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Influence of impurity on isotope coefficient of superconductors

P.Udomsamuthirun (1)

(1) Department of Physics, Faculty of Science, Srinakharinwirot University, Bangkok 10110,Thailand.E-mail: udomsamut55@https://www.sodocs.net/doc/20673067.html,

Keywords: isotope coefficient, impurity , impure superconductors

PACS : 74.20.Fg, 74.70.Ad.

Abstract

We study influence of anisotropy of the impurity potential on isotope coefficient of impure superconductor in Born limit .The isotope coefficient of nonmagnetic and magnetic impurities including an angle-resoled Fermi surface density of state, anisotropy order parameter and impure potential that we make the assumption that the form of them are in the harmonic form are investigated . Our numerical calculation of isotope coefficient can fit well with experimental data of high-T c superconductors.

1.Introduction

One of the important parameters in the experiment and theory of

superconductor is the isotope coefficient α defined by αα?M T c where c T is the superconducting transition temperature and M is the isotope mass of the element. In the BCS theory it has been predicted that 5.0=α suggesting phonon-mediated pairing in superconductors that verified experimentally for conventional

superconductors. In high temperature superconductors the isotope effect shows complex behavior : some measurements[1,2] have shown near zero isotope effect coefficient and some measurements[3,4] have shown the values greater than 0.5 .To explain the unusual isotope effect in high temperature superconductors, many models have been proposed such as the van Hove singularity[5-7], anharmonic phonon[8,9],pairing-breaking effect[10] and pseudogap[11,12], nonadiabaticity[13] and impurity effect[14,15].

The results of impurities that introduced into the superconductors are found in a change of the superconducting transition temperature and the isotope coefficient.They modify the quasi-particle spectrum, interaction parameters and induce pair breaking in superconducting state. The non-magnetic impurities have little effect on transition temperature in s-wave superconductors [16] but they exhibit a strong pair breaking effect in the high temperature superconductors [17] . Shi and Li[18] study the influence of nonmagnetic impurities on transition temperature of the layered superconductors by using an superconducting-normal layer model with d-wave pairing in the superconducting layer. Mierzynska and Wysokinski[15] study the effect of in-plane and out-of-plane impurities on isotope effect coefficient in layered superconductors. Haran and Nagi [14,19,20,21] proposed a theory of nonmagnetic and magnetic impurities in an anisotropic superconductors including the effect of anisotropic (momentum-dependent) impurities scattering. Openov , Semenihin and Kishore [22] study the effect of impurities, both magnetic and nonmagnetic, on the isotope effect coefficient in high temperature superconductors in the framework of Abrikosov-Gorkov approach. But Openov et al.[22] do not include the effect of momentum-dependent of scattering potential. Although Haran and Nagi consider the effect of anisotropic (momentum-dependent) impurities scattering, they do not consider the form of anisotropic function and apply to experimental data.

The purpose of this work is to apply the model of nonmagnetic and magnetic impurity of Haran and Nagi to investigate the influence of an angle-resoled Fermi surface density of state, anisotropic order parameter and anisotropic impure potential on isotope effect coefficient of superconductors. To apply this model, we make the assumption that an angle-resoled Fermi surface density of state and all of potentials of this model are in the harmonic form. Finally we show the numerical calculation of isotope coefficient that fit well with experimental data of high-T c superconductors.

2.The isotope coefficient

Haran and Nagi [14,19,20,21] consider the problem of nonmagnetic and

magnetic impurities in an anisotropic superconductor for the case of anisotropic (momentum-dependent) impurity scattering in weak-coupling approximation. They consider an anisotropic superconductor with randomly distributing impurities,treating the electron-impurity scattering within second Born approximation, and

neglecting the impurity-impurity interaction. The normal and anomalous temperature Green’s function averaged over the impurity position read

222)

(~~~),(k i k G k k ?+++?=ξωξωω(1)222)(~~)(~),(k k k F k ?++?=ξωω(2)where the renormalized Matsubara frequency )(~k ω

and the renormalized order parameter )(~k ? are given by

332)2(),(),()(~πωωωk d k G k k u in k i ′′′+=∫(3)

