Have you ever heard the phrase men want to be accepted for who they are? All right. Well, that's true for men and women alike. So this isn't singular to one gender. However, this makes me think of the Marilyn Monroe quote and how this might affect your relationship to actually Feel accepted for who you are. So I just want to read this quote really quickly for everybody. It says I'm selfish impatient a little insecure I make mistakes I'm out of control and at times hard to handle but if you can't handle me at my worst Then sure as hell don't deserve me at my best. Okay? Well, as I look at that, you know, I look at a human being and certainly as a human being We should from a compassionate place from an unconditional place Accept human beings for who they are. However, I don't think bad behavior is something you should accept So I want to pick this quote apart for a moment I want to share with you some things about men that you shouldn't accept and then lean into the conversation of how Accepting men for who they are in a different category might make you might might Lead you down a path to a relationship It's success and let me just say this when men do feel accepted just like you ladies We also feel appreciated and respected for who we are.  They're self centric to some degree We even in the dating process oftentimes focus on our own needs first We're not actually absolutely selfless in the early stage of dating So I can kind of accept that and even in relationships We can sometimes be more focused on our own needs than another's needs However, if it doesn't have the corresponding self lists Added to it if someone is completely selfish. Well, that's to me Unacceptable behavior in relationship. I don't suspect you'd want to be in relationship with a man who is absolutely selfish Now a little impatient. Hey, look, I have a capacity to be a little bit impatient in life Sometimes I get a little annoyed that it's taking forever for the waiter or waitress To come back to get my order and that sort of thing I have a little bit of habit of impatience, but that doesn't necessarily rule my entire life I'm a little insecure at times. I'm sure you're a little as insecure at times whether you're a man or woman You do want to feel accepted When you're feeling some doubt doubt is a natural Human need to some degree and what I mean by doubt, you know under the six basic human needs Certainty is really important But sometimes Uncertainty is important too so we can and I'm using insecurities as uncertainty because the reality is is humans aren't perfect and actual Unconditional love accepting someone for who they are is accepting that they might be insecure at times now if someone is Habitually insecure habitually in fear. It makes them less attractive to be in relationship The quote goes on to say I make mistakes. Okay, we all make mistakes. I give you that one I'm out of control and times hard to handle That's not necessarily an attractive quality be out of control and hard to handle at times I would say someone who is out of control and hard to handle at times doesn't know how to regulate their emotions and When someone is habitually in the space and by the way, this was a quote from a woman You know, there's this expectation that men are supposed to be the strong person The masculine energy needs to hold space when someone is out of control Well, do you want to hold space for a man who's out of control? Someone who's hard to handle at times that can be exhausting By the way, good morning. I hope you my coffee mug says I don't want to work anymore I just want to be put up on a pedestal Cherish or to be cherished put up on a pedestal and taken care of a friend of mine got me that Excuse my slurping the coffee is hot If you can't handle me at my worst then you don't deserve me at my best Well, to some degree we're all going to have bad days. So that's certainly an important aspect of a relationship But to lean into that as an expectation That you're allowed to fly off the handle whenever you want and I have to accept it I don't believe is a healthy way to be in relationship. So let's talk about let's for a moment Talk about ways behavior men's behavior you oftentimes accept But I don't believe is healthy for the relationship. Let's start with he's a lazy person His top priority is leisure or playing games Well, if someone's top priority is leisure playing games and isn't necessarily focused on Responsibility of supporting oneself as an example And there's an expectation that you accept this man for is and you are the one that fully supports the relationship And he just gets to have fun and play video games all day long. I'm not so certain. That's a healthy behavior to accept. Okay How about he's a flaky person? He's hot or cold ladies Many of you are accepting behavior from men who are hot and cold because you see that glimpse of where they're hot And you hyper focused on that and yet they can be corresponding cold at the same moment And inconsistent behavior doesn't allow one to feel safe in a relationship Men aren't attracted to women who are hot and cold and corresponding I don't believe you should accept that kind of behavior from someone who is both hot and they're cold But jonathan, he's the most amazing man I ever met. There's no other man like him I have to accept this behavior Because he's the prize. He's the golden ticket that will he wonka was giving Ladies, let me just say something about this idea that one person is perfect And yet you have to accept all of their bad behavior. They're inconsistent behavior. They're hot