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Product Equation for Riemann Zeta Function

Theorem

There exists a constant $B$ such that

$\displaystyle \frac{\zeta'(s)}{\zeta(s)} = B - \frac 1{s-1} +\frac12 \log \pi - \frac12 \frac{\Gamma'(s/2+1)}{\Gamma(s/2+1)} + \sum_\rho \left( \frac 1{s-\rho} + \frac1\rho \right)$

where $\zeta$ is the Riemann zeta function, $\rho$ runs over the non-trivial zeros of $\zeta$, and $\Gamma$ is the gamma function.

Proof

Let $\xi$ be the completed Riemann zeta function

$\displaystyle \xi(s) = \frac 12 s(s-1) \pi^{-s/2} \Gamma\left(\frac s2\right) \zeta(s)$

By Gamma Difference Equation, this gives

$\displaystyle \xi(s) = (s-1) \pi^{-s/2} \Gamma\left(\frac s2 + 1\right) \zeta(s)$

So taking the logarithm,

$\displaystyle \log \xi(s) = \log (s - 1) - \frac s2 \log\pi + \log\Gamma\left(\frac s2 + 1\right) + \log \zeta(s)$

And taking the derivative we obtain

$\displaystyle \frac{\xi'(s)}{\xi(s)} = \frac1{s-1} -\frac12 \log \pi + \frac{\Gamma'\left(s/2 + 1\right)}{\Gamma\left(s/2 + 1\right)} + \frac{\zeta'(s)}{\zeta(s)} \qquad (1)$

We have that the Completed Riemann Zeta Function has Order One, so by the Hadamard Factorisation Theorem, there exist constants $A$, $B$ such that

$\displaystyle \xi(s) = \exp(A+Bs)\: \prod_\rho \left( 1 - \frac s{\rho} \right) \exp\left( \frac s{\rho} \right)$

where $\rho$ runs over the zeros of $\xi$, that is, the nontrivial zeros of $\zeta$.

Taking the logarithm, we obtain

$\displaystyle \log \xi(s) = A+Bs + \sum_\rho \left[ \log \left( 1 - \frac s{\rho} \right) + \frac s{\rho} \right]$

which has derivative

$\displaystyle \frac{\xi'(s)}{\xi(s)} = B + \sum_\rho \left( \frac 1{s - \rho} + \frac 1\rho\right) \qquad (2)$

Combining $(1)$ and $(2)$ we have

$\displaystyle B + \sum_\rho \left( \frac 1{s - \rho} + \frac 1\rho\right) = \frac1{s-1} -\frac12 \log \pi + \frac{\Gamma'\left(s/2 + 1\right)}{\Gamma\left(s/2 + 1\right)} + \frac{\zeta'(s)}{\zeta(s)}$

from which the result follows by rearranging the terms.

$\blacksquare$

Note

The sum $\displaystyle \sum_\rho \frac 1{|\rho|}$ diverges, so we must be careful of the order in which we take the terms.

By the functional equation for $\zeta$, the zeros occur in complex conjugate pairs, and

$\displaystyle \frac 1\rho + \frac1{\overline{\rho}} = \frac{2 \Re(\rho)}{|\rho|^2} \leq \frac 2{|\rho|^2}$

and we see by zeros of functions of finite order that a sum of such terms does converge.

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# Showing that the powerset of odd numbers is uncountable

• April 20th 2011, 11:02 AM
StaryNight
Showing that the powerset of odd numbers is uncountable
I have already proved that the powerset of the natural numbers is uncountable. To show that P(M) is uncountable where M={n∈N | 2 divides n) I think I need to construct a bijection between P(N) and P(M)? I'm unsure of exactly how to achieve this - I can only imagine a bijection between N and M.

