From b72ed1c49444ab0b4c0d7b9073bced7ffa220488 Mon Sep 17 00:00:00 2001 From: Kees van Kempen Date: Wed, 18 May 2022 16:41:41 +0200 Subject: [PATCH] ass5: Is this 17? --- ...es-for-printing-slide-13-gap-parameter.pdf | Bin 0 -> 4116 bytes ...es-for-printing-slide-13-gap-parameter.svg | 148 ++++++++++++++++++ references.bib | 2 +- superconductivity_assignment5_kvkempen.tex | 63 +++++++- 4 files changed, 210 insertions(+), 3 deletions(-) create mode 100755 Lecture-7-slides-for-printing-slide-13-gap-parameter.pdf create mode 100755 Lecture-7-slides-for-printing-slide-13-gap-parameter.svg diff --git a/Lecture-7-slides-for-printing-slide-13-gap-parameter.pdf b/Lecture-7-slides-for-printing-slide-13-gap-parameter.pdf new file mode 100755 index 0000000000000000000000000000000000000000..f99fa411cf147284333346b8912dc66e13a36fa3 GIT binary patch literal 4116 zcmb7Hc{r478@JS?kwTW_;7!@G%s%#|>=827P{tUHG^3dzOGjjBGnNPmHIW>$9U@WG z8B4a(W=oP%vK+=TgnaKvoz6Mmb$!?OUhnhX@AG?>`@VnA^T+*Duro1JM`&Q63U?+- zmY`?=4p6-Op?Z1%%$i8{rTYOG5VD5?001*11<;99&<5b?L=z%`;!T7a7(i)sDiMDi z8j^)_fb;D4Fb42`p83AsWPcxgK~h$mqd?4;^APB0BlL^74^(5(J<3Rc=WWO4WhjyC zy}lOw<9g+~;#$xdTAMq}?Vx^yv=)SE2;Ce<2Lk}~^tb@{!0>Re1Hg>;0SEv_z=K~f zEO6l47v^H?E+FD}@nGPsCDnHI?PObGoBTLN@y5b=A@7@TE73?>nX_U8M!;CjCHL>eWC zN+8kzE_e1n<;hK*%jr+YtMwzS|8~4wcE5$UP9oP03>V}@Ut`IPLOs5Q?EjezE(@CL z>Higr=J!~B+?QCF+5~k=k=ZfTKC^oV@5>#HjEO1tl;1-4oQHPosNJz_ASL2Dpnp<= zi7-3lk=Wi+(0+Ty%EG*|-f(qjVP$Z5dZymkpsnS@g6+9xxryQWM@LtKSt!|60TtQP zIuY8HZ?As6Z?NCrcXDMRt-m)RF*hDHcc70QIvLfbU%T)(MwR)soE z<0`EyYiV%uws&WMCxk_>GYnpu`?!MeI%1gKyRm36(ZMgTfismcKVRFQwtA8}@iB1r z!)uGu(olf(f%c504^QvkA}A}^9_KwqSvzl&ISriBrn=aOk<?}u zDkZ92z52v40NPNiJ|6VtRw-`KeGz!!ZM;36I-PuD;c0_ztdc=%?}uN1>1@{jJu5NX zQBv*+FT1NCLKFw8I3D( zI@gw&=qXK}sE|kJy`MJD4-s@a%T&I6)JkTo{Y1cWgm@BpZttm*8x?b4P*fg-i7+KYdw)%R8@d4HS)n#gTf|?LZ{Jo&`=iKd5FXNodnl5PY z_Un(L_`C|>SCITW3oV!;nh7{v@iv-jiKOPEfym^;1?^F6|0Q+BovQW9`4>ke8a{8i zdRbDKx(OdSkz=LtF}sdYf13GRplWZnlCrsMrSvgT-`<6}Yx8%6Z0oF@<;+zN1YiA{ zX3B{zcB!gVdCPer6DJ7UcK6Q*B@3$|S##o-8*OVi!+||^s^NzipOuG<)Pe{8q;b@| z({55@wqCI@dR?1yJOAjXjah`62Tj{$ZZYd1c13+HpWQ?7k!nJ!zUFR)cG(?~%qZLJ z^Aj80RcvNqLP9Wxxsw1Sm%J&DNK+S%zLtyd^XnP9D;;d8iOgo*tVzI?7oNTOikXW| zkE6+3JSkDk>fg0+fM06W}tgZ=URX;-*&gf7{L^s zV@mVZFXXpw(K>WkP3@CU0u&bRQfKq5r@JC|utmCS$Up{iUF+uO&Hne;jb1^~I`T6Y z8T)1K&PqTx#)-_Q$$M&qos$04cm9pHlaq0({m8)0&9aUei2`Cb;`O`&S?}x~3>h`M zEzLi9F^u5b8F488fz#}U>m+2cCTk0Wuh`RdB8MgR_{g4G`TGu11rkCW)`56vYh~Lh zkN56!*DJbL65b}=X$_tkzKdY*GZ*@7bUsr3SZ_=&!L=dxa01?CQA$QA%XA5sk*HOZ zW|^W?&nDi$isq3BwV(E-qK?GghoIq~p!Xxy+WEC`#A< zQTY^ItML00!n5~9XqRM7Olhlr1=~8vQZr|ccRZME*QH<_$}vJI@T6}&dcb`lzhD7eOK9K@k%Ug7c*DOplH0B&XF)Xv>`b57gy(F&Uc3znPL2f zIdoU(VPy96L(^X>t4F?6%06;^I$Mb_638f!V$jW z)vt{aGk)ymSafo7J5$i>7*J`L5(a0D_-5CsLz3?b<&&DFSsyS0V_T+88_ey|Y{$!;q$1m|2UJe~Fw@ z3x9OyM`;P#SuG!C}PUk5e3s&6(2pCIxL$ZJQss!q6?SAQDv6B0Z4mZv<5;;r!O}=7_iHX=){{CG%mi;?