1. A Series Journal of the Chinese Physical Society
Distributed by IOP Publishing
Online: http://iopscience.iop.org/cpl
http://cpl.iphy.ac.cn
CHINESE PHYSICAL SOCIET Y
ISSN: 0256-307X
中国物理快报
Chinese
Physics
Letters
Volume 29 Number 3 March 2012
2. CHIN. PHYS. LETT. Vol. 29, No. 3 (2012) 037401
Zero-Bias Conductance versus Potential Strength of Interface in Ferromagnetic
Superconductors
Hamidreza Emamipour1*
, Jafar Emamipour2
1
Department of Physics, Ilam University, Ilam, Iran
2
Young Researchers Club, Ilam Branch, Islamic Azad University, Ilam, Iran
(Received 8 October 2011)
We study zero-bias conductance (ZBC) spectra of a normal-metal/insulator/singlet (and triplet) ferromagnetic
superconductor as a function of potential strength of interface in the Blonder–Tinkham–Klapwijk (BTK) theory
framework. We consider possible pairing states including spin singlet s-wave pairing (SWP), spin triplet opposite
spin pairing (OSP) and spin triplet equal spin pairing (ESP). It is found that ZBC as a function of potential
strength of interface shows a clear difference between SWP, OSP and ESP states. These results may serve as a
useful tool for discriminating pairing states in ferromagnetic superconductors.
PACS: 74.20.Rp, 74.50.+r, 74.70.Dd DOI:10.1088/0256-307X/29/3/037401
After the discovery of each novel superconduc-
tor material, the most important question which
attracts the attention of physicists to itself is the
mechanism of Cooper pairing and symmetries of or-
der parameter in the newly discovered superconduc-
tors. The study of tunneling spectroscopy in normal-
metal/superconductor junctions provides much infor-
mation including excitation spectrum, symmetry of
pair potential and other interesting properties of su-
perconductors. Nowadays it is well cleared that An-
dreev reflection (AR) plays a fundamental role in
transport of N/S junctions.[1]
In 1982, Blonder, Tin-
kham and Klapwijk have presented the theory of tun-
neling spectroscopy for conventional s-wave supercon-
ductors (BTK theory) in which AR has been taken
into account.[2]
After the discovery of high tempera-
ture superconductors, BTK theory has been extended
to unconventional d-wave superconductors. The exis-
tence of zero bias conductance peaks (ZBCPs) in tun-
neling spectroscopy of normal-metal/unconventional
superconductor junctions, which arises from the for-
mation of mid-gap Andreev resonant states, has been
accepted as a strong indication of d-wave symmetry
in high-temperature cuprate superconductors.[3−9]
Discovery of superconductivity in some ferro-
magnetic materials such as UGe2,[10]
ZrZn2,[11]
and
URhGe[12]
opened a new chapter in the history of co-
existence between ferromagnetism and superconduc-
tivity and it has attracted much attention in low tem-
perature physics. One of the fundamental questions
about these materials is the mechanism of Cooper pair
formation. Because of the presence of ferromagnetism
in such materials, it seems that the symmetry of pair
potential should be spin triplet equal spin pairing.
Nevertheless, some research groups have shown that
one can not ignore spin singlet s-wave pairing.[13−20]
In any case, as was mentioned in Ref. [21], the conduc-
tance spectroscopy may serve as a useful tool which
can make a clear distinction between spin singlet and
spin triplet superconductors.
In this Letter, we calculate zero bias conductance
(ZBC) as a function of potential strength of normal
metal/ferromagnetic superconductor (N/FS) junction
where the FS is in three different Cooper pairing
states: s-wave spin singlet pairing (SWP), opposite-
spin triplet pairing (OSP) and equal-spin triplet pair-
ing (ESP). The results show that ZBC versus 𝑧 (poten-
tial strength of N/FS junction) shows a clear difference
for three different Cooper pairing symmetries: SWP,
OSP and ESP, in a way that for SWP, ZBC is smaller
than 2 and is a decreasing function of 𝑧. For OSP
and ESP states, with the increase of 𝑧, ZBC reaches
a minimum value. This minimum value for the OSP
state varies with the increase of the FS exchange field
while for the ESP state, ZBC does not show any sen-
sitivity toward the FS exchange field. The obtained
results may serve as a tool in determining the type of
pairing symmetry in ferromagnetic superconductors.