32)2(),(),()()(~πωk d k F k k u in k k i ′′′+?=?∫(4)Here )

12(+=n T πω,T is temperature ,n is integer number. k ξ is the quasiparticle energy, i n is impurity concentration, ),(k k u ′ is a momentum-dependent impurity potential, and )(k ? is the orbital part of the singlet superconducting order parameter defined as )()(k e k ?=? where )(k e is a real basis function and 12>=<.. denotes the average value over the Fermi surface that

∫∫∫→FS k k d k n dS N k d ξπ)()2(0

33 and (..))(..∫=>

resolved Fermi surface density of state.0N is the overall density of state at Fermi surface. The momentum-dependent impurity potential is

v v ?′+′=′S k k J k k v k k u ),(),(),( (5)where S v is a classical spin of impurity and σr is the electron spin density and

assumed a separable form of scattering probabilities.

)

()(),(21202k f k f v v k k v ′+=′(6))()(),(21202k g k g J J k k J ′+=′(7)

where )(),(1010J J v v are isotropic (anisotropic) scattering amplitudes for non-

magnetic and magnetic potential. )(),(k g k f are the momentum-dependent anisotropy function in the nonmagnetic and magnetic scattering channel and let )()(k f k g ±=The averaged over the Fermi surface of )(k f and )(k g vanish ,

0)()(>=>=<=

2101v N n i π=Γ,)1(2000+=S S J N n G i π, )1(2101+=S S J N n G i π are scattering rate of isotropic non-magnetic, anisotropic non-magnetic, isotropic magnetic, and anisotropic magnetic channel ,respectively.

Substitution Eqs.(5-7) into Eqs.(3,4) , we can get the transition temperature as [14]

])22(2121[])2(2121)[1()ln(0200220???????

?+Ψ???????Ψ><+????????+Γ+Ψ???????Ψ><+C T G G ef π(8)

Here T C (T C0 ) is the critical temperature of impure(clean) superconductors.

After some calculation, the isotope effect coefficient of impure superconductors in the Born approximation is found as[14]

?????

?+Ψ′????????>

Here α(0α) is the isotope coefficient of impure(clean) superconductors.

Eq.(8) and Eq.(9) are the general equation of T C and isotope coefficient αthat contain both nonmagnetic and magnetic impurities, anisotropy of order

parameter, and anisotropy of scattering potential of impurities. They can be reduced to nonmagnetic impurity case [19] by set 010==G G . In the work of Opennov,Semenihim and Kishore [22], they also study the critical temperature and isotope coefficient of anisotropy superconductor containing both nonmagnetic and magnetic impurities. The coefficient FS

FS p p >?<>?

)()(1χ of their model can be reduced to 21>

02G +Γ=τ and the electron relaxation time due to exchange scattering by magnetic impurities 02G ex m =

τ . The second term on the right-side of Eq.(8) do not appear in Opennov,Semenihim and Kishore’s T C equation because they do not consider the momentum-dependent of impurity scattering potential .

3.Calculation and discussion

To show the numerical calculation of this model , we must know the form of function )(k e , )(k f and )(k n . Firstly, we consider the form of order parameters .The order parameters of s-wave superconductors are corresponded to 0)(?=?k that give 1)(=k e . And for the high temperature superconductors , they are most likely layered d x 2-y 2-wave pairing symmetry . In d x 2-y 2-wave superconductor the order parameters are corresponded to φ2cos )(0?=?k ,that give φ2cos )(≡k e . Secondly,we consider the form of the anisotropic impurity scattering potential. The possible of the anisotropy impurity scattering potential was suggested in a discussion of the

irradiation data by Giapintzakis et al.[23]. Haran and Nagi[21] take the function )(k f in the scattering potential proportional to the harmonic functions which in a polar

notation )sin(φn and )cos(φn , where n is an integer number. They use φ2sin )(≡k f ,φ3sin ,φ2cos ,and higher-order harmonics to study the critical temperature of impure superconductor in Born limit. Thirdly, we consider the form of an angle-resoled Fermi surface density of state. The model of an angle-resoled Fermi surface density of state,

1)(≡φn , φφ2cos )(≡n and φφ4cos )(≡n ,are used by Mierzynska and Wysokinski

[15] in the study of the effect of in-plane and out-of-plane impurities on isotope effect coefficient in layered superconductors.

