheng Y and Yang S J 2014 Physica B 454 93 [20] Leek P, Buitelaar M, Talyanskii V, Smith C, Anderson D, Jones G, Wei J and Cobden D 2005 Phys. Rev. Lett. 95 256802 [21] Connolly M, Chiu K, Giblin S, Kataoka M, Fletcher J, Chua C, Griffiths J, Jones G, Fal’Ko V and Smith C 2013 Nat. Nanotechnol. 8 417 [22] Thouless D J 1983 Phys. Rev. B 27 6083 [23] Gumbs G, Aˇızin G and Pepper M 1999 Phys. Rev. B 60 13954 [24] Nakahara M 2003 Geometry, Topology and Physics, 2nd edn. (London: Institute of Physics Publishing) pp. 428–437 [25] Berry M V 1984 Proc. Royal Soc. Lond. A: Mathematical and Physical Sciences 392 45 [26] Wang Z C, Li L and Gao J 2004 Phys. Lett. A 331 337 [27] Xiao D, Chang M C and Niu Q 2010 Rev. Mod. Phys. 82 1959 [28] Flensberg K, Niu Q and Pustilnik M 1999 Phys. Rev. B 60 R16291 [29] Pustilnik M, Flensberg K and Niu Q 2000 J. Low Temp. Phys. 118 571 [30] Fletcher N, Ebbecke J, Janssen T, Ahlers F, Pepper M, Beere H and Ritchie D 2003 Phys. Rev. B 68 245310 [31] Shabani J, Kim Y, Lutchyn R and Nayak C 2014 APS Meeting Abstracts 1 50003 [32] Bloch I 2005 Nat. Phys. 1 23
067301-6
Chinese Physics B Volume 25
Number 6
June 2016
TOPICAL REVIEW ł Low-dimensional complex oxide structures 066803
Aberration-corrected scanning transmission electron microscopy for complex transition metal oxides Qing-Hua Zhang, Dong-Dong Xiao and Lin Gu
067303
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067504
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060502
Interaction function of coupled bursting neurons Xia Shi and Jiadong Zhang
060503
Exploring the relationship between fractal features and bacterial essential genes Yong-Ming Yu, Li-Cai Yang, Qian Zhou, Lu-Lu Zhao and Zhi-Ping Liu
060504
𝐻∞ synchronization of the coronary artery system with input time-varying delay Xiao-Meng Li, Zhan-Shan Zhao, Jing Zhang and Lian-Kun Sun (Continued on the Bookbinding Inside Back Cover)
060505
An exclusion process with dynamic roadblocks Ning Guo, Jin-Yong Chen, Mao-Bin Hu and Rui Jiang
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Spectroscopic measurements and terahertz imaging of the cornea using a rapid scanning terahertz time domain spectrometer Wen-Quan Liu, Yuan-Fu Lu, Guo-Hua Jiao, Xian-Feng Chen, Zhi-Sheng Zhou, Rong-Bin She, Jin-Ying Li, Si-Hai Chen, Yu-Ming Dong and Jian-Cheng Lv
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Structures, stabilities, and magnetic properties of the Fe𝑛 Au (nn = 11–12) clusters Jin Lv, Jiang-Yan Zhang, Rui-Rui Liang and Hai-Shun Wu
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Photoionization microscopy of Rydberg hydrogen atom in a non-uniform electrical field Shao-Hao Cheng, De-Hua Wang, Zhao-Hang Chen and Qiang Chen
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064101
Electromagnetic backscattering from one-dimensional drifting fractal sea surface I: Wave–current coupled model Tao Xie, Shang-Zhuo Zhao, William Perrie, He Fang, Wen-Jin Yu and Yi-Jun He (Continued on the Bookbinding Inside Back Cover)
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Optical bistability and multistability via double dark resonance in graphene nanostructure Seyyed Hossein Asadpour, G Solookinejad, M Panahi and E Ahmadi Sangachin
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Optical control of light propagation in photonic crystal based on electromagnetically induced transparency Dan Wang, Jin-Ze Wu and Jun-Xiang Zhang
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Single-photon scattering by two separated atoms in a supercavity Wei Zhu, Xiao Xiao, Duan-Lu Zhou and Peng Zhang
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Control of dispersion in fiber coupled resonator-induced transparency structure He Tian, Yun-Dong Zhang, Da-Wei Qi, Run-Zhou Su, Yan Bai and Qiang Xu