Any help would be appreciated.
• April 20th 2011, 11:19 AM
Plato
Quote:

Originally Posted by StaryNight
I have already proved that the powerset of the natural numbers is uncountable. To show that P(M) is uncountable where M={n∈N | 2 divides n) I think I need to construct a bijection between P(N) and P(M)? I'm unsure of exactly how to achieve this - I can only imagine a bijection between N and M.

This is a well-known result: If $A\sim B$, equipotent, then http://quicklatex.com/cache3/ql_5da9...68b5a03_l3.png.
After all there is a bijection http://quicklatex.com/cache3/ql_cfdb...b9342b1_l3.png.
So result follows that subsets of $A$ biject to subsets of $B$.
• April 22nd 2011, 07:05 AM
MoeBlee
Let f be a bijection between M and N.

Define g:PM -> PN by

g(x) = the f-image of x.

It's trivial to verify that g is a bijection onto PN.


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# Chapter 18 Worksheet

1. (6 pts) Define the following terms in your own words. Don't just copy/paste from the book:
• Exception
• Exception Handling
• Try block
• Catch block
• Unhandled exception
• Throw
2. Show how to modify the following code so that an error is printed if the number conversion is not successful. Modify this code, don't just copy the example from the text. Do NOT ask again if the conversion is unsuccessful.
user_input_string = input("Enter a number:")
user_value = int(user_input_string)

3. What will the following code output? Predict, and then run the code to see if you are correct. Write your prediction here and if you are right. If you aren't, make sure you understand why. (Make sure to write both the prediction, and the actual results. No credit will be given if you just list the results or the prediction. If the program raises an error, list that fact for this and the next problem as well.)
x = 5
y = 0
print("A")
try:
print("B")
a = x / y
print("C")
except:
print("D")
print("E")
print(a)

4. What will the following code output? Predict, and then run the code to see if you are correct. Write your prediction here and if you are right. If you aren't, make sure you understand why.
x = 5
y = 10
print("A")
try:
print("B")
a = x / y
print("C")
except:
print("D")
print("E")
print(a)


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# Homework Set 5

For several of these problems you'll need to look up some standard moments of inertia.

## Problem 9.28

Part A

Recall the definition of impulse from lecture 14.

Part B

Us the definition of momentum to get the velocity given that you know the mass.

## Collisions in one dimension

Use conservation of momentum for Part A. Then check the kinetic energy before and after to see what the change is. You can't assume from the start that this is an elastic collision!

## Problem 9.100

Part A and B: The actual collision is a one dimensional elastic collision where one body is at rest so it obeys the equations we derived for this kind of collision. To find the velocity of the object moving before the collision use conservation of mechanical energy for the motion down the slope.

Part C: Use conservation of energy to find the height the block goes up and convert to distance

## Problem 9.88

For a 2 dimensional elastic collision you can apply 3 equations two of which represent conservation of momentum and one of which corresponds to conservation of energy.

The problem is similar to our billiards example from the lecture. The difference is the masses are not the same, but as they are related to each other you'll find that the actual value of the mass is not important (the ratio however, is)

## Problem 9.55

Use conservation of momentum (initial momentum of the system is zero). Remember that momentum is a vector and should add as such!

## Problem 9.102

Draw a triangle to represent the vector sum of the the two initial momenta (which are equal in magnitude). The third side of this triangle should have length $2mv_{f}$, (from conservation of momentum). Use trigonometry to find the angle you need.

## Problem 10.12

Don't forget to convert your angular velocity from $\mathrm{rpm}$ to $\mathrm{rad/s}$.

To find the radial component of the acceleration you'll need to know the tangential velocity, which you should get by first finding the angular velocity at that time and then converting.

The tangential component of the acceleration can be found from you answer to part A.

## Problem 10.16

As before you need to do some unit conversion. Once you have the angular acceleration you can use one of the rotational motion kinematic equations.

## Problem 10.28

Remember that only force applied tangentially will produce a torque, if you work out this component and multiply that by the radius you'll have your answer.