and cold behavior Have some self-respect Do not accept that kind of behavior when I say self-respect. I'm saying self-worth self-esteem self-confidence Self-reliance if you're not familiar with my book What the heck is self-love anyway a journey of personal development self-help and spiritual work By the way, there's a link below to get a copy of my book I'm here to say Self-respect says i'm going to stand up for myself when this behavior is unacceptable Because many of you are accepting behavior bad behavior Feeling like that's how you respect men and i'm here to say stop it Does his work or children take a priority in his life? Now first off I put this down You might scoff a little bit at the children part But I want to lean into this for a moment. Okay, certainly someone's work and their capacity to support themselves Is rather important and yet a healthy balance to life is a is a life where you're spending time with those people important to you and you don't continually make work your precedent over and over and over again and Dropping the people in your life is most important down to third fourth or fifth In your level of priority and that includes children as well. I'm here to say children are important Work is important. Those are those are important things So work's important your children important your relationship is important And when you set one so higher above in importance as a priority And by the way, I've seen this happen with men and women where they've actually engaged in what's known as covert incest where their children are such a priority in their life They put their children on such a pedestal that it compromises all areas of their life I know I was speaking to a woman recently.   I was thinking of the maryland manro quote my guess is she probably lost her temper a lot I shouldn't laugh because she also had a lot of hardship in her life I suspect that if someone's temper if they're not able to regulate their emotions and they go off the handle They're hyper emotional. They lose their temper. Is that behavior you want to accept? Can you respect someone who hand who operates that way? I invite you to look inward on that one Are his hobbies his top priority are his hobbies his top priority Look it I love to play golf. Okay, I haven't played. I actually played the other day a little nine hole that was Only took one hour. Okay, for some men some of their hobbies are such an extraordinary priority in their life That it actually compromises the relationship there in and I'm here to say where someone puts something so far above is their highest importance And not there remember I said work is important Children are important Hobbies are important your relationship are important But to to put them on such a high importance that it compromises What's what to me is a healthy balanced life? Isn't behavior that you necessarily should accept And lastly, what about men who are high functioning alcoholics women as well? Is that behavior you want to accept? because i'm here to say acceptance within within the scope of healthy acceptance For understandable acceptance. Like I said moments of making mistake. Of course, we all make mistakes. Are there moments where we're insecure? Absolutely. I believe those are things we should accept And sometimes we're self-centric. We want our needs met first. That's a very natural thing in that in a healthy happy relationship But I want to talk about acceptance in a different form that probably affects many of you today And men who are accepted in these areas feel quite respected in their lives so I'm going to talk about financial acceptance because the reality is today Women are now in a completely different position that they ever were In society where they actually can For really in the last 20 to 30 years fully support themselves There isn't the dependent need to be with a man for financial. Now, that's not always the case But i'm saying today we are we have seen this Explode in the last 20 years Unlike the 200,000 years before I want you to think about neanderthals went back 200,000 years for a hundred and 99,050 years Most every woman was completely dependent upon a man Actually, that's not true because women gave birth so big little boys were dependent upon women But they were dependent upon men And so I'm seeing a shift where there are women who have a significantly higher nest egg They have higher earning capacity and some men that enter their life may not meet that social status or financial status And I know three couples in particular where the women are significant have significant worth net worth I should say And these are men who are great men Who are good men? They're good. They are good communicators. They are they are good fathers to their children These are all divorced people Actually, one couple doesn't neither one of them has children in fact one of the men gave a kidney to a total stranger And yet his financial earnings and net worth was nowhere near his partner See these are women who accepted men. These are men that could take care of themselves They're good responsible human beings But they accepted that they may be the major breadwinner in the relationship and these are relationships that are flourishing And yet a lot of women will look at men like in these men's cases and say I reject them Because I want someone better than me. That's just a that's just an observation I make made Now there's the the the area that I want to lean in for this conversation most Centers around the book the five love languages the five love languages If you're not familiar