064205
Analysis of algebraic reconstruction technique for accurate imaging of gas temperature and concentration based on tunable diode laser absorption spectroscopy Hui-Hui Xia, Rui-Feng Kan, Jian-Guo Liu, Zhen-Yu Xu and Ya-Bai He
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Spectral broadening induced by intense ultra-short pulse in 4H–SiC crystals Chun-hua Xu, Teng-fei Yan, Gang Wang, Wen-jun Wang, Jing-kui Liang and Xiao-long Chen
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Effect of turbulent atmosphere on the on-axis average intensity of Pearcey–Gaussian beam
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Laser-induced breakdown spectroscopy applied to the characterization of rock by support vector machine combined with principal component analysis Hong-Xing Yang, Hong-Bo Fu, Hua-Dong Wang, Jun-Wei Jia, Markus W Sigrist and Feng-Zhong Dong
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Investigation of impurity transport using laser blow-off technique in the HL-2A Ohmic and ECRH plasmas Kai Zhang, Zheng-Ying Cui, Ping Sun, Chun-Feng Dong, Wei Deng, Yun-Bo Dong, Shao-Dong Song, Min Jiang, Yong-Gao Li, Ping Lu and Qing-Wei Yang
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066101
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066102
Thermodynamic and transport properties of spiro-(1,1’)-bipyrrolidinium tetrafluoroborate and acetonitrile mixtures: A molecular dynamics study Qing-Yin Zhang, Peng Xie, Xin Wang, Xue-Wen Yu, Zhi-Qiang Shi and Shi-Huai Zhao
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Phenomenological description of semi-soft nematic elastomers Wen-Wen Diao, Qing-Tian Meng and Fang-Fu Ye
066104
Finite temperature effect on mechanical properties of graphene sheets with various grain boundaries Yong Ge, Hong-Xiang Sun, Yi-Jun Guan and Gan-He Zeng
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Understanding of surface pit formation mechanism of GaN grown in MOCVD based on local thermodynamic equilibrium assumption Zhi-Yuan Gao, Xiao-Wei Xue, Jiang-Jiang Li, Xun Wang, Yan-Hui Xing, Bi-Feng Cui and De-Shu Zou
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Comparative study on beryllium and magnesium as a co-doping element for ZnO:N Yu-Quan Su, Ming-Ming Chen, Long-Xing Su, Yuan Zhu and Zi-Kang Tang
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066802
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066804
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067101
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Numerical simulation study of organic nonvolatile memory with polysilicon floating gate Zhao-wen Yan, Jiao Wang, Jian-li Qiao, Wen-jie Chen, Pan Yang, Tong Xiao and Jian-hong Yang
067103
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(Continued on the Bookbinding Inside Back Cover)
067104
First principles study of the diffusional phenomena across the clean and Re-doped 𝛾-Ni/𝛾’-Ni3 Al interface of Ni-based single crystal superalloy Min Sun and Chong-Yu Wang
067105
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067106
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067201
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067203
Control of Hall angle of Skyrmion driven by electric current Gao-Bin Liu, Da Li, de Chatel P F, Jian Wang, Wei Liu and Zhi-Dong Zhang
067204
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A self-powered sensitive ultraviolet photodetector based on epitaxial graphene on silicon carbide Jiao Huang, Li-Wei Guo, Wei Lu, Yong-Hui Zhang, Zhe Shi, Yu-Ping Jia, Zhi-Lin Li, Jun-Wei Yang, HongXiang Chen, Zeng-Xia Mei and Xiao-Long Chen
067206
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067301