## Problem 10.38

Part A. Find moment of inertia of the ball and angular acceleration, then use the rotational form of Newton's Second Law to find the torque.

Part B. The torque needs to come from somewhere, but as the distance at which the force is applied is quite different the force will be different too.

## Problem 10.67

One object gains potential energy, another loses it. All three gain kinetic energy.

## Problem 10.92

This problem is the opposite of the hoop and disk demo we did in the lecture. There we saw that some of the potential energy needed to be used to turn the hoop, here we will get it back when we go up the incline.

For the time taken, if at any point on the slope the kinetic energy is equally divided in to translational and kinetic energy, what must the translational acceleration be?


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#### Current Issue

Volume 37 Issue 12
Online Date:
 For Selected: Download Citations
GENERAL
 Select Long-Range Interaction Enhanced Adiabatic Quantum Computers Anqi Shi , Haoyu Guan , Jun Zhang , and Wenxian Zhang Chin. Phys. Lett. 2020, 37 (12): 120301 .   DOI: 10.1088/0256-307X/37/12/120301 Abstract   PDF(pc) (1229KB) ( 137 ) PDF(mobile)(1412KB) ( 19 ) HTML ( 100 )  A quantum computer is not necessarily alone, e.g., thousands and millions of quantum computers are simultaneously working together for adiabatic quantum computers based on nuclear spins. Long-range interaction is inevitable between these nuclear spin qubits. Here we investigate the effect of long-range dipolar interaction between different adiabatic quantum computers. Our analytical and numerical results show that the dipolar interaction can enhance the final fidelity in adiabatic quantum computation for solving the factorization problem, when the overall interaction is negative. The enhancement will become more prominent if a single quantum computer encounters an extremely small energy gap which occurs more likely for larger-size systems.
 Select Active Thermal Wave Cloak Liu-Jun Xu and Ji-Ping Huang Chin. Phys. Lett. 2020, 37 (12): 120501 .   DOI: 10.1088/0256-307X/37/12/120501 Abstract   PDF(pc) (2531KB) ( 89 ) PDF(mobile)(2514KB) ( 9 ) HTML ( 41 )  Active metamaterials have shown huge advantages to control electromagnetic and acoustic waves. However, how to use active metamaterials to control thermal waves has not been explored, though thermal waves are significant in various fields. To address the problem, here we report an active scheme for thermal wave cloaks. The thermal waves are based on conduction and convection, which are dominated by the Fourier and Darcy laws, respectively. By calculating the propagation of thermal waves in a free space, we can derive the global temperature and pressure distributions. We then apply these calculation results to actively control the boundary temperature and pressure, and active thermal wave cloaks can be obtained. Compared with existing passive schemes to control thermal waves, the present active scheme is more flexible for switching on/off and changing geometries. This work provides active and controllable components to thermal wave cloaks, which can be further used to design more active thermal wave metamaterials.
