Document that explains the chosen concept to the animator 1

Cooper Pairs Authors: Anura.B.Kenkre Course Name: Superconductivity In superconductors,there exists an electron-lattice-electron interaction. This indirect interaction proceeds when one electron interacts with the lattice and deforms it;a second electron sees the deformed lattice and adjusts itself to take advantage of the deformation to lower its energy. Thus,the second electron interacts with the first electron via the lattice deformation. The pair of electrons thus formed are known as Cooper Pairs. 2

Learning Objectives After interacting with this Learning Object, the learner will be able to : Describe the formation of a cooper pair. Explain the relation between the intrinsic coherence length and the size of a cooper pair. 3

Definitions of the components/Keywords: Certain metals and alloys exhibit almost zero electrical resistivity when they are cooled to sufficiently low temperatures.This reduction in the value of electrical resistivity to zero is known as superconductivity. The temperature at which the transition to the superconducting state takes place is called the critical temperature(Tc)or the superconducting transition temperature. In superconductors,there exists an electron-lattice-electron interaction. This indirect interaction proceeds when one electron interacts with the lattice and deforms it;a second electron sees the deformed lattice and adjusts itself to take advantage of the deformation to lower its energy. Thus,the second electron interacts with the first electron via the lattice deformation. The pair of electrons thus formed are known as Cooper Pairs. Superconducting coherence length(ξ)gives approximate spatial dimension (size) of the Cooper pair. 4

The coherence length (ξo) determined by the energy gap at zero temperature,Eg(T=0)is called Intrinsic coherence Length and it is given by the formula: where is the fermi velocity(on the fermi surface) and Is the Planck constant. Definitions of the components/Keywords continued: 5

6 IMPORTANT NOTE TO THE ANIMATOR: All the instructions/labels or anything WRITTEN in blue are CONTENT NOT TO BE DISPLAYED! All the instructions WRITTEN in black are CONTENT TO BE DISPLAYED! This is not applicable for images as there can be overlapping of these colours there. This should be followed for all the instructions,labels,etc… Kindly keep a note of this while displaying text in the animation.

Master layout or diagram Make a schematic diagram of the concept Explain to the animator about the beginning and ending of the process. Draw image big enough for explaining. In above image, identify and label different components of the process/phenomenon. (These are like characters in a film)‏ Illustrate the basic flow of action by using arrows. Use BOLD lines in the diagram, (minimum 2pts.)‏ In the slide after that, provide the definitions of ALL the labels used in the diagram INSTRUCTIONS SLIDE 7

Master layout or diagram You may have multiple master layouts. –In this case, number the master layout. ( e.g. Master layout 1)‏ – Each Master layout should be followed by the stepwise description of the animation stages related to it INSTRUCTIONS SLIDE 8

Master Layout : electrons :Atoms of lattice At T>Tc. At T=Tc. For animation refrence see this link: Zoomed in 9

Master Layout2: ANIMATION AREA :electrons:Atoms of lattice 10 COHERENCE LENGTH Aluminium(Al) Tin(Sn) TEMPERATURE DEPENDANCE Niobium(Nb) Lower THigher T Interactivity:

Animation design Please see the design template provided in the next slide. This is a sample template, and you are free to change as per your design requirements. Try and recreate the sections/subsections as shown in the template

Play/pauseRestart Credits :electrons:Atoms of lattice 12 Want to know more… (Further Reading) ‏ Definitions Test your understanding (questionnaire) ‏ Lets Learn! Lets Sum up (summary ) ‏ What will you learn COHERENCE LENGTH Aluminium(Al) Tin(Sn) TEMPERATURE DEPENDANCE Niobium(Nb) Lower THigher T Action buttons

Explain the process In this step, use an example to explain the concept. It can be an analogy, a scenario, or an action which explains this concept/process/topic Try to use examples from day-to-day life to make it more clear You have to describe what steps the animator should take to make your concept come alive as a series of moving images. Keep the examples simple to understand, and also to illustrate/animate. 13

