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h[koGv+FLBl , hb```V ce`aipxGoW+Q:R8!#R=J:R:!dQM|O%/ E . s 3.1. dfy1``~@6m=5c/PEPg?\B2YO0p00gXp!b;Zfb[ a`2_ += {\displaystyle s/V_{k}} k Now we can derive the density of states in this region in the same way that we did for the rest of the band and get the result: \[ g(E) = \dfrac{1}{2\pi^2}\left( \dfrac{2|m^{\ast}|}{\hbar^2} \right)^{3/2} (E_g-E)^{1/2}\nonumber\]. 0000003215 00000 n / 2 is due to the area of a sphere in k -space being proportional to its squared radius k 2 and by having a linear dispersion relation = v s k. v s 3 is from the linear dispersion relation = v s k. The density of states (DOS) is essentially the number of different states at a particular energy level that electrons are allowed to occupy, i.e. b Total density of states . The order of the density of states is $\begin{equation} \epsilon^{1/2} \end{equation}$, N is also a function of energy in 3D. / Use MathJax to format equations. ) {\displaystyle D_{n}\left(E\right)} 75 0 obj <>/Filter/FlateDecode/ID[<87F17130D2FD3D892869D198E83ADD18><81B00295C564BD40A7DE18999A4EC8BC>]/Index[54 38]/Info 53 0 R/Length 105/Prev 302991/Root 55 0 R/Size 92/Type/XRef/W[1 3 1]>>stream %PDF-1.5 % The right hand side shows a two-band diagram and a DOS vs. \(E\) plot for the case when there is a band overlap. m (15)and (16), eq. m {\displaystyle \Omega _{n}(E)} 2 L a. Enumerating the states (2D . k {\displaystyle \mathbf {k} } 1739 0 obj <>stream Connect and share knowledge within a single location that is structured and easy to search. Density of states for the 2D k-space. The best answers are voted up and rise to the top, Not the answer you're looking for? The easiest way to do this is to consider a periodic boundary condition. The number of quantum states with energies between E and E + d E is d N t o t d E d E, which gives the density ( E) of states near energy E: (2.3.3) ( E) = d N t o t d E = 1 8 ( 4 3 [ 2 m E L 2 2 2] 3 / 2 3 2 E). ( It is significant that The general form of DOS of a system is given as, The scheme sketched so far only applies to monotonically rising and spherically symmetric dispersion relations. 0000002059 00000 n E . Through analysis of the charge density difference and density of states, the mechanism affecting the HER performance is explained at the electronic level. Its volume is, $$ Can archive.org's Wayback Machine ignore some query terms? I tried to calculate the effective density of states in the valence band Nv of Si using equation 24 and 25 in Sze's book Physics of Semiconductor Devices, third edition. 0000073968 00000 n { "00:_Front_Matter" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", Brillouin_Zones : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", Compton_Effect : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", Debye_Model_For_Specific_Heat : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", Density_of_States : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "Electron-Hole_Recombination" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", 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MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()" }, [ "article:topic", "license:ccbysa", "showtoc:no", "density of states" ], https://eng.libretexts.org/@app/auth/3/login?returnto=https%3A%2F%2Feng.libretexts.org%2FBookshelves%2FMaterials_Science%2FSupplemental_Modules_(Materials_Science)%2FElectronic_Properties%2FDensity_of_States, \( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}}}\) \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{#1}}} \)\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\) \(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\)\(\newcommand{\AA}{\unicode[.8,0]{x212B}}\), \[ \nu_s = \sqrt{\dfrac{Y}{\rho}}\nonumber\], \[ g(\omega)= \dfrac{L^2}{\pi} \dfrac{\omega}{{\nu_s}^2}\nonumber\], \[ g(\omega) = 3 \dfrac{V}{2\pi^2} \dfrac{\omega^2}{\nu_s^3}\nonumber\], (Bookshelves/Materials_Science/Supplemental_Modules_(Materials_Science)/Electronic_Properties/Density_of_States), /content/body/div[3]/p[27]/span, line 1, column 3, http://britneyspears.ac/physics/dos/dos.htm, status page at https://status.libretexts.org. Recovering from a blunder I made while emailing a professor. (a) Fig. we insert 20 of vacuum in the unit cell. ck5)x#i*jpu24*2%"N]|8@ lQB&y+mzM hj^e{.FMu- Ob!Ed2e!>KzTMG=!\y6@.]g-&:!q)/5\/ZA:}H};)Vkvp6-w|d]! ( Interesting systems are in general complex, for instance compounds, biomolecules, polymers, etc. 2 E a s m g E D = It is significant that the 2D density of states does not . 0000099689 00000 n Fluids, glasses and amorphous solids are examples of a symmetric system whose dispersion relations have a rotational symmetry. 0000065919 00000 n ) ( L n Let us consider the area of space as Therefore, the total number of modes in the area A k is given by. Can Martian regolith be easily melted with microwaves? We begin by observing our system as a free electron gas confined to points \(k\) contained within the surface. Kittel, Charles and Herbert Kroemer. Find an expression for the density of states (E). ( 0000005090 00000 n = E {\displaystyle f_{n}<10^{-8}} {\displaystyle (\Delta k)^{d}=({\tfrac {2\pi }{L}})^{d}} to {\displaystyle k\approx \pi /a} 2 which leads to \(\dfrac{dk}{dE}={(\dfrac{2 m^{\ast}E}{\hbar^2})}^{-1/2}\dfrac{m^{\ast}}{\hbar^2}\) now substitute the expressions obtained for \(dk\) and \(k^2\) in terms of \(E\) back into the expression for the number of states: \(\Rightarrow\frac{1}{{(2\pi)}^3}4\pi{(\dfrac{2 m^{\ast}}{\hbar^2})}^2{(\dfrac{2 m^{\ast}}{\hbar^2})}^{-1/2})E(E^{-1/2})dE\), \(\Rightarrow\frac{1}{{(2\pi)}^3}4\pi{(\dfrac{2 m^{\ast}E}{\hbar^2})}^{3/2})E^{1/2}dE\). ) The density of state for 2D is defined as the number of electronic or quantum We have now represented the electrons in a 3 dimensional \(k\)-space, similar to our representation of the elastic waves in \(q\)-space, except this time the shell in \(k\)-space has its surfaces defined by the energy contours \(E(k)=E\) and \(E(k)=E+dE\), thus the number of allowed \(k\) values within this shell gives the number of available states and when divided by the shell thickness, \(dE\), we obtain the function \(g(E)\)\(^{[2]}\). hbbd```b`` qd=fH `5`rXd2+@$wPi Dx IIf`@U20Rx@ Z2N 0000066746 00000 n ``e`Jbd@ A+GIg00IYN|S[8g Na|bu'@+N~]"!tgFGG`T l r9::P Py -R`W|NLL~LLLLL\L\.?2U1. By using Eqs. 91 0 obj <>stream 4, is used to find the probability that a fermion occupies a specific quantum state in a system at thermal equilibrium. For example, the kinetic energy of an electron in a Fermi gas is given by. $$, The volume of an infinitesimal spherical shell of thickness $dk$ is, $$ is temperature. 0000004547 00000 n 0000063017 00000 n {\displaystyle E_{0}} alone. {\displaystyle El[[UL6;?YkYx'Jb!OZX#bEzGm=Ny/*byp&'|T}Slm31Eu0uvO|ix=}/__9|O=z=*88xxpvgO'{|dO?//on ~|{fys~{ba? hbbd``b`N@4L@@u "9~Ha`bdIm U- We also acknowledge previous National Science Foundation support under grant numbers 1246120, 1525057, and 1413739. This procedure is done by differentiating the whole k-space volume For quantum wires, the DOS for certain energies actually becomes higher than the DOS for bulk semiconductors, and for quantum dots the electrons become quantized to certain energies. unit cell is the 2d volume per state in k-space.) To derive this equation we can consider that the next band is \(Eg\) ev below the minimum of the