The interface of N/FS is located at 𝑥 = 0 and
perpendicular to the 𝑥 = 0-axes. The interface po-
tential has a 𝛿-function form [𝑉 (𝑥) = 𝑉 0 𝛿(𝑥)], the FS
and normal metal are considered in 𝑥 > 0 and 𝑥 < 0
regions, respectively. The motion of quasiparticles is
described by spin generalized Bogoliubov de Gennes
(SBdG) equations.[22,23]
Effective pair potential in the
FS region is determined by Δ(𝑘, 𝑟) = Δ(𝑘)Θ(𝑥), in
which Θ(𝑥) is a step function. This illustration for
pair potential can help us make an analytical solution
of the SBdG equations. Spin singlet state is described
by Δ(𝑘) = 𝑖Δ𝜎𝑦 and spin triplet state in terms of
the 𝑑(𝑘) = (𝑑 𝑥(𝑘), 𝑑 𝑦(𝑘), 𝑑 𝑧(𝑘)) vector is described
by Δ(𝑘) = 𝑖(𝑑(𝑘) · 𝜎)𝜎𝑦.
*Email: h_emamipour@yahoo.com
c○ 2012 Chinese Physical Society and IOP Publishing Ltd
037401-1
3. CHIN. PHYS. LETT. Vol. 29, No. 3 (2012) 037401
Following Refs. [22,23], the spin triplet is divided
into two different states: OSP state and ESP state.
In the OSP state the exchange field (ℎ) and 𝑑-vector
of the FS are parallel (i.e. ℎ × 𝑑(𝑘) = 0). Thus, as
a model for calculation, we consider ℎ and 𝑑 for the
OSP state as ℎ = ℎˆ𝑧 and 𝑑 = Δ
𝑘 𝑥+𝑖𝑘 𝑦
𝑘 ˆ𝑧 .
In the ESP state, ℎ and 𝑑(𝑘) are perpendicular
to each other (i.e. ℎ · 𝑑(𝑘) = 0). Thus, as a model
for calculation, we consider ℎ and 𝑑(𝑘) for the ESP
state as ℎ = ℎˆ𝑧 and 𝑑 = Δ
𝑘 𝑥+𝑘 𝑦
𝑘 ˆ𝑥, where ˆ𝑥 and ˆ𝑧
are the unit vectors in the direction of 𝑥 and 𝑧 axes,
respectively.
Now we are ready to apply the BTK model.[2]
Based on the BTK model, the incoming electrons
transmit to the FS region as electron-like and hole-
like quasiparticles. The electrons can undergo the
Andreev reflection coefficient 𝐴 and the normal reflec-
tion coefficient 𝐵 and conductance can be found from
(1+𝐴−𝐵). By solving the SBdG equations, the eigen-
values and corresponding eigenvectors of SWP, OSP
and ESP states are found, and by applying boundary
conditions in the interface, the tunneling conductance
is found (for more details see Refs. [24,25]).
The results are as follows:
𝜎 𝜌[𝜌′
]
𝑠 ={4𝜆 𝜌′ } ·
{︁[︁
4𝑧2
𝜃 + (1 + 𝜆 𝜌′ )2
+ |𝜂 𝜌|{4𝜆 𝜌′
− |𝜂 𝜌|[4𝑧2
𝜃 + (1 − 𝜆 𝜌′ )2
]}
]︁
·
[︁
|4𝑧2
𝜃 + (1 + 𝜆 𝜌′ )2
− 𝜂 𝜌[4𝑧2
𝜃 + (1 − 𝜆 𝜌′ )2
]𝑒𝑖(𝜑−−𝜑+)
|2
]︁−1}︁
, (1)
with
𝜂 𝜌 =
[︁ |Δ|
|𝐸 − 𝜌ℎ| +
√︀
(𝐸 − 𝜌ℎ)2 − |Δ|2
]︁2
, (2)
𝜆2
𝜌′ = 1 − 𝜌′ ℎ
𝐸 𝐹 cos2 𝜃
, 𝑧 𝜃 =
𝑧
cos 𝜃
, (3)
where 𝑧 (= 𝑚*
𝑒 𝑉0/¯ℎ2
𝑘 𝐹 ) is a dimensionless parameter
that indicates the strength of potential at interface, 𝜌
and 𝜌′
denote spin indices.