Analogy / Scenario / Action Often this pairing of electrons is visualised in terms of ball bearings (the “electrons”) resting on a rubber sheet (the “lattice”). Putting one ball bearing on the sheet will cause it to stretch creating a depression in which the ball sits. This lowers the gravitational potential energy of the ball by making it lower down. If another ball is placed on the sheet, it too will form a depression, but if it is placed near enough to the first the two will roll together and form a deeper depression. This lowers the overall gravitational potential energy of the two balls and causes there to be a coupling between them that would otherwise not be there without the rubber sheet. The animation below gives an idea of how this occurs. In practice, this is only a schematic representation of the microscopics of the interaction within electron pairs. Note for the animator:You can have a look at this video link for the animation of this analogy: ANALOGY FOR FORMATION OF COOPER PAIR: 14

Stepwise description of process The goal of the document is to provide instructions to an animator who is not a expert. You have to describe what steps the animator should take to make your concept come alive as a moving visualization. Use one slide per step. This will ensure clarity of the explanation. Add a image of the step in the box, and the details in the table below the box. You can use any images for reference, but mention about it's copyright status The animator will have to re-draw / re-create the drawings Add more slides as per the requirement of the animation

Step 1: Refer to master layout 2. Description of the action/ interactivity Initially, show a pop up near the interactivity asking the user to select either ‘temperature dependence’ or ‘coherence length'. Until they click on either, the options below each will remained greyed out. When they select either, the corresponding option will be clickable..For example, If they select coherence length, only then the options for Aluminium, tin and niobium will get clickable. Keep the action buttons all of the same size and you can use a suitable colour scheme for them as well. Include a play, pause and restart button so that the user has control. When the user clicks on temperature dependence, go to slide 17 and let the options below ‘temperature dependence' get clickable. If the user clicks on Coherence length go to slide 26,and at the same time the options for Aluminium, tin and niobium will get clickable. If the user clicks on niobium. go to slide 27. If the user clicks on tin, go to slide 28. If the user clicks on Aluminium. go to slide

Step 2: Description of the action/ interactivity Text to be displayed (if any) ‏ (DT) ‏ Put a blank slide and let the text in Fig 1 appear. Let this slide remain for 2 seconds. Then go to step 3. If the user clicks on lower T action button then go to step 4. Do not allow the user to click on higher T action button. FOR TEMPERATURE: T 1 >>T c Fig 1 17 For the action buttons refer to master layout.

Step 3: Title: Motion of electrons at temperature T 1 >>T c. Description of the action/ interactivity Always keep the size of the atoms of the lattice a lot larger than the size of the electrons. Move the electrons along the paths shown by the arrows in Fig2. The arrow shows the direction of motion and entire motion of one electron is depicted in one colour. For example,in Fig 2,entire motion of electron 1 is depicted by black coloured arrow. The electrons hit the atoms of the lattice and move along the path shown by the respective arrows. Show the movement of the electrons along the different coloured arrows simultaneously. Show this animation step for a time length of 8 seconds. Do not let the electrons pass through the atoms of the lattice or through each other while moving in their respective paths indicated by arrows. REPEAT THE ANIMATION TILL THE TEMPERATURE VALUE CHANGES. The arrows are there to show the path along which the electron will move. The arrows are not to be shown on screen. Include + and – signs on the atoms of lattice and the electrons respectively. If the user clicks on lower T action button then go to step 4. Do not allow the user to click on higher T action button. :electrons :Atoms of lattice Fig 2 Electron 1 18 For the action buttons refer to master layout.

Step 4: Description of the action/ interactivity Text to be displayed (if any) ‏ (DT) ‏ Put a blank slide and let the text in Fig 3 appear. Let this slide remain for 2 seconds. Then go to step 5. If the user clicks on lower T action button then go to step 6. If the user clicks on Higher T action button then go to step2. FOR TEMPERATURE: T 2 >>T c (T 2 <T 1 ) Fig 3 19 For the action buttons refer to master layout.