with the book the five love languages, I highly recommend checking this out the links below The five love language. This is the way we connect with our partner In our own unique way of the way we give love and show love and the five love languages are words of affirmation although if you're leo, it's words of adoration Physical touch quality time acts of service and gifts Now what's interesting is I notice that A significant percentage of men Their love language is acts of service And a significant percentage of women their love language is words Okay, this is where oftentimes there's a disconnect between the two people The partners and in these particular cases A lot of times women because they desiring of the words don't accept the man that his love language might be acts of service And I'm just using this as the example Okay, I know a woman. Um, I was talking to Potential client was telling me about our boyfriend Who does all these wonderful things for her? That's all these wonderful things for her Fixes her sink Fix put a tire on her car Made sure her car was running properly hung up a tv just things like that And yet she was desiring of the words And yet all of his actions demonstrated that he loved her It's just words were difficult for him because in his childhood He was stifled by speaking from an emotional level And rather than say i'm done with this relationship or create demands what we just what I encouraged her to do Was to accept him for who he is and begin a process of leading by example leading by example by introducing words That were important to her in the relationship making requests And then watching little by little to see if you see see a shift in behavior If you experience a shift in behavior folks acceptance is also the recognition that they human beings aren't exactly the way you wish they were being And yet we men are trainable. We are trainable human beings are trainable. They have a capacity for growth Now look many of you are in relationship with what I call the users Or spenders if you're not familiar with my graph the three types of people actively dating They're the users spenders and growers and by the way, they could be moving in any one of these directions And i'm going to take this down because it's the glare, but the users They are the seek the short they they seek short term game. They're the love bombers They're the players the gold diggers the entitled people by the way. I was told gold diggers without a date Selfish people only caring for those needs. We're talking about not accepting that The spenders they seek comp connection companionship coupling and sex No direction uncertainty fearful have usually a dysfunctional life and lastly lastly the builders They seek long-term commitment. They're emotionally grown up good relationship skills They have their act together and some men are in the spender category These are the dysfunctional men and they're either moving towards a grower and builder Or they're moving towards using you if you continually accept behavior Like what we just laid out Especially the high functioning alcoholics this came out during covid We saw a significant percentage of men and women that have now become high functioning alcoholics Is that behavior you want to accept? Folks i'm here to say if you really want a relationship that flourishes That where a man feels respected for who he is and you feel respected for who you are It's going to start by from the very first date leading by example And having deeper conversations being radically honest with one another laying your cards on the table very early on And establishing the rules of engagement what i mean by rules of engagement is what is your standard?  Why didn't they teach me this in school? Why didn't my parents teach me this? I didn't realize I didn't know what I wanted And if you need help with that folks right here schedule the discovery call with me to see if working with the Coach is right for you. There's a link below Folks my area of expertise is to help you vet for these grower and builder So you're not continually dating these spenders over and over and over again All right, you know what it is men feel respected when they're accepted However, I'm here to say only respect those men with good behavior. I'm good emotional maturity good relationship skills They have a level of empathy. They're transparent in their lives their actions match their words They have victor consciousness and not victim consciousness and most importantly They have good communication skills that so when there's differences You're able to navigate them with a lot more ease this sinking in is this resonating. Please let me know Please hit that thumbs up button. Please share this video. Please subscribe to my channel if you're brand new And by the way again all the links below to a discovery call with me to join my groups Check out the description in the show notes All right, those who know my format know I'm this is time for q and a if you have a question for me Write the word question then post the question thereafter or you can purchase a super sticker super chat All the money's from the super sticker super chat goes to a scholarship fund in the name of my son conor asley That's a picture of him right there He's my son who passed away almost five years ago his birthday is coming up soon And his honor we donate to causes like the hoffman process and insight institute just to name a few Andrea says it's a shit show in the dating world. Yes, it is for men and women alike Linda asked question. What is the definition of a