Normalized ZBC as a function of 𝑧 is obtained from
the equation
ZBC(𝑧) =
1
𝜎0
∑︁
𝜌[𝜌′]
+∞∫︁
−∞
𝑑𝐸
+ 𝜋/2∫︁
− 𝜋/2
𝑑𝜃 cos 𝜃
𝜎
𝜌[𝜌′
]
𝑠
cosh2
(𝐸/2𝑘B
𝑇)
,
(4)
where 𝜎0 is the normalized factor found from the tun-
neling conductance of 𝑁/𝑁 junction and is in the form
𝜎0 = 2
+𝜋/2∫︁
−𝜋/2
𝑑𝜃 cos 𝜃
1 + 𝑧2
𝜃
+∞∫︁
−∞
𝑑𝐸
cosh2
(𝐸/2𝑘B
𝑇)
. (5)
Normalized ZBC for SWP, OSP and ESP states are
found by setting quantities {𝜌′
= 0, 𝜑+ = 𝜑− = 0, 𝜌 =
±}, {𝜌′
= 0, 𝜑+ = 𝜃, 𝜑− = 𝜋 − 𝜃, 𝜌 = ±}, and
{𝜌′
= ±, 𝜑+ = 𝜃, 𝜑− = 𝜋 − 𝜃, 𝜌 = 0} in Eq. (1), re-
spectively.
0 1 2 3 4 5
0
0.5
1
2
ZBC
(a) SWP
1.5
2
2.5
ZBC
(b) OSP
0 2 4 6 8 10
1.8
2
2.2
2.4
z
ZBC
(c) ESP
h/∆0=0
h/∆0=0.1
h/∆0=0.2
h/∆0=0.3
h/∆0=0
h/∆0=0.1
h/∆0=0.2
h/∆0=0.3
h/∆0=0
h/∆0=0.1
h/∆0=0.2
h/∆0=0.3
1.5
Fig. 1. Normalized zero bias conductance (ZBC) versus
potential strength of interface (𝑧) for (a) SWP, (b) OSP
and (c) ESP with ℎ/Δ0 = 0, 0.1, 0.2, 0.3.
0 1 2 3 4 5
0
0.5
1
1.5
2
ZBC
(a) SWP
1
1.5
2
2.5
ZBC
(b) OSP
1.8
2
2.2
2.4
ZBC
(c) ESP
0 2 4 6 8 10
z
h/∆0=0.4
h/∆0=0.5
h/∆0=0.6
h/∆0=0.4
h/∆0=0.5
h/∆0=0.6
h/∆0=0.4
h/∆0=0.5
h/∆0=0.6
Fig. 2. Normalized zero bias conductance (ZBC) versus
potential strength of interface (𝑧) for (a) SWP, (b) OSP
and (c) ESP with ℎ/Δ0 = 0.4, 0.5, 0.6.
Figures 1 and 2 show the dependence of ZBC on
𝑧 at temperature
𝑘B
𝑇
Δ0
= 0.05 for SWP, OSP and ESP
states at ℎ/Δ0 = 0−0.3 and ℎ/Δ0 = 0.4−0.6 respec-
tively, where Δ0 is the pair potential at zero temper-
ature and zero exchange field, Δ0 = Δ(𝑇 = 0, ℎ = 0)
and 𝑘B is the Boltzman constant. Figure 1(a) shows
that with increasing 𝑧 the value of ZBC decreases very
fast in a way that at 𝑧 = 4 the value of ZBC is approx-
imately zero. This is because ZBC at the N/FS junc-
tion behaves proportional to 1/𝑧4
while the behavior
of ZBC at N/N junction is proportional to 1/𝑧2
. Thus,
normalized ZBC defined in Eq. (4) is proportional to
1/𝑧2
, as shown in Figs. 1(a) and 2(a) numerically. In
the high transparent limit 𝑧 → 0, the maximum value
of ZBC(=2) takes place. Also, Figs. 1(a) and 2(a)
show that ZBC versus 𝑧 does not change qualitatively
for different exchange fields ℎ.
Figures 1(b) and 2(b) show the 𝑧-dependence of
037401-2
4. CHIN. PHYS. LETT. Vol. 29, No. 3 (2012) 037401
the ZBC in the OSP state. As shown in Fig. 1(b),
ZBC at first decreases and then shows an increasing
behavior with 𝑧 in a way that a minimum is formed
in the ZBC-𝑧 curves. This minimum varies with ex-
change field and in large ℎ (see Fig. 2(b)) it disappears.
In fact, as it can be seen from Fig. 2(b), it can be de-
duced that for ℎ
Δ0
> 0.3 ZBC splits for different ℎ.