 Select Three-Terminal Thermionic Heat Engine Based on Semiconductor Heterostructures Yun-Yun Yang , Shuai Xu , and Ji-Zhou He Chin. Phys. Lett. 2020, 37 (12): 120502 .   DOI: 10.1088/0256-307X/37/12/120502 Abstract   PDF(pc) (1211KB) ( 45 ) PDF(mobile)(1205KB) ( 6 ) HTML ( 11 )  We propose a model for three-terminal thermionic heat engines based on semiconductor heterostructures. According to electron transport theory, we drive the formulas for the charge current and energy current flowing from the electron reservoir and we then obtain the power output and efficiency in the linear and nonlinear regimes. Furthermore, we analyze the performance characteristic of the thermionic heat engine and get the maximum power output by optimizing the performance parameters. Finally, we optimize the thermodynamic performance of the thermionic heat engine by maximizing the product of the power output and efficiency.
THE PHYSICS OF ELEMENTARY PARTICLES AND FIELDS
 Select Impressions of the Continuum Bound State Problem in QCD Si-Xue Qin and C. D. Roberts Chin. Phys. Lett. 2020, 37 (12): 121201 .   DOI: 10.1088/0256-307X/37/12/121201 Abstract   PDF(pc) (928KB) ( 28 ) PDF(mobile)(893KB) ( 3 ) HTML ( 9 )  Modern and anticipated facilities will deliver data that promises to reveal the innermost workings of quantum chromodynamics (QCD). In order to fulfill that promise, phenomenology and theory must reach a new level, limiting and overcoming model-dependence, so that clean lines can be drawn to connect the data with QCD itself. Progress in that direction, made using continuum methods for the hadron bound-state problem, is sketched herein.
ATOMIC AND MOLECULAR PHYSICS
 Select Molecular Opacity Calculations for Lithium Hydride at Low Temperature Gui-Ying Liang, Yi-Geng Peng, Rui Li, Yong Wu, and Jian-Guo Wang Chin. Phys. Lett. 2020, 37 (12): 123101 .   DOI: 10.1088/0256-307X/37/12/123101 Abstract   PDF(pc) (1015KB) ( 31 ) PDF(mobile)(1255KB) ( 2 ) HTML ( 14 )  The opacities of the lithium hydride molecule are calculated for temperatures of 300 K, 1000 K, 1500 K, and 2000 K, at a pressure of 10 atm, in which the contributions from the five low-lying electronic states are considered. The ab initio multi-reference single and double excitation configuration interaction (MRDCI) method is applied to compute the potential energy curves (PECs) of the $^{7}$LiH, including four $^{1}\!\varSigma^{+}$ states and one $^{1}\!\varPi$ state, as well as the corresponding transition dipole moments between these states. The ro-vibrational energy levels are calculated based on the PECs obtained, together with the spectroscopic constants. In addition, the partition functions are also computed, and are provided at temperatures ranging from 10 K to 2000 K for $^{7}$LiH, $^{7}$LiD, $^{6}$LiH, and $^{6}$LiD.
 Select Observation of Cesium ($nD_{5/2}$+$6S_{1/2}$) Ultralong-Range Rydberg-Ground Molecules Su-Ying Bai, Jing-Xu Bai, Xiao-Xuan Han, Yue-Chun Jiao, Jian-Ming Zhao, and Suo-Tang Jia Chin. Phys. Lett. 2020, 37 (12): 123201 .   DOI: 10.1088/0256-307X/37/12/123201 Abstract   PDF(pc) (890KB) ( 25 ) PDF(mobile)(883KB) ( 5 ) HTML ( 10 )  Ultralong-range Cs$_2$ Rydberg-ground molecules ($nD_{5/2} + 6S_{1/2}F$) ($33 \leq n \leq 39$, $F=3$ or 4) are investigated by a two-photon photo-association spectroscopy of an ultracold Cs gas. Two vibrational ground molecular spectra of triplet $^{3}\!\varSigma$ and hyperfine mixed singlet-triplet $^{1,3}\!\varSigma$ molecular states and their corresponding binding energies are attained. The experimental observations are simulated by an effective Hamiltonian including low energy electron scattering pseudopotentials, the spin-orbit interaction of the Rydberg atom, and the hyperfine interaction of the ground-state atom. The zero-energy singlet and triplet s-wave scattering lengths are extracted by comparing the experimental observations and calculations. Dependences of the measured binding energies on the effective principal quantum number, $n_{\rm eff} = n - \delta_D$ ($\delta_D$ is the quantum defect of Rydberg $D$ state), yield the scaling of $n_{\rm eff}^{-5.60 \pm 0.16}(^{3}\!\varSigma,F=3)$, $n_{\rm eff}^{-5.62 \pm 0.16}(^{3}\!\varSigma,F=4)$ for deep triplet potential and $n_{\rm eff}^{-5.65 \pm 0.38}(^{1,3}\!\varSigma, F=3)$, $n_{\rm eff}^{-6.19 \pm 0.14}(^{1,3}\!\varSigma,F=4)$ for shallow mixed singlet-triplet potential well. The simulations of low-energy Rydberg electron scattering show agreement well with the experimental measurements.