As corresponding to that mention in above paragraph, we make the

assumption that the form of order parameters , the form of the anisotropic impurity scattering potential and the form of an angle-resoled Fermi surface density of state ()(k e , )(k f and )(k n ) are in the harmonic form . The constraint conditions are 12>===

k k n dS . Within these constraints

and our assumptions, we can show the values of >

Substitution the values of >=<>==<>==<>=

4.Conclusion

We have applied the model of nonmagnetic and magnetic impurity of Haran and Nagi to investigate the influence of an angle-resoled Fermi surface density of state, anisotropy order parameter and impure potential on isotope effect coefficient of impure superconductors. We make the assumption that the form of order parameters ,the form of the anisotropic impurity scattering potential and the form of an angle-resoled Fermi surface density of state ()(k e , )(k f and )(k n ) are in the harmonic form . Our numerical calculation can fit well with the experimental data of high-T c superconductors. We find that there is an effect of nonmagnetic impurity on isotope coefficient in d-wave superconductors but no effect on s-wave superconductors and an effect of magnetic impurity on isotope coefficient are found both in s-wave and d-wave superconductors .

Acknowledgement

The author would like to thank Professor Dr.Suthat Yoksan for the useful discussion and also thank Thai Research Fund and Office of Higher Education Commission for the financial support.

Reference

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00.20.4

0.60.8 1.0

T c /T c0α/α0Figure. 1 .We show the numerical calculation of isotope coefficient fitted to experimental data [3,4,24].

情态动词表推测用法总结及专项练习

情态动词表推测用法总结及专项练习 1.can / could用于表推测的用法 (1) 从使用句型上看，can 通常只用于否定句或疑问句，一般不用于肯定句，而could 可用于肯定句、否定句和疑问句。两者没有时间上的差别，只是could 比can 更委婉，更不确定。如：It can’t [couldn’t] be true. 那不可能是真的。 What can [could] they be doing? 他们会在干什么呢? We could go there this summer. 今年夏天我们可能要去那儿。注：can 有时也用于肯定句中表示推测，主要用于表示理论上的可能性(即从理论上看是可能的，但实际未必会发生)，或表示“有时”之意。如： Even experienced teachers can make mistakes. 即使是有经验的教师也可能出错。 She can be very unpleasant. 她有时很令人讨厌。 (2) 从时间关系看，对现在或将来情况作推测，后接动词原形；对正在进行的情况作推测，后接be doing 结构；对过去情况作推测，后接动词完成式。如： He could have gone home. 他可能已经回家了。 He can’t [couldn’t] have understood. 他不可能理解了。

Why does he know this? Can [Could] someone have told him about it? 他怎么知道? 会是哪个人告诉他了吗? (3) “could+完成式”除表示对过去的推测外，还有以下重要用法： ①表示过去没有实现的可能性，常译为“本来可以”。如： I could have lent you the money．Why didn’t you ask me? 我本来可以借这笔钱给你的。你为什么不向我提出? ②用来委婉地责备某人过去应该做某事而没有去做，常译为“本来应该”。如： You could have helped him. 你本来应该帮助他的。 ③表示“差点儿就要”。如： I could have died laughing. 我差点儿笑死了。 2. may / might用于表推测的用法表示推测，两者都可用，只是might 比may 语气更不确定，表示的可能性更小。 (1) 在句型使用方面：两者均可用于肯定句和否定句，但用于疑问句时，may通常不用于句首，但可用于疑问句的句中（如特殊疑问句等），而might尽管可以用于疑问句的句首，但不算普通，通常会改用其他句式（如用could等）。如： He may [might] know the answer. 他可能知道答案。