This is because the wave functions of the OSP state
depend on ℎ, so ZBC depends on ℎ (Ref. [23]) and it
splits. The magnitude of ZBC for large ℎ decreases
because the absolute value of pair potential(|Δ(𝑇, ℎ)|)
in the OSP state decreases with increase of ℎ in a way
that ℎ/Δ0 ≃ 0.7, it goes to zero.[24]
Therefore ZBC
tends to 1 with increase of ℎ.
By comparing Fig. 1(a) with Fig. 1(b), we can see
a different behavior in ZBC for SWP and OSP states,
which stems from their pair potentials. In the OSP
state, the sign change of pair potential on Fermi sur-
face leads to the formation of mid-gap Andreev res-
onant state and thus ZBC for large 𝑧 can be larger
than 2, while in the case of SWP, in which the pair
potential has no sign change on Fermi surface, ZBC
for large 𝑧 goes to zero.
Figures 1(c) and 2(c) show that for the ESP state,
ZBC at first decreases (for 𝑧 < 0.5) and then shows
an increasing function with 𝑧 (for 𝑧 > 0.5). This is be-
cause of the phase dependence of pair potential in the
ESP state. Phase dependence of pair potential in the
ESP state leads to the formation of mid-gap Andreev
resonant (MAR) states and MAR states are respon-
sible for the formation of zero bias conductance. As
mentioned in Ref. [23] the effects of MAR states in low
𝑧 (𝑧 < 0.5) is very weak and so in 𝑧 < 0.5 with the
increase of 𝑧, ZBC decreases. However, the effects of
MAR states in large 𝑧 is strong. Thus, in 𝑧 > 0.5
with the increase of 𝑧, ZBC increases. However, be-
cause the wave functions of ESP state do not have any
ℎ-dependence, ZBC does not vary with ℎ (unlike OSP
state).
It is noticeable that in high transparent 𝑧 = 0, the
value of ZBC for all these pairing states is equal to
2. This is because in 𝑧 = 0 there is not any mid-
gap Andreev resonant state for OSP and ESP states.
However, for large 𝑧, the effect of mid-gap Andreev
resonant state is strong.
Thus in large 𝑧, we find a clear difference in ZBC
curves between spin singlet SWP, spin triplet OSP and
spin triplet ESP states. In SWP state, ZBC is a fast
decreasing function of 𝑧. For OSP and ESP states the
minimum appears in ZBC curves which varies with ex-
change field of 𝐹 𝑆 while for ESP it does not change.
In summary we have studied zero bias conduc-
tance in normal metal/ferromagnetic superconductor
junctions as a function of strength potential of junc-
tion. We have found a clear difference in ZBC curves
between three different pairing SWP, OSP and ESP
states. The results can be used for discriminating
between pairing states in ferromagnetic superconduc-
tors.
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037401-3
5. Chinese Physics Letters
Volume 29 Number 3 March 2012
GENERAL
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CAI Jia-Xiang, MIAO Jun
030202 Application of the Homotopy Perturbation Method to the Burgers Equation with Delay
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030301 Entanglement of the Thermal State of an Anisotropic XY Z Spin Chain in an
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030306 Pulse Designed Coherent Dynamics of a Quantum Dot Charge Qubit
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030307 Spin Dynamics in the XY Model
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030308 Quantum State Tomography and Quantum Games
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030401 Anisotropic Bianchi Type-I Model with a Varying Λ Term
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030501 Noise-Induced Voltage Collapse in Power Systems