FUNDAMENTAL AREAS OF PHENOMENOLOGY(INCLUDING APPLICATIONS)
 Select Generation and Tunable Focal Shift of the Hybridly Polarized Vector Optical Fields with Parabolic Symmetry Xu-Zhen Gao, Meng-Shuai Wang, Jia-Hao Zhao, Peng-Cheng Zhao, Xia Zhang, Yue Pan, Yongnan Li, Chenghou Tu, and Hui-Tian Wang Chin. Phys. Lett. 2020, 37 (12): 124201 .   DOI: 10.1088/0256-307X/37/12/124201 Abstract   PDF(pc) (898KB) ( 74 ) PDF(mobile)(896KB) ( 6 ) HTML ( 60 )  Based on a parabolic coordinate system, we theoretically design and experimentally generate hybridly polarized vector optical fields with parabolic symmetry of the first and second kinds, which can further enrich the family of vector optical fields. The wavefront of this new-kind vector optical field contains circular, elliptic and linear polarizations, and the polarizations can keep the same or change along the parabolic curves. Then we present the realization of tunable focal shift with the hybridly polarized vector optical field, and show a specific law of the focal shift of the focused hybridly polarized vector optical field with the parabolic symmetry. We hope these results can provide a new way to flexibly modulate focal fields, which can be applied in realms such as optical machining, optical trapping and information transmission.
 Select Hermite Non-Uniformly Correlated Array Beams and Its Propagation Properties Xue-Chun Zhao, Lei Zhang, Rong Lin, Shu-Qin Lin, Xin-Lei Zhu, Yang-Jian Cai, and Jia-Yi Yu Chin. Phys. Lett. 2020, 37 (12): 124202 .   DOI: 10.1088/0256-307X/37/12/124202 Abstract   PDF(pc) (1957KB) ( 113 ) PDF(mobile)(1957KB) ( 13 ) HTML ( 111 )  We study the evolution of spectral intensity and degree of coherence of a new class of partially coherent beams, Hermite non-uniformly correlated array beams, in free space and in turbulence, based on the extended Huygens–Fresnel integral. Such beams possess controllable rectangular grid distributions due to multi-self-focusing propagation property. Furthermore, it is demonstrated that adjusting the initial beam parameters, mode order, shift parameters, array parameters and correlation width plays a role in resisting intensity and degree of coherence degradation effects of the turbulence.
 Select Extreme Ultraviolet Frequency Comb with More than 100 µW Average Power below 100 nm Jin Zhang, Lin-Qiang Hua, Zhong Chen, Mu-Feng Zhu, Cheng Gong, and Xiao-Jun Liu Chin. Phys. Lett. 2020, 37 (12): 124203 .   DOI: 10.1088/0256-307X/37/12/124203 Abstract   PDF(pc) (766KB) ( 107 ) PDF(mobile)(755KB) ( 5 ) HTML ( 43 )  Extreme ultraviolet (XUV) frequency comb is a powerful tool in precision measurement. It also brings many new opportunities to the field of strong field physics since high harmonic generation related phenomena can be studied with high repetition rate. We demonstrate the generation of an XUV frequency comb with the aid of intra-cavity high harmonic generation process. The setup is driven by a high power infrared frequency comb, and an average power of 4.5 kW is reached in the femtosecond enhancement cavity. With Xe gas as the working media, harmonics up to the 19th order are observed. Power measurement indicates that as much as 115.9 μW (1.3 mW) are generated at $\sim$94 nm ($\sim$148 nm). The shortest wavelength we