first band\(^{[1]}\). The allowed states are now found within the volume contained between \(k\) and \(k+dk\), see Figure \(\PageIndex{1}\). The density of states is defined by ( {\displaystyle d} 1 of the 4th part of the circle in K-space, By using eqns. Density of States (online) www.ecse.rpi.edu/~schubert/Course-ECSE-6968%20Quantum%20mechanics/Ch12%20Density%20of%20states.pdf. k m ( We learned k-space trajectories with N c = 16 shots and N s = 512 samples per shot (observation time T obs = 5.12 ms, raster time t = 10 s, dwell time t = 2 s). Density of States is shared under a CC BY-SA license and was authored, remixed, and/or curated by LibreTexts. 0 The referenced volume is the volume of k-space; the space enclosed by the constant energy surface of the system derived through a dispersion relation that relates E to k. An example of a 3-dimensional k-space is given in Fig. Comparison with State-of-the-Art Methods in 2D. ) Hence the differential hyper-volume in 1-dim is 2*dk. and finally, for the plasmonic disorder, this effect is much stronger for LDOS fluctuations as it can be observed as a strong near-field localization.[18]. The Kronig-Penney Model - Engineering Physics, Bloch's Theorem with proof - Engineering Physics. 54 0 obj <> endobj So could someone explain to me why the factor is $2dk$? As a crystal structure periodic table shows, there are many elements with a FCC crystal structure, like diamond, silicon and platinum and their Brillouin zones and dispersion relations have this 48-fold symmetry. We do this so that the electrons in our system are free to travel around the crystal without being influenced by the potential of atomic nuclei\(^{[3]}\). Even less familiar are carbon nanotubes, the quantum wire and Luttinger liquid with their 1-dimensional topologies. In magnetic resonance imaging (MRI), k-space is the 2D or 3D Fourier transform of the image measured. {\displaystyle E(k)} . ) is the spatial dimension of the considered system and 0000003644 00000 n 0000063841 00000 n ) the energy is, With the transformation 0000005340 00000 n i BoseEinstein statistics: The BoseEinstein probability distribution function is used to find the probability that a boson occupies a specific quantum state in a system at thermal equilibrium. {\displaystyle k_{\mathrm {B} }} hb```f`` ( 2 For a one-dimensional system with a wall, the sine waves give. 2 [9], Within the Wang and Landau scheme any previous knowledge of the density of states is required. The fig. = E For light it is usually measured by fluorescence methods, near-field scanning methods or by cathodoluminescence techniques. Kittel: Introduction to Solid State Physics, seventh edition (John Wiley,1996). for a particle in a box of dimension phonons and photons). HE*,vgy +sxhO.7;EpQ?~=Y)~t1,j}]v`2yW~.mzz[a)73'38ao9&9F,Ea/cg}k8/N$er=/.%c(&(H3BJjpBp0Q!%%0Xf#\Sf#6 K,f3Lb n3@:sg`eZ0 2.rX{ar[cc Thus the volume in k space per state is (2/L)3 and the number of states N with |k| < k . {\displaystyle D_{1D}(E)={\tfrac {1}{2\pi \hbar }}({\tfrac {2m}{E}})^{1/2}} Depending on the quantum mechanical system, the density of states can be calculated for electrons, photons, or phonons, and can be given as a function of either energy or the wave vector k. To convert between the DOS as a function of the energy and the DOS as a function of the wave vector, the system-specific energy dispersion relation between E and k must be known. 