WEI Du-Qu, LUO Xiao-Shu, ZHANG Bo
030502 Quantifying Information Flow between Two Chaotic Semiconductor Lasers Using Symbolic
Transfer Entropy
LI Nian-Qiang, PAN Wei, YAN Lian-Shan, LUO Bin, XU Ming-Feng, TANG Yi-Long
030701 Temperature Uniformity of Wafer on a Large-Sized Susceptor for a Nitride Vertical MOCVD
Reactor
LI Zhi-Ming, JIANG Hai-Ying, HAN Yan-Bin, LI Jin-Ping, YIN Jian-Qin, ZHANG Jin-Cheng
030702 A Single-Transistor Active Pixel CMOS Image Sensor Architecture
ZHANG Guo-An, ZHANG Dong-Wei, HE Jin, SU Yan-Mei, WANG Cheng, CHEN Qin, LIANG Hai-Lang,
YE Yun
THE PHYSICS OF ELEMENTARY PARTICLES AND FIELDS
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031302 Effects of Anomalous Tensor Couplings in B0
s − ¯B0
s Mixing
CHANG Qin, HAN Lin, YANG Ya-Dong
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033101 An Effective Eigenchannel R-Matrix Method for Calculating Electron-Ion Scattering
Processes with Spectroscopic Precision
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FUNDAMENTAL AREAS OF PHENOMENOLOGY(INCLUDING
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034101 Grating-Coupled Waveguide Cloaking
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ZHOU Hang
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WANG Yi-Ze, LI Feng-Ming
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XU Yong-Jian, HU Chun-Dong, LIU Sheng, XIE Ya-Hong, LIANG Li-Zhen, JIANG Cai-Chao
035202 Quasi-monoenergetic Tens-of-MeV Proton Beams by a Laser-Illuminated Funnel-Like Target
BAN Hong-Ye, GU Yan-Jun, KONG Qing, LI Ying-Ying, ZHU Zhen, S. Kawata
7. 035203 Competition between Buneman and Langmuir Instabilities
GUO Jun, YU Bin
CONDENSED MATTER: STRUCTURE, MECHANICAL AND THERMAL
PROPERTIES
036101 Crystallization Characteristics of SiNx-Doped SbTe Films for Phase Change Memory
WAN Qi-Jian, FENG Jie, GUO Gang
036102 High-Quality Single Crystalline Ge(111) Growth on Si(111) Substrates by Solid Phase
Epitaxy
SUN Bing, CHANG Hu-Dong, LU Li, LIU Hong-Gang, WU De-Xin
036103 Generation of Nitrogen-Vacancy Centers in Diamond with Ion Implantation
CUI Jin-Ming, CHEN Xiang-Dong, FAN Le-Le, GONG Zhao-Jun, ZOU Chong-Wen, SUN Fang-Wen,
HAN Zheng-Fu, GUO Guang-Can
036104 B–C–N Compounds with Mixed Hybridization of sp2
-Like and sp3
-Like Bonds
LUO Xiao-Guang, HE Ju-Long
036105 Fabrication of GaAs Nanowires by Colloidal Lithography and Dry Etching
CHEN Ke, HE Jian-Jun, LI Ming-Yu, LaPierre R
036801 Adsorption, Diffusion, and Dissociation of H2O on Kaolinite (001): a Density Functional
Study
HE Man-Chao, ZHAO Jian
036802 Influence of Heating Rate on Morphologies and Magnetic Properties of α-Fe2O3
LI Zi-Yue, ZHANG Hui-Min, LIU Li-Hu, SUN Hui-Yuan
036803 Effect of Multiple Depositions and Annealing Treatments on the Erbium Silicide Nanoislands
Self-Assembled on Si(001) Substrates
DING Tao, SONG Jun-Qiang, CAI Qun
CONDENSED MATTER: ELECTRONIC STRUCTURE, ELECTRICAL,
MAGNETIC, AND OPTICAL PROPERTIES
037101 First-Principles Study on Electronic Structures and Optical Properties of Doped Ag Crystal
CAO Can, CHEN Ling-Na, JIA Shu-Ting, ZHANG Dan, XU Hui
037102 Fabrication and Electrical Characteristics of Individual ZnO Submicron-Wire Field-Effect
Transistor
JIANG Wei, GAO Hong, XU Ling-Ling
037201 Preparation and Photocatalytic Activity of ZnO/Fe2O3 Nanorod Arrays and ZnO/NiO
Nanotube Arrays
CUI Yin-Fang, WANG Cong, WU Su-Juan, LIU Yu, WANG Tian-Min
037301 Effect of Rapid Thermal Annealing Ambient on Photoluminescence of ZnO Films