can reach is $\sim$55 nm. The coherence of the generated light is tested with an optical-heterodyne-based measurement of the third harmonic. The resulted line width is $\sim$3 Hz. In addition, with this system, we also observe a strong suppression of below threshold harmonics from O$_2$ compared to that from Xe. These results suggest that the current system is ready for precision spectroscopic measurements with few-electron atomic and molecular systems in XUV region as well as the study of strong field physics with an unprecedented 100 MHz repetition rate.
PHYSICS OF GASES, PLASMAS, AND ELECTRIC DISCHARGES
 Select Energetic Particles in Magnetic Confinement Fusion Plasmas Wei Chen and Zheng-Xiong Wang Chin. Phys. Lett. 2020, 37 (12): 125001 .   DOI: 10.1088/0256-307X/37/12/125001 Abstract   PDF(pc) (372KB) ( 101 ) PDF(mobile)(385KB) ( 20 ) HTML ( 14 )
CONDENSED MATTER: STRUCTURE, MECHANICAL AND THERMAL PROPERTIES
 Select Accuracy of Machine Learning Potential for Predictions of Multiple-Target Physical Properties Yulou Ouyang, Zhongwei Zhang, Cuiqian Yu, Jia He, Gang Yan, and Jie Chen Chin. Phys. Lett. 2020, 37 (12): 126301 .   DOI: 10.1088/0256-307X/37/12/126301 Abstract   PDF(pc) (1244KB) ( 138 ) PDF(mobile)(1247KB) ( 21 ) HTML ( 90 )  The accurate and rapid prediction of materials' physical properties, such as thermal transport and mechanical properties, are of particular importance for potential applications of featuring novel materials. We demonstrate, using graphene as an example, how machine learning potential, combined with the Boltzmann transport equation and molecular dynamics simulations, can simultaneously provide an accurate prediction of multiple-target physical properties, with an accuracy comparable to that of density functional theory calculation and/or experimental measurements. Benchmarked quantities include the Grüneisen parameter, the thermal expansion coefficient, Young's modulus, Poisson's ratio, and thermal conductivity. Moreover, the transferability of commonly used empirical potential in predicting multiple-target physical properties is also examined. Our study suggests that atomic simulation, in conjunction with machine learning potential, represents a promising method of exploring the various physical properties of novel materials.
CONDENSED MATTER: ELECTRONIC STRUCTURE, ELECTRICAL, MAGNETIC, AND OPTICAL PROPERTIES
 Select HfX$_{2}$ (X = Cl, Br, I) Monolayer and Type II Heterostructures with Promising Photovoltaic Characteristics Xingyong Huang, Liujiang Zhou, Luo Yan, You Wang, Wei Zhang, Xiumin Xie, Qiang Xu, and Hai-Zhi Song Chin. Phys. Lett. 2020, 37 (12): 127101 .   DOI: 10.1088/0256-307X/37/12/127101 Abstract   PDF(pc) (1540KB) ( 39 ) PDF(mobile)(1530KB) ( 4 ) HTML ( 7 )  Two-dimensional (2D) materials and their corresponding van der Waals (vdW) heterostructures are considered as promising candidates for highly efficient solar cell applications. A series of 2D HfX$_{2}$ (X = Cl, Br, I) monolayers are proposed, via first-principle calculations. The vibrational phonon spectra and molecular dynamics simulation results indicate that HfX$_{2}$ monolayers possess dynamical and thermodynamical stability. Moreover, their electronic structure shows that their Heyd–Scuseria–Ernzerhof(HSE06)-based band values (1.033–1.475 eV) are suitable as donor systems for excitonic solar cells (XSCs). The material's significant visible-light absorbing capability (${\sim}10^{5}$ cm$^{-1}$) and superior power conversion efficiency (${\sim}$20%) are demonstrated by establishing a reasonable type II vdW heterostructure. This suggests the significant potential of HfX$_{2}$ monolayers as a candidate material for XSCs.