Step 5: Title: Motion of electrons at a lower temperature T 2 >T c. (T 2 <T 1 ) Description of the action/ interactivity Always keep the size of the atoms of the lattice a lot larger than the size of the electrons. Move the electrons along the path shown by the arrows. In Fig4. Compared to previous figure, show fewer number of collisions (because temp is lower). The arrow shows the direction of motion and entire motion of one electron is depicted in one colour. The electrons hit the atoms of the lattice and move along the path shown by the respective arrows. Show the movement of the electrons along the different coloured arrows simultaneously. Show this animation step for a time length of 10 seconds. Do not let the electrons pass through the atoms of the lattice or through each other while moving in their respective paths indicated by arrows. REPEAT THE ANIMATION TILL THE TEMPERATURE VALUE CHANGES. The arrows are there to show the path along which the electron will move. The arrows are not to be shown on screen. Include + and – signs on the atoms of lattice and the electrons respectively. If the user clicks on lower T action button then go to step 6. If the user clicks on Higher T action button then go to step2. IMPORTANT NOTE TO THE ANIMATOR: Keep the speed of the electrons in Fig 4 slower as compared to their speed in Fig 2. : electrons Fig 4 20 :Atoms of lattice For the action buttons refer to master layout.

Step 6: Description of the action/ interactivityText to be displayed (if any) ‏ (DT) ‏ Put a blank slide and let the text in Fig 5 appear. Let this slide remain for 2 seconds. Then go to step 7. If the user clicks on Higher T action button then go to step 4. Do not allow the user to click on lower T action button. FOR TEMPERATURE: T 3 =T c (T 3 <T 2 ) Fig 5 21 For the action buttons refer to master layout.

Step 7: Title: At T=T c, Lattice distortion. Description of the action/ interactivity Audio naration Always keep the size of the atoms of the lattice a lot larger than the size of the electrons. First show the lattice without electrons and arrows and then zoom in to take a close up so that only 6 atoms of the lattice as shown in Fig6,can be seen. Move the electron 1 along the BLUE arrow as shown in Fig6 above. When electron1 moves along the blue arrow, bend the atoms of the lattice as shown in the above Fig6. Show this animation step for about 2 seconds. Keep the shape of the atoms spherical even when the lattice bends. Include + and – signs on the atoms of lattice and the electrons respectively. If the user clicks on Higher T action button then go to step 4. Do not allow the user to click on lower T action button. :Atom of the lattice :Electron NET POSITIVE CHARGE ( Fig 6 ELECTRON 1 ) 22 Note for the animator:The black arrow is there to show the path along which the electron will move and the blue arrow is for naming the electron..The arrows are not to be shown on screen. For the action buttons refer to master layout.

Step 8: Title: At T=T c,Lattice distortion Description of the action/ interactivity Audio narration As soon as the atoms of the lattice bend towards electron 1,show electron 2 coming in and moving along the black arrow as shown in Fig7. Show this animation step for about 2 seconds. Keep the shape of the atoms spherical even when the lattice bends. Include + and – signs on the atoms of lattice and the electrons respectively. If the user clicks on Higher T action button then go to step 4. Do not allow the user to click on lower T action button. :Atom of the lattice :Electron NET POSITIVE CHARGE Fig 7 () ELECTRON 2 ELECTRON 1 23 Note for the animator:The black arrow is there to show the path along which the electron will move and the red arrow is for naming the electron..the arrows are not to be shown on screen. For the action buttons refer to master layout.

Step 9: Title: cooper pair getting formed. Description of the action/ interactivity Audio narration Always keep the size of the atoms of the lattice a lot larger than the size of the electrons. Show the 2 electrons getting paired by means of a dotted elliptical shape between them, as shown in Fig8 above. The atoms of the lattice remain bent as in previous slide. Show this animation step for about 2 seconds. Keep the shape of the atoms spherical even when the lattice bends. Include + and – signs on the atoms of lattice and the electrons respectively. If the user clicks on Higher T action button then go to step 4. Do not allow the user to click on lower T action button. Note for the animator:The black arrow is there for naming the cooper pair..the arrow is not to be shown on screen. :Electron Fig 8 :Atom of the lattice 24 COOPER PAIR For the action buttons refer to master layout.