high functioning alcoholic? This is a person who can drink habitually and still maintain some level of Consciousness throughout the day. This is someone who can drink habitually and still maintain some level of consciousness In other words, they're not passed out drunk That's what I mean by a high functioning alcoholic And I am nowhere near an expert on this as well Ah, let's keep swimming bump bump bump bump bump bump bear with me See if there's any questions if you have a question post the word quit write the word question So it's easier for me to find ah Let's keep going bump bump bump bump bump Mary says yes that all you have been talking about today resonates with me. I'm happy to hear that Folks, you know, something's interesting to me. I want to share something else. Uh question hasn't come in yet You know, when you think about Relationship success, I want you to reverse engineer everything Marriage roughly here in the United States have a as a 50 success rate. Okay um so 50 success rate and it's 35 less than you know, and if for second and third marriages over two thirds and in divorce, okay Now probably two people living together like my sweetheart and I there's a picture of Marie. She's uh at a funeral in Chicago this weekend or this week she um Or we and I live together and probably our success rate is one out of four Why don't you think how many relationships never get to a point of living together? People are in relationship today And probably one out of ten will never make it to living together getting married And of those people that were in relationship, how many first dates have you gone on to get into a relationship? You might have gone on 50 first dates just to get into a relationship. So dating has the least success Uh success rate relationships have the second worst success rate living together probably as the third and marriage is the fourth Okay, so my point in bringing this up is dating is not A guarantee to relationship success. It's a vetting process And many of you do a poor job at vetting men So you can put the odds in your favor and that's what I do as a coach. So anyways, just wanted to say that All right, we had a question come in Oh really quickly. I want to thank angela for becoming a new member. You can join right here to join my group here on YouTube by hitting that join button right next to the share button and the like button I want to thank roller girl for the dollar 99 super sticker for collar. Thank you so much. I appreciate that And we got two questions that came in one from michelle Question why as a single person would I feel so leery of dating sites never seen one and don't want to Well, it is a very scary thing for many people To put themselves out there in a forum where you're literally seen by millions of people I can understand the fear around that the reality is these days though most people don't for those of us in midlife Our daily life doesn't put us in Surrounded by single eligible people Once you think about this when you're in high school most everybody was single and not quite eligible but single In college is the first time in your life if you went to college Where you're surrounded by single eligible people of your own relatively your own age And the minute you graduate from college, that's where everything changes It used to be the workplace was the most popular place to meet people back in the 60s And certainly by the 70s it was night clubs and bars 70s and 80s and 90s And now since the 2000s it's the internet happens to be the number one place where people are meeting today I went to a wedding last weekend where the couple met on match.com There was over 10 couples there at the wedding like marie and I where we all met through a dating site Just happens to be the number one place to meet people today I understand your fear But I want you to maybe overcome your fear because it happens to be the best place to meet people to go out on dates And we need dates to end up being in a relationship All right. Thank you, michelle for your question Melissa writes I broke up with an avoidant. Do they ever come back? Yes, they do It is very common for men and women to feel a sense of loss Uh, wait, did you say you broke up with uh, I broke up with and it's very common for someone to feel a desire to come back Now again people that have an avoidant love attachment style means whenever there is some fear in the relationship They retreat. Okay. Just like the anxious people fight for a relationship avoidance retreat now That's something that can be healed And I'd like you to check out the book Attached by amirah levine and rachel heller highly recommend checking out this book So you can understand your love attachment style in depth and maybe understand someone else love attachment style as well okay Let's keep swimming cc writes question Do you think two people need to have the same or different attachment styles for a relationship to be success? You know, it's interesting I've been thinking a lot about this. I I suspect we all have a default love attachment style for example My default is anxious now. I'm as close to secure as I can possibly be but my default is anxious Now what's interesting is I have a belief my Marie would kill me for saying this. I think she's a secure attachment style But her default is to run And we talked about this the other day It's her natural inclination to run when there's maybe not a conflict You know resolving a conflict like where to go to dinner kind of thing And we have two different opinions. I'm talking about where there's some level of