 Select Inversion/Mirror Symmetry-Protected Dirac Cones in Distorted Ruby Lattices Lei Sun, Xiaoming Zhang, Han Gao, Jian Liu, Feng Liu, and Mingwen Zhao Chin. Phys. Lett. 2020, 37 (12): 127102 .   DOI: 10.1088/0256-307X/37/12/127102 Abstract   PDF(pc) (4800KB) ( 70 ) PDF(mobile)(5788KB) ( 2 ) HTML ( 10 )  The exotic electronic band structures of Ruby and Star lattices, characterized by Dirac cone and nontrivial topology, offer a unique platform for the study of two-dimensional (2D) Dirac materials. In general, an ideal isotropic Dirac cone is protected by time reversal symmetry and inversion, so that its robustness against lattice distortion is not only of fundamental interest but also crucial to practical applications. Here we systematically investigate the robustness of Dirac cone in a Ruby lattice against four typical lattice distortions that break the inversion and/or mirror symmetry in the transition from Ruby to Star. Using a tight-binding approach, we show that the isotropic Dirac cones and their related topological features remain intact in the rotationally distorted lattices that preserve the inversion symmetry ($i$-Ruby lattice) or the in-plane mirror symmetry ($m$-Ruby lattice). On the other hand, the Dirac cones are gapped in the $a$- and $b$-Ruby lattices that break both these lattice symmetries or inversion. Furthermore, a rotational unitary matrix is identified to transform the original into the distorted lattice. The symmetry-protected Dirac cones were also verified in photonic crystal systems. The robust Dirac cones revealed in the non-mirror symmetric $i$-Ruby and non-centrosymmetric $m$-Ruby lattices provide a general guidance for the design of 2D Dirac materials.
 Select Fano Effect and Spin-Polarized Transport in a Triple-Quantum-Dot Interferometer Attached to Two Ferromagnetic Leads Jiyuan Bai, Kongfa Chen, Pengyu Ren, Jianghua Li, Zelong He, and Li Li Chin. Phys. Lett. 2020, 37 (12): 127301 .   DOI: 10.1088/0256-307X/37/12/127301 Abstract   PDF(pc) (871KB) ( 36 ) PDF(mobile)(864KB) ( 2 ) HTML ( 10 )  We report the conductance and average current through a triple-quantum-dot interferometer coupled with two ferromagnetic leads using the nonequilibrium Green's function. The results show that the interference between the resonant process and the non-resonant process leads to the formation of Fano resonance. More Fano resonances can be observed by applying a time-dependent external field. As a Zeeman magnetic field is applied, the spin-up electron transport is depressed in a certain range of electron energy levels. A spin-polarized pulse device can be realized by adjusting the spin polarization parameters of ferromagnetic leads. Moreover, the $I$–$V$ characteristic curves show that under the influence of Fano resonance, the spin polarization is significantly enhanced by applying a relatively large reverse bias voltage. These results strongly suggest that the spin-polarized pulse device can be potentially applied as a spin-dependent quantum device.
 Select Gap Structure of 12442-Type KCa$_2$(Fe$_{1-x}$Co$_{x}$)$_4$As$_{4}$F$_2$ ($x$ = 0, 0.1) Revealed by Temperature Dependence of Lower Critical Field Jianan Chu, Teng Wang, Han Zhang, Yixin Liu, Jiaxin Feng, Zhuojun Li, Da Jiang, Gang Mu, Zengfeng Di, and Xiaoming Xie Chin. Phys. Lett. 2020, 37 (12): 127401 .   