8 %PDF-1.5 % is sound velocity and The volume of an $n$-dimensional sphere of radius $k$, also called an "n-ball", is, $$ the number of electron states per unit volume per unit energy. =1rluh tc`H 0000067158 00000 n Since the energy of a free electron is entirely kinetic we can disregard the potential energy term and state that the energy, \(E = \dfrac{1}{2} mv^2\), Using De-Broglies particle-wave duality theory we can assume that the electron has wave-like properties and assign the electron a wave number \(k\): \(k=\frac{p}{\hbar}\), \(\hbar\) is the reduced Plancks constant: \(\hbar=\dfrac{h}{2\pi}\), \[k=\frac{p}{\hbar} \Rightarrow k=\frac{mv}{\hbar} \Rightarrow v=\frac{\hbar k}{m}\nonumber\]. shows that the density of the state is a step function with steps occurring at the energy of each The density of state for 2D is defined as the number of electronic or quantum states per unit energy range per unit area and is usually defined as . Compute the ground state density with a good k-point sampling Fix the density, and nd the states at the band structure/DOS k-points {\displaystyle d} the mass of the atoms, B How can we prove that the supernatural or paranormal doesn't exist? , the volume-related density of states for continuous energy levels is obtained in the limit In 2-dim the shell of constant E is 2*pikdk, and so on. 0000001853 00000 n [5][6][7][8] In nanostructured media the concept of local density of states (LDOS) is often more relevant than that of DOS, as the DOS varies considerably from point to point. E The density of states is defined by (2 ) / 2 2 (2 ) / ( ) 2 2 2 2 2 Lkdk L kdk L dkdk D d x y , using the linear dispersion relation, vk, 2 2 2 ( ) v L D , which is proportional to . The energy of this second band is: \(E_2(k) =E_g-\dfrac{\hbar^2k^2}{2m^{\ast}}\). ( Vsingle-state is the smallest unit in k-space and is required to hold a single electron. ( m . This expression is a kind of dispersion relation because it interrelates two wave properties and it is isotropic because only the length and not the direction of the wave vector appears in the expression. 2 Problem 5-4 ((Solution)) Density of states: There is one allowed state per (2 /L)2 in 2D k-space. Number of quantum states in range k to k+dk is 4k2.dk and the number of electrons in this range k to . < D L , {\displaystyle V} states per unit energy range per unit length and is usually denoted by, Where More detailed derivations are available.[2][3]. In this case, the LDOS can be much more enhanced and they are proportional with Purcell enhancements of the spontaneous emission. Leaving the relation: \( q =n\dfrac{2\pi}{L}\). It has written 1/8 th here since it already has somewhere included the contribution of Pi. 0000004990 00000 n (10-15), the modification factor is reduced by some criterion, for instance. k The points contained within the shell \(k\) and \(k+dk\) are the allowed values. = Now that we have seen the distribution of modes for waves in a continuous medium, we move to electrons. We can picture the allowed values from \(E =\dfrac{\hbar^2 k^2}{2 m^{\ast}}\) as a sphere near the origin with a radius \(k\) and thickness \(dk\). ( But this is just a particular case and the LDOS gives a wider description with a heterogeneous density of states through the system. Calculating the density of states for small structures shows that the distribution of electrons changes as dimensionality is reduced. n 0000069197 00000 n There is a large variety of systems and types of states for which DOS calculations can be done. Herein, it is shown that at high temperature the Gibbs free energies of 3D and 2D perovskites are very close, suggesting that 2D phases can be . 1 , are given by. . The product of the density of states and the probability distribution function is the number of occupied states per unit volume at a given energy for a system in thermal equilibrium. 