Topological charge pump by surface acoustic waves Yi Zheng, Shi-Ping Feng and Shi-Jie Yang
067302
Quantum transport through a multi-quantum-dot-pair chain side-coupled with Majorana bound states Zhao-Tan Jiang and Cheng-Cheng Zhong
067304
Piezoelectric polarization and quantum size effects on the vertical transport in AlGaN/GaN resonant tunneling diodes Dakhlaoui H and Almansour S
067305
Influence of surface states on deep level transient spectroscopy in AlGaN/GaN heterostructure Qing Zhu, Xiao-Hua Ma, Wei-Wei Chen, Bin Hou, Jie-Jie Zhu, Meng Zhang, Li-Xiang Chen, Yan-Rong Cao and Yue Hao
067306
Heterogeneous integration of GaAs pHEMT and Si CMOS on the same chip Li-Shu Wu, Yan Zhao, Hong-Chang Shen, You-Tao Zhang and Tang-Sheng Chen
067401
Critical current density behaviors across a grain boundary inclined to current with different angles in YBa2 Cu3 O7−𝛿 bicrystal junctions Tao Hua, Wei-Wei Xu, Zheng-Ming Ji, Da-Yuan Guo, Qing-Yun Wang, Xiang-Rong Ma and Rui-Yu Liang (Continued on the Bookbinding Inside Back Cover)
067402
Structural stabilities and electrical properties of Ba8 Ga16−𝑥 Cu𝑥 Sn30 single crystals under high temperatures Jin-Song Wang, Feng Cheng, Hong-Xia Liu, De-Cong Li, Lan-Xian Shen and Shu-Kang Deng
067501
Pressure effect on magnetic phase transition and spin-glass-like behavior of GdCo2 B2 Guang-Hui Hu, Ling-Wei Li and Umehara Izuru
067503
Role of vacancy-type defects in magnetism of GaMnN Hai-Ying Xing, Yu Chen, Chen Ji, Sheng-Xiang Jiang, Meng-Yao Yuan, Zhi-Ying Guo, Kun Li, Ming-Qi Cui and Guo-Yi Zhang
067801
First-principles studies of electronic, optical, and mechanical properties of γ-Bi2 Sn2 O7 Chao-Hao Hu, Xue-Hui Yin, Dian-Hui Wang, Yan Zhong, Huai-Ying Zhou and Guang-Hui Rao
067802
Broadband, polarization-insensitive, and wide-angle microwave absorber based on resistive film Dan-Dan Bu, Chun-Sheng Yue, Guang-Qiu Zhang, Yong-Tao Hu and Sheng Dong
067803
Raman scattering studies on the collapsed phase of CaCo2 As2 Jianting Ji, Anmin Zhang, Run Yang, Yong Tian, Feng Jin, Xianggang Qiu and Qingming Zhang INTERDISCIPLINARY PHYSICS AND RELATED AREAS OF SCIENCE AND TECHNOLOGY
068101
All-dielectric frequency selective surface design based on dielectric resonator Zheng-Bin Wang, Chao Gao, Bo Li, Zhi-Hang Wu, Hua-Mei Zhang and Ye-Rong Zhang
068102
Coalbed methane adsorption and desorption characteristics related to coal particle size Yan-Yan Feng, Wen Yang and Wei Chu
068103
Electronic transport properties of silicon junctionless nanowire transistors fabricated by femtosecond laser direct writing Liu-Hong Ma, Wei-Hua Han, Hao Wang, Qi-feng Lyu, Wang Zhang, Xiang Yang and Fu-Hua Yang
068104
Generalized model for laser-induced surface structure in metallic glass Lin-Mao Ye, Zhen-Wei Wu, Kai-Xin Liu, Xiu-Zhang Tang and Xiang-Ming Xiong
068105
Bound states of Dirac fermions in monolayer gapped graphene in the presence of local perturbations Mohsen Yarmohammadi and Malek Zareyan
068401
An efficient multipaction suppression method in microwave components for space application Wan-Zhao Cui, Yun Li, Jing Yang, Tian-Cun Hu, Xin-Bo Wang, Rui Wang, Na Zhang, Hong-Tai Zhang and Yong-Ning He
068402
Improvement of sintering, nonlinear electrical, and dielectric properties of ZnO-based varistors doped with TiO2 Osama A Desouky and K E Rady
068403
Perpendicularly oriented barium ferrite thin films with low microwave loss, prepared by pulsed laser deposition Da-Ming Chen, Yuan-Xun Li, Li-Kun Han, Chao Long and Huai-Wu Zhang
068501
An efficient calibration method for SQUID measurement system using three orthogonal Helmholtz coils Hua Li, Shu-Lin Zhang, Chao-Xiang Zhang, Xiang-Yan Kong and Xiao-Ming Xie
068901
A local fuzzy method based on “p-strong” community for detecting communities in networks Yi Shen, Gang Ren, Yang Liu and Jia-Li Xu