DOI: 10.1088/0256-307X/37/12/127401 Abstract   PDF(pc) (697KB) ( 36 ) PDF(mobile)(674KB) ( 2 ) HTML ( 10 )  We report an in-depth investigation on the out-of-plane lower critical field $H_{\rm c1}$ of the KCa$_2$(Fe$_{1-x}$Co$_{x}$)$_4$As$_{4}$F$_2$ (12442-type, $x$ = 0, 0.1). The multi-gap feature is revealed by the kink in the temperature-dependent $H_{\rm c1}(T)$ curve for the two samples with different doping levels. Based on a simplified two-gap model, the magnitudes of the two gaps are determined to be $\varDelta_1$ = 1.2 meV and $\varDelta_2$ = 5.0 meV for the sample with $x$ = 0, $\varDelta_1$ =  0.86 meV and $\varDelta_2$ = 2.8 meV for that with $x$ = 0.1. With the cobalt doping, the ratio of energy gap to critical transition temperature ($\varDelta/k_{\rm B}T_{\rm c}$) remains almost unchanged for the smaller gap and is suppressed by 20% for the larger gap. For the undoped KCa$_2$Fe$_4$As$_{4}$F$_2$, the obtained gap sizes are generally consistent with the results of angle-resolved photoemission spectroscopy experiments.
 Select Low Voltage Reversible Manipulation of Ferromagnetic Resonance Response in CoFeB/HfO$_{2}$ Heterostructures Yangping Wang, Hongyan Zhou, Yibing Zhao, Fufu Liu, and Changjun Jiang Chin. Phys. Lett. 2020, 37 (12): 127501 .   DOI: 10.1088/0256-307X/37/12/127501 Abstract   PDF(pc) (962KB) ( 32 ) PDF(mobile)(953KB) ( 3 ) HTML ( 13 )  We report that the ferromagnetic resonance (FMR) response of the CoFeB/HfO$_{2}$ heterostructures is stabilized and reversibly manipulated by ionic gel. Ionic gel with excellent flexibility is used as a medium to form an electric field. When a 4 V gate voltage is applied, the resonance field $H_{\rm r}$ and peak-to-peak linewidth $\Delta H_{\rm pp}$ at different angles are regulated. When $\theta = 20^{\circ}$, the $H_{\rm r}$ is regulated up to 82 Oe. When $\theta = 70^{\circ}$, $\Delta H_{\rm pp}$ is tuned up to 75 Oe. When the gate voltage is repeatedly applied, the FMR spectra can be freely switched between the initial state and the gated state. Our study provides an effective method to manipulate the damping of the magnetic film stably and reversibly.
 Select Trion-Facilitated Dexter-Type Energy Transfer in a Cluster of Single Perovskite CsPbBr$_{3}$ Nanocrystals Zengle Cao, Fengrui Hu, Zaiqin Man, Chunfeng Zhang, Weihua Zhang, Xiaoyong Wang, and Min Xiao Chin. Phys. Lett. 2020, 37 (12): 127801 .   DOI: 10.1088/0256-307X/37/12/127801 Abstract   PDF(pc) (1544KB) ( 80 ) PDF(mobile)(1529KB) ( 8 ) HTML ( 16 )  Semiconductor colloidal nanocrystals (NCs) can interact with each other to profoundly influence the charge transfer, transport and extraction processes after they have been assembled into a high-density film for optoelectronic device applications. These interactions normally occur among several nearby single colloidal NCs, which should be effectively separated from their surroundings to remove the ensemble average effect for fine optical characterizations. By means of atomic force microscopy (AFM) nanoxerography, here we prepare individual clusters of perovskite CsPbBr$_{3}$ NCs and perform single-particle measurements on their optical properties at the cryogenic temperature. While discrete photoluminescence bands can be resolved from the several single CsPbBr$_{3}$ NCs that are contained within an individual cluster, the shorter- and longer-wavelength bands are dramatically different in that their intensities show sub- and superlinear dependences on the laser excitation powers, respectively. This can be explained by the generation of charged excitons (trions) at high laser excitation powers, and their subsequent Dexter-type energy transfer from smaller- to larger-sized CsPbBr$_{3}$ NCs. Our findings not only suggest that these individual clusters prepared by AFM nanoxerography can serve as a potent platform to explore few-NC interactions but they also reveal the long-neglected role played by trions in channeling photo-excited energies among neighboring NCs.
17 articles