~ zjT?L^1e;qn7VJLJnT37t2=H#chb<-KGUhI#z&$FFpbw8R;j*_U)*d&cTZ#&@BJ87pI{LsNKwa~7T)9-*B$M= zToO!sbBZk^k|@%T_XuoiIYGXxjLK7?!X#uap1fUh>SkH2hV1WF-qEZthoVzo^I}v~ zUf=8rU+xLgr62_}$9F&{$1zc2WkZ8|Qf0O`sG5vjfrN2*ZTUlrOI3wZP4}2Si#HuK zUGjWYA?rW_5OPc-#=eMT(ex$FjFB55il~&?fx79(i(yZ7TBE0{My9o8>&2PXa9!OE z^xOQqE6S8;%MsRNb9Xn{75g4y5JC3%^&Ld+3AIwIxRac5S>^RYK~VsBU??zR-yyOKfe{0F<$<) z7`Hs7L^0~_9ws=`-sr zKvl&!?gsCXM=y1vYQoD^{hk-aH9I7A&(A3QDrgwH%J*9>o&9fh!L4-H)ddoZ!F;d% zxpnEcns}{}{Xg}^UmAj|4!+mce`yHX+NM?{ZyKl-xH7~3hb{=>N{XL~bM2otu%lAE zg9xD5QYPR@R0@F5Kxk^90cAfrJ&>jY14Ryq`cmo zuQw3m?uFV1j)6Y74g0_RaAN@HX@VPKLFY*AHGKwK3ztM8n{f4VaFw4ZLU7gA-$^#c)A0cm-?i+yqK&pL)BZY`5Gv6J3W_uY6pVcR0l*7@ z!T>(NI)*~S!NcJ`0QoxxM`HhtA^r;nM}WfhA8Qe4@IL$lLu-Q?@b4H3g#uIhCkCpg ze`09tzcEb=c%{FurBm^w03vlw+TloHL~g!Zt!7W5fY*%6+c)uPN%o-t+^e>x3u$yb SmA*z0N()?|pkQii2K^7Kxty>7 literal 0 HcmV?d00001 diff --git a/Lecture-7-slides-for-printing-slide-13-gap-parameter.svg b/Lecture-7-slides-for-printing-slide-13-gap-parameter.svg new file mode 100755 index 0000000..6cdd5c4 --- /dev/null +++ b/Lecture-7-slides-for-printing-slide-13-gap-parameter.svg @@ -0,0 +1,148 @@ + +image/svg+xmlTTc= 1.76kBTc diff --git a/references.bib b/references.bib index ea7ac07..a32fa26 100755 --- a/references.bib +++ b/references.bib @@ -111,7 +111,7 @@ Results from previous neutron measurements are found to be consistent with the X @book{annett, edition = {1}, - series = {{OXFORD} {MASTER} {SERIES} {IN} {CONDENSED} {MATTER} {PHYSICS}}, + series = {Oxford Master Series in Condensed Matter Physics}, title = {Superconductivity, {Superfluids}, and {Condensates}}, volume = {5}, isbn = {978-0-19-850756-7 0-19-850756-9}, diff --git a/superconductivity_assignment5_kvkempen.tex b/superconductivity_assignment5_kvkempen.tex index f0cfabb..a1d0979 100755 --- a/superconductivity_assignment5_kvkempen.tex +++ b/superconductivity_assignment5_kvkempen.tex @@ -58,12 +58,71 @@ with $V$ Cooper's approximate potential and $g(\epsilon_F)$ the density of state A thorough discussion can be found in Annett's book \cite[chapter 6]{annett} and in the slides of week 6 of this course. The binding energy of the Cooper pairs (i.e. the energy gain of forming these pairs) is \[ - -E = 2\hbar\omega_De^{-1/\lambda}. + -E = 2\hbar\omega_De^{-1/\lambda} =: \Delta_0, \] +which is also called the gap parameter $\Delta_0$ at zero temperature for BCS. In the weak coupling limit of the BCS theory, the case we have considered so far, it is assumed that $\lambda << 1$. +It should be noted that this weak limit also means that the gap is smaller than the thermal energy of the highest excited energy phonon, which corresponds to the Debye temperature +\[ + \Delta < k_B\Theta_D. +\] It is when this assumption breaks down, BCS does not work and we find an upper limit to the critical temperature $T_c$. -We will look at a way to express the critical temperature in terms we can derive, and than look at the values that maximize this critical temperature whilst still following BCS theory. +We will look at a way to express the critical temperature in terms we can derive, and then look at the values that maximize this critical temperature whilst still following BCS theory. +From the derivation of the BCS coherent state, this gap parameter at finite temperature is found. +There is a temperature dependence $\Delta(T)$ as in figure \ref{fig:gap-T}. +\begin{figure} + \centering + \includegraphics[width=.4\textwidth]{Lecture-7-slides-for-printing-slide-13-gap-parameter.pdf} + \caption{By taking the gap parameter to zero, we find the critical temperature. Figure from the slides of lecture 7.