Step 10: Title: cooper pair drifting through the material. Description of the action/ interactivity Audio narration Always keep the size of the atoms of the lattice a lot larger than the size of the electrons. Show the 2 electrons travelling together along the direction shown by the orange arrow. This movement is shown in Fig9. The atoms of the lattice remain bent as in previous slide. Show this animation step for about 2 seconds. Keep the shape of the atoms spherical even when the lattice bends. Include + and – signs on the atoms of lattice and the electrons respectively. If the user clicks on Higher T action button then go to step 4. Do not allow the user to click on lower T action button. Note for the animator:The orange arrows are there to show the path along which the electron will move..The arrows are not to be shown on screen. :Atom of the lattice :Electron Fig 9 25 For the action buttons refer to master layout.

We will now explore the relation between Intrinsic Coherence length and the Cooper Pair size... Description of the action/ interactivity Audio narration When the user selects the radio button for ‘coherence length’ display the text above. Step 11: 26 For the action buttons refer to master layout.

Step12:Intrinsic Coherence length for Niobium(Nb) Niobium(Nb) metal sheet. ξ Description of the action/ interactivity Text to be displayed (if any) ‏ (DT) ‏ Show the niobium metal sheet appearing first. Show the animation given in slides 22 to 25. Then show the double headed arrow inFig7 above appearing along with the symbol ξ 0 Then display the text given beside. Make the cooper pair move along the path shown by the blue arrow.(blue arrow not to be displayed) Always keep the size of the atoms of the lattice a lot larger than the size of the electrons. Include + and – signs on the atoms of lattice and the electrons respectively.  The intrinsic coherence length for Niobium is 3.8× cm. ξo Fig 7 :Electron :Atom of the lattice 27 Nb

Step13:Intrinsic Coherence length for Tin(Sn) Tin(Sn) metal sheet. Description of the action/ interactivity Text to be displayed (if any) ‏ (DT) ‏ Show the tin metal sheet appearing first. Show the animation given in slides 22 to 25. Then show the double headed arrow inFig7 above appearing along with the symbol ξ 0 Then display the text given beside. Make the cooper pair move along the path shown by the blue arrow. (blue arrow not to be displayed) Always keep the size of the atoms of the lattice a lot larger than the size of the electrons. Include + and – signs on the atoms of lattice and the electrons respectively.  The intrinsic coherence length for Tin is 23× cm. Fig 8 :Electron :Atom of the lattice ξo 28 Sn

Step14:Intrinsic Coherence length for Aluminium(Al) Aluminium(Al) Metal sheet. Description of the action/ interactivity Text to be displayed (if any) ‏ (DT) ‏ Show the aluminiuim metal sheet appearing first. Show the animation given in slides 22 to 25. Then show the double headed arrow inFig7 above appearing along with the symbol ξ 0 Then display the text given beside. Make the cooper pair move along the path shown by the blue arrow. (blue arrow not to be displayed) Always keep the size of the atoms of the lattice a lot larger than the size of the electrons. Include + and – signs on the atoms of lattice and the electrons respectively.  The intrinsic coherence length for Aluminium is 160× cm. ξo Fig 7 :Electron :Atom of the lattice 29 Al

Interactivity and Boundary limits In this section, you will add the ‘Interactivity’ options to the animation. Use the template in the next slide to give the details. Insert the image of the step/s (explained earlier in the Section 3) in the box, and provide the details in the table below. The details of Interactivity could be: Types: Drop down, Slider bar, Data inputs etc. Options: Select one, Multiple selections etc Boundary Limits: Values of the parameters, which won’t show results after a particular point Results: Explain the effect of the interaction in this column Add more slides if necessary