on the contrary的解析

On the contrary Onthecontrary, I have not yet begun. 正好相反，我还没有开始。 https://www.sodocs.net/doc/20673067.html, Onthecontrary, the instructions have been damaged. 反之，则说明已经损坏。 https://www.sodocs.net/doc/20673067.html, Onthecontrary, I understand all too well. 恰恰相反，我很清楚 https://www.sodocs.net/doc/20673067.html, Onthecontrary, I think this is good. ⑴我反而觉得这是好事。 https://www.sodocs.net/doc/20673067.html, Onthecontrary, I have tons of things to do 正相反，我有一大堆事要做 Provided by jukuu Is likely onthecontrary I in works for you 反倒像是我在为你们工作 https://www.sodocs.net/doc/20673067.html, Onthecontrary, or to buy the first good. 反之还是先买的好。 https://www.sodocs.net/doc/20673067.html, Onthecontrary, it is typically american. 相反，这正是典型的美国风格。 222.35.143.196 Onthecontrary, very exciting.

恰恰相反，非常刺激。 https://www.sodocs.net/doc/20673067.html, But onthecontrary, lazy. 却恰恰相反，懒洋洋的。 https://www.sodocs.net/doc/20673067.html, Onthecontrary, I hate it! 恰恰相反，我不喜欢！ https://www.sodocs.net/doc/20673067.html, Onthecontrary, the club gathers every month. 相反，俱乐部每个月都聚会。 https://www.sodocs.net/doc/20673067.html, Onthecontrary, I'm going to work harder. 我反而将更努力工作。 https://www.sodocs.net/doc/20673067.html, Onthecontrary, his demeanor is easy and nonchalant. 相反，他的举止轻松而无动于衷。 https://www.sodocs.net/doc/20673067.html, Too much nutrition onthecontrary can not be absorbed through skin. 太过营养了反而皮肤吸收不了. https://www.sodocs.net/doc/20673067.html, Onthecontrary, I would wish for it no other way. 正相反，我正希望这样 Provided by jukuu Onthecontrary most likely pathological. 反之很有可能是病理性的。 https://www.sodocs.net/doc/20673067.html, Onthecontrary, it will appear clumsy. 反之，就会显得粗笨。 https://www.sodocs.net/doc/20673067.html,

不及物动词归纳

1.只是不及物的： faint,hesitate,lie,occur,pause,rain,remain,sleep,sneeze. 2.常见的及物，不及物的： answer,ask,begin,borrow,choose,climb,dance,eat,enter,fail,f ill,grow,help,hurry,jump,know,leave,marry,meet,obey,pull,re ad,see,sell,touch,wash,watch,win,write 3.及物不及物意义变化的lift.升高beat vi.跳动vt. 敲、打; grow vi.生长vt. 种植play vi.玩耍vt. 打（牌、球）,演奏smell vi.发出（气味）vt. 嗅ring vi.（电话、铃）响vt.打电话speak vi.讲话vt. 说（语言）hang vi. 悬挂vt. 绞死operate vi.动手术vt. 操作 4.意义不变的 start, answer, sing, close, consider, insist, read, learn, prepare, pay, hurt, improve....

live, go, work, listen, look, come, die, belong, fa ll, exist, rise, arrive, sit, sail, hurry, fail, su cceed. agree... 不及物动词 agree, go, work, listen, look, come, die, belong, f all, exist, rise, arrive, sit, sail, hurry, fail, s ucceed、beat、buy, catch, invent, found, like, obs erve, offer, prevent, promise, raise, find, forget, receive, regard, see, say, seat, supply, select, s uppose, show, make, take, tell 6.不及物动词短语 down (stop functioning 坏了，不好使了) That old Jeep had a tendency to break down just w hen I needed it the most. on (become popular 出名) Popular songs seem to catch on in California first and then spread eastward. 3. come back ( return to a place 返回)