XU Xiao-Yan, MA Xiang-Yang, JIN Lu, YANG De-Ren
037302 Structural and Optical Properties of GaAs0.5Sb0.5 and In0.5Ga0.5As0.5Sb0.5: ab initio
Calculations for Pure and Doped Materials
Mazin SH. Othman, Khudheir A. Mishjil, Nadir F. Habubi
037303 Effect of Electronic Correlations on Magnetotransport through a Parallel Double Quantum
Dot
FANG Dong-Kai, WU Shao-Quan, ZOU Cheng-Yi, ZHAO Guo-Ping
037401 Zero-Bias Conductance versus Potential Strength of Interface in Ferromagnetic
Superconductors
Hamidreza Emamipour, Jafar Emamipour
037402 Interface-Induced High-Temperature Superconductivity in Single Unit-Cell FeSe Films on
SrTiO3
WANG Qing-Yan, LI Zhi, ZHANG Wen-Hao, ZHANG Zuo-Cheng, ZHANG Jin-Song, LI Wei, DING Hao,
OU Yun-Bo, DENG Peng, CHANG Kai, WEN Jing, SONG Can-Li, HE Ke, JIA Jin-Feng, JI Shuai-Hua,
WANG Ya-Yu, WANG Li-Li, CHEN Xi, MA Xu-Cun, XUE Qi-Kun
8. 037601 Electron Paramagnetic Resonance of Mn2+
-Doped Cadmium Formate Dihydrate Single
Crystals
Indrajeet Mishra, Ram Kripal, Sanjay Misra
037701 Size Effects of the Properties in a Ferroelectric Bilayer Film with Surface Transition Layers
CUI Lian, XU Quan, HAN Zhi-You, XU Xu
037801 Wavelength Dependence of Laser-Induced Bulk Damage Morphology in KDP Crystal:
Determination of the Damage Formation Mechanism
HU Guo-Hang, ZHAO Yuan-An, LI Da-Wei, XIAO Qi-Ling
037802 Optical Bistability in Graded Core-Shell Granular Composites
WU Ya-Min, CHEN Guo-Qing, MA Chao-Qun, XUE Si-Zhong, ZHU Zhuo-Wei
CROSS-DISCIPLINARY PHYSICS AND RELATED AREAS OF SCIENCE AND
TECHNOLOGY
038101 Removing Impurity of cBN Crystal Prepared at High Pressure and High Temperature
JI Xiao-Rui, YANG Xiao-Hong
038102 Phase Evolution of Cubic ZnS Annealed in Mild Oxidizing Atmosphere
XUE Shu-Wen, ZHANG Jun, SHAO Le-Xi
038103 Optical and Mechanical Properties of Transparent Conductive Al-Doped ZnO Films
Deposited by the Sputtering Method
ZHU Yun, WANG Yue, WAN Peng-Fei, LI Hong-Yu, WANG Shou-Yu
038104 An Integrated Phase Change Memory Cell with Dual Trench Epitaxial Diode Selector
ZHANG Chao, SONG Zhi-Tang, WU Guan-Ping, LIU Bo, WANG Lian-Hong, XU Jia, LIU Yan,
WANG Lei, YANG Zuo-Ya, FENG Song-Lin
038201 Characterization of a ZnO Epilayer Grown on Sapphire by using Rutherford
Backscattering/Channeling and X-Ray Diffraction
Ding Bin-Feng
038301 Chemical Mechanical Polishing of Ge2Sb2Te5 Using Abrasive-Free Solutions of Iron
Trichloride
YAN Wei-Xia, WANG Liang-Yong, ZHANG Ze-Fang, HE Ao-Dong, ZHONG Min, LIU Wei-Li,
WU Liang-Cai, SONG Zhi-Tang
038401 A New Method to Calculate the Degree of Electromagnetic Impedance Matching in
One-Layer Microwave Absorbers
MA Zhi, CAO Chen-Tao, LIU Qing-Fang, WANG Jian-Bo
038501 Tunable Metamorphic Resonant Cavity Enhanced InGaAs Photodetectors Grown on GaAs
Substrates
LIU Shao-Qing, HAN Qin, ZHU Bin, YANG Xiao-Hong, NI Hai-Qiao, HE Ji-Fang, WANG Win,
NIU Zhi-Chuang
038502 A Novel CMOS Device Capable of Measuring Near-Field Thermal Radiation
FENG Chong, TANG Zhen-An, YU Jun
038701 Relaxation of Evolutionary Dynamics on the Bethe Lattice
TIAN Liang, LIN Min
038901 Search for Directed Networks by Different Random Walk Strategies
ZHU Zi-Qi, JIN Xiao-Ling, HUANG Zhi-Long
038902 Hyperedge Communities and Modularity Reveal Structure for Documents
XIE Zheng, YI Dong-Yun, OUYANG Zhen-Zheng, LI Dong
038903 Formation Mechanism of the Accumulative Magnification Effect in a Financial Time Series
DUAN Wen-Qi
038904 Mandelbrot Law of Evolving Networks
REN Xue-Zao, YANG Zi-Mo, WANG Bing-Hong, ZHOU Tao