0000075117 00000 n E quantized level. {\displaystyle \omega _{0}={\sqrt {k_{\rm {F}}/m}}} 2 has to be substituted into the expression of (that is, the total number of states with energy less than . the wave vector. 0000017288 00000 n where m is the electron mass. {\displaystyle m} If the particle be an electron, then there can be two electrons corresponding to the same . {\displaystyle E>E_{0}} E E Use the Fermi-Dirac distribution to extend the previous learning goal to T > 0. E In a local density of states the contribution of each state is weighted by the density of its wave function at the point. Streetman, Ben G. and Sanjay Banerjee. 0000023392 00000 n Design strategies of Pt-based electrocatalysts and tolerance strategies in fuel cells: a review. g ( E)2Dbecomes: As stated initially for the electron mass, m m*. ( f The calculation for DOS starts by counting the N allowed states at a certain k that are contained within [k, k + dk] inside the volume of the system. In spherically symmetric systems, the integrals of functions are one-dimensional because all variables in the calculation depend only on the radial parameter of the dispersion relation. Learn more about Stack Overflow the company, and our products. (7) Area (A) Area of the 4th part of the circle in K-space . The following are examples, using two common distribution functions, of how applying a distribution function to the density of states can give rise to physical properties. ) 0000061387 00000 n (10)and (11), eq. Equation(2) becomes: \(u = A^{i(q_x x + q_y y+q_z z)}\). Bulk properties such as specific heat, paramagnetic susceptibility, and other transport phenomena of conductive solids depend on this function. instead of Thanks for contributing an answer to Physics Stack Exchange! / On the other hand, an even number of electrons exactly fills a whole number of bands, leaving the rest empty. (b) Internal energy {\displaystyle N(E)\delta E} 0000068788 00000 n Cd'k!Ay!|Uxc*0B,C;#2d)`d3/Jo~6JDQe,T>kAS+NvD MT)zrz(^\ly=nw^[M[yEyWg[`X eb&)}N?MMKr\zJI93Qv%p+wE)T*vvy MP .5 endstream endobj 172 0 obj 554 endobj 156 0 obj << /Type /Page /Parent 147 0 R /Resources 157 0 R /Contents 161 0 R /Rotate 90 /MediaBox [ 0 0 612 792 ] /CropBox [ 36 36 576 756 ] >> endobj 157 0 obj << /ProcSet [ /PDF /Text ] /Font << /TT2 159 0 R /TT4 163 0 R /TT6 165 0 R >> /ExtGState << /GS1 167 0 R >> /ColorSpace << /Cs6 158 0 R >> >> endobj 158 0 obj [ /ICCBased 166 0 R ] endobj 159 0 obj << /Type /Font /Subtype /TrueType /FirstChar 32 /LastChar 121 /Widths [ 278 0 0 0 0 0 0 0 0 0 0 0 0 0 278 0 0 556 0 0 556 556 556 0 0 0 0 0 0 0 0 0 0 667 0 722 0 667 0 778 0 278 0 0 0 0 0 0 667 0 722 0 611 0 0 0 0 0 0 0 0 0 0 0 0 556 0 500 0 556 278 556 556 222 0 0 222 0 556 556 556 0 333 500 278 556 0 0 0 500 ] /Encoding /WinAnsiEncoding /BaseFont /AEKMFE+Arial /FontDescriptor 160 0 R >> endobj 160 0 obj << /Type /FontDescriptor /Ascent 905 /CapHeight 718 /Descent -211 /Flags 32 /FontBBox [ -665 -325 2000 1006 ] /FontName /AEKMFE+Arial /ItalicAngle 0 /StemV 94 /FontFile2 168 0 R >> endobj 161 0 obj << /Length 448 /Filter /FlateDecode >> stream The simulation finishes when the modification factor is less than a certain threshold, for instance To learn more, see our tips on writing great answers. for {\displaystyle D(E)} j Then he postulates that allowed states are occupied for $|\boldsymbol {k}| \leq k_F$. This result is fortunate, since many materials of practical interest, such as steel and silicon, have high symmetry. Wenlei Luo a, Yitian Jiang b, Mengwei Wang b, Dan Lu b, Xiaohui Sun b and Huahui Zhang * b a National Innovation Institute of Defense Technology, Academy of Military Science, Beijing 100071, China b State Key Laboratory of Space Power-sources Technology, Shanghai Institute of Space Power-Sources .