英语造句

一般过去式时间状语：yesterday just now (刚刚) the day before three days ag0 a week ago in 1880 last month last year 1. I was in the classroom yesterday. I was not in the classroom yesterday. Were you in the classroom yesterday. 2. They went to see the film the day before. Did they go to see the film the day before. They did go to see the film the day before. 3. The man beat his wife yesterday. The man didn’t beat his wife yesterday. 4. I was a high student three years ago. 5. She became a teacher in 2009. 6. They began to study english a week ago 7. My mother brought a book from Canada last year. 8.My parents build a house to me four years ago . 9.He was husband ago. She was a cooker last mouth. My father was in the Xinjiang half a year ago. 10.My grandfather was a famer six years ago. 11.He burned in 1991

lie-lie-lay三个动词的区别

动词lie，lie，lay的区别 1)lie用作规则动词(lie，lied，lied)时意为“说谎”。如： ①He wasn't telling the truth. He lied again/He was lying. 他没讲实话，他又撒谎了。/他在撒谎。 ②She lied to us about her job. 她就她的工作对我们撒了谎。 ③Your watch must be lying. 你的表肯定不准。【注意】表示这一意思时，lie可用作名词。我们一般说tell a lie或tell lies，而不说say/talk/speak a lie(或lies)。再如：Lies cannot cover up facts.(谎言掩盖不了事实。)如要说“我痛恨说谎话”，可以说I hate lying或I hate telling lies，一般不说I hate lie或I hate to lie。hate to lie尽管句法正确，但似乎含有“虽然不愿，但仍不得不说谎”之意。

2)lie用作不规则动词(lie，lay，lain)时，表示“躺”，“(东西)平放”，“位于”等意。如： ①He's still lying in bed. 他还躺在床上。 ②He felt tired, so he went and lay down for a rest. 他感到疲劳，所以去躺下休息了。 ③His books lay open on the desk when I went in. 我进去时，发现他的书平摊在书桌上。 ④Snow lay thick on the fields. 田野里覆盖着厚厚的一层雪。 ⑤Don't leave your things lying about. 别把东西四处乱丢。

英语情态动词的用法大全附解析

英语情态动词的用法大全附解析一、初中英语情态动词 1．—We've got everything ready for the picnic. —Do you mean I __________ bring anything with me? A. can't B. mustn't C. couldn't D. needn't 【答案】 D 【解析】【分析】句意：—我们已经为野餐准备好了一切。—你的意思是我不必带任何东西吗？A. can't 不能；B. mustn't 表示禁止，一定不要；C. couldn't不能，表示过去时态；D. needn't不必。结合句意，故选D。【点评】本题考查情态动词的用法。 2．—How amazing this robot is! —Wow, it has video cameras in its eyes, so it “see” and interact with pe ople. A. may B. can C. must D. should 【答案】 B 【解析】【分析】句意：—这个机器人多么惊人啊！—哇，在它的眼睛里有摄像机，因此它能看见和人打交道。A. may 可以，可能；表示许可，B. can 能；表示能力，C. must 必须；D. should应该；根据it has video cameras in its eyes,可知是有能力看见，故选B。【点评】考查情态动词辨析。熟记情态动词的含义和用法。 3．—Where is George? —He _______ be here just now. His coffee is still warm. A. need B. can't C. must D. shouldn't 【答案】 C 【解析】【分析】句意：——乔治在哪里？——他刚才一定在这里，他的咖啡还是热的。A. need需要； B. can't 不能，不可能（表示推测）； C. must 必须，一定（表示推测）； D. shouldn't不应该；根据His coffee is still warm.可知表示肯定推测一定在这儿；故答案为C。【点评】考查情态动词。掌握情态动词表推测时的意义和用法。 4．— Sorry, I forgot to take money with me. Maybe I can't buy the book you like. — Mum, you ______ worry about it. We can pay by Alipay （支付宝）. A. can't B. needn't C. mustn't D. shouldn't 【答案】 B 【解析】【分析】句意：——对不起，我忘记带钱了。也许我不能买你喜欢的书。——妈妈，你不用担心。我们可以用支付宝支付。A. can't不能，指不允许或否定推测；B. needn't 不需，指没必要；C. mustn't不能，表禁止；D. shouldn't不应该，表建议。根据句意语境，本句是说妈妈不需要担心，故答案为B。【点评】考查情态动词。理解句意并掌握情态动词的意义和用法区别。

英语情态动词用法总结(完整)

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