} + \label{fig:gap-T} +\end{figure} +For larger temperatures, thermal energy is increased, and less energy is required to break up Cooper pairs, thus degrading the superconductivity. +This puts a limit $T_c$. + +We will mostly follow the derivation by Waldram \cite[paragraph 7.9, mostly p.128--130]{waldram}. +The superconducting state breaks down at high temperature, at which also $\Delta$ vanishes so that the gap parameter is a good order parameter for the state. + +Let's consider the gap parameter +\[ + \Delta_{\vec{k}} = -\sum_{\vec{k'}}(1-2f_{\vec{k'}})u_{\vec{k'}}v_{\vec{k}}V_{\vec{k'}\vec{k}}, +\] +with $u$ and $v$ occupation functions for the BCS state, $f$ the Fermi occupation number, and $V$ the potential between the states. +Minimizing $\Delta_{\vec{k}}$ and taking that $V_{\vec{k'}\vec{k}} = -V$ is constant gives us a self-consistent relation for the gap parameter. +We also recognize that the states that we sum over all all those states such that they have smaller energy than the highest excited phonon. +\[ + \Delta_{\vec{k}} = V\sum_{\epsilon_{\vec{k'}}}(1-2f_{\vec{k'}})\frac{\Delta_{\vec{k'}}}{2E_{\vec{k'}}}. +\] +Now the right-hand side is independent of $\vec{k}$ but does contain $\Delta_{\vec{k'}}$. +We can thus conclude that the gap parameter should be constant over all states $\vec{k}$! +That means we can divide both sides by it, giving us +\[ + 1 = V\sum_{\epsilon_{\vec{k'}}}(1-2f_{\vec{k'}})\frac{1}{2E_{\vec{k'}}}. +\] +Converting the equation to an integral, and substituting in $f(E) = [\exp{(E/(k_BT))}+1]^{-1}$ and $E = \sqrt{\epsilon^2 + \Delta(T)^2}$ yields +\[ + 1 = 2g(\epsilon_F)V\int_0^{k_B\Theta_D}\frac{1-2[\exp{(E/(k_BT))}+1]^{-1}}{2\sqrt{\epsilon^2 + \Delta(T)^2}} \textup{d}\epsilon. +\] + +I believe Waldram that one could find that +\[ + T_c = 1.14\Theta_D\exp{(-1/(g(\epsilon_F)V))} = 1.14\Theta_D\exp{(-1/(\lambda)} +\] +from this nice equation. + +As limiting value, we take $\lambda = 0.3$, as was posed as a reasonable limit for the weak coupling by Alix in lecture 7, +although Waldram \cite{waldram} thinks it is more like $\lambda \approx 0.4$. + +For metals, Waldram thinks $\Theta_D \leq \SI{300}{\kelvin}$ is a good limit. +This leads to our final maximum +\[ + T_c \leq 1.14 \cdot 300 \cdot \exp{(-1/0.4)} \approx \SI{28}{\kelvin}. + %https://www.wolframalpha.com/input?i=1.14*300*e%5E%28-1%2F.4%29 +\] +(Using $\lambda = 0.3$ yields an even lower $T_c \leq \SI{12}{\kelvin}$.) + +For larger $T_c$ values, larger binding of Cooper pairs would be needed to overcome the thermal energy. +This means our assumption of weak coupling breaks down, making most of the derivation invalid without further arguments. \section{Energy gap $\Delta$ et al.}