INSTRUCTIONS SLIDE Self- Assessment Questionnaire for Learners Please provide a set of questions that a user can answer based on the LO. They can be of the following types: –These questions should be 5 in number and can be of objective type (like MCQ, Match the columns, Yes or No, Sequencing, Odd One Out). –The questions can also be open-ended. The user would be asked to think about the question. The author is requested to provide hints if possible, but a full answer is not necessary. –One can include questions, for which the user will need to interact with the LO (with certain parameters) in order to answer it. 31

INSTRUCTIONS SLIDE Please make sure that the questions can be answered by interacting with the LO. It is better to avoid questions based purely on recall. Questionnaire for users to test their understanding 32

Questionnaire 1. As the temperature decreases what happens to the number of collisions? Answers: a)Increaseb)decrease c) no change d) ‏ becomes zero. Correct Answers:1)b Feedback: If user clicks correct answer then display “Correct! Make sure you can explain the reasoning!” If user clicks incorrect answer then display “Play with the simulation and Try again!” Display one question on one page..When the user selects his answer, compare with the correct answer and display the appropriate feedback. 33

Questionnaire 2.cooper pairs are formed at what temperature? Answers: a)above T c b)below T c c)At all timesd) ‏ at T= T c. Correct Answers: 2)d Feedback: If user clicks correct answer then display “Correct! Make sure you can explain the reasoning!” If user clicks incorrect answer then display “Play with the simulation and Try again!” 34

Questionnaire Feedback: If user clicks correct answer then display “Correct! Make sure you can explain the reasoning!” If user clicks incorrect answer then display “Play with the simulation and Try again!” 3. Match the following: metal Intrinsic coherence length (cm) 1)Aluminium(Al) 1) 3.8 × )Tin(Sn) 2)160 × )Niobium(Nb) 3)23× ) Correct Answers: *Note:allow the user to match the coloums by means of arrows. 35

Questionnaire Feedback: If user clicks correct answer then display “Correct! Make sure you can explain the reasoning!” If user clicks incorrect answer then display “Play with the simulation and Try again!” 4. For different metals, as the intrinsic coherence length increases, what happens to the size of cooper pairs? Answers:a)Increasesb)decreasesc)remains same d)becomes zero. Correct Answers:4)a 36

Questionnaire Feedback: If user clicks correct answer then display “Correct! Make sure you can explain the reasoning!” If user clicks incorrect answer then display “Play with the simulation and Try again!” 5.When the cooper pairs are formed, what happens to the shape of the lattice? Answers:a)bends away from the electronsb)bends towards the electronsc)stays as it isd)distorts unevenly. Correct Answers:5)b 37

Questionnaire Explore: What could be the effect of the value of the band gap energy(Eg),on the existence of the cooper pair? 38

Links for further reading Reference websites: ooper.php Books: Introduction to Solid state physics-Charles Kittel(chapter 12) Solid state physics-MA Wahab.(chapter 17) Solid state physics-Ashcroft/Mermin. (Chapter 34) 39

INSTRUCTIONS SLIDE Please provide points to remember to understand the concept/ key terms of the animation The summary will help the user in the quick review of the concept. Summary 40

Summary Certain metals and alloys exhibit almost zero electrical resistivity when they are cooled to sufficiently low temperatures. This reduction in the value of electrical resistivity to zero is known as superconductivity. The temperature at which the transition to the superconducting state takes place is called the critical temperature(Tc)or the superconducting transition temperature. In superconductors, there exists an electron-lattice-electron interaction. This indirect interaction proceeds when one electron interacts with the lattice and deforms it; a second electron sees the deformed lattice and adjusts itself to take advantage of the deformation to lower its energy. Thus, the second electron interacts with the first electron via the lattice deformation and gets paired with it. The pair of electrons thus formed are known as Cooper Pairs. The coherence length (or the Cooper-pair size) ξo determined by the energy gap at zero temperature,Eg(T=0)iscalled Intrinsic coherence Length. 41