discourse in the relationship And her natural indication Is to run a little bit and I think we're probably attracted to one another because of this I'm not saying this is an absolute. I'm just speculating here Now in her in both our cases I don't fight that hard and she doesn't run that far away and this is our healing to build trust with one another I think ultimately when two couples build a significant amount of trust Our attachment styles begin to get healed. Okay So I do believe that avoidance tend to attract anxious anxious tend to attract avoidance when two anxious together get together They probably one cancels not cancels the other out But the one person feels a little bit overwhelmed and when two avoidance together, they can't ever make it work out It's just my speculation here. So cc. I hope I answered your question with that one. Thank you so much Shaw writes question My friends started Dating a guy that she slept with on the first date since then he invites her over to his place Which follows by time in the sack. However, she wants to be wind and dine Okay, it's not really a question but she wants to be wind and dine Listen, certainly there are men that are only in it for the sex Um, I don't like the idea of being wind and dined What I would prefer is couples do social activities hobbies and mutual interests social activities might include going out to dinner Um, but the end of the day It sounds like this is a relationship based on sex and not necessarily Uh a relationship built on true Co-creation and getting to know one another. So I'm sorry that your friend is experiencing what she doesn't want Maybe she should speak up for what she does want But I don't think being wind and dined is going to solve this problem Because my suspicion is all he cares about is the sex in the case of your friend. That's just my gut Reaction to this. So thank you Shaw for that question Let's keep going Jennifer says dating apps sites and apps are the way to meet I lived in New York City and I've met men organically sometimes They're you know, again organically Meeting online. They're all spokes in a wheel. So thank you so much for that Uh Maryland says I'm sick of men who play golf and have hobbies. Yes, when their hobbies and priorities exceed Their importance of the relationship. I can understand why you feel that way Lillian Lillia says question What are the differences between love bombing or someone who's really into you and wants to be with you more frequently? Okay, great question So love bombing to me is trying to convince you to like them Okay, it's coming from a place of convincing you to like them. It's oftentimes used in Sometimes nefarious ways like the tinder swindler where they're they're giving you so much love and attention And these are usually women deprived of love and attention that they are so appreciative of it That they'll hand over their wallet to a man. Okay now I know as a man When I am feeling a sense of limerence or lust I could love bomb lust is I want to get in their pants and I'll say whatever it takes to do it I've experienced that I've done it. Okay Or I've experienced experience extreme infatuation. I for some reason I put this person up on a pedestal and I just went overboard giving compliments compliments and compliments At the same time when a man likes a woman the way I did when I met Marie A healthy form of appreciation Doesn't feel yucky a man can demonstrate a healthy form of appreciation And it doesn't feel yucky when it feels yucky. That's typically love bombing and when it's uh when it's when it feels This is where it's tricky because if you're deprived for love It's going to feel good But that's actually coming from a depleted place within yourself And this is why I highly recommend going back to my book what the heck is self love anyway So you're not dependent upon someone to give you validation Okay, I hope that question helps Lillian. Thank you so much Shannon writes What should I do when my boyfriend has several other female friends, but he won't introduce them to me He texts them helps them when their car breaks down, but I don't get to go and meet them Well, I think it's these days most men and women have created friendships with the opposite sex in many cases We appreciate our friendships with the opposite sex those heterosexual folks.  That's not a good sign So what should you do about it make a request? To meet these people or maybe you cut back on this relationship may not be that he if someone is secretive in their life And unwilling to introduce you in their life. How can you fully trust this person? How can you trust this person when they're going and now again? I have no I certainly respect people that go out of their way to help other people There's nothing wrong with that But if they're not willing to introduce you then it's going to be difficult to build the deep roots of trust Um in the relationship. So Shannon, I hope that helps you. So thank you so much uh Angel mama bears says I don't trust any dating sites. I played around with a few years and Half 95 what I found were catfish I'm sorry. You've experienced that. I met my sweet art Marie through a dating app I went to a wedding as I said to you earlier. There were 10 couples who met through dating sites so I'm sorry you're having that experience and yes, that's a common experience that doesn't put you that's certainly a very Common experience and yet if you learn the techniques to do a better job bedding and screening You may not experience that and again. I teach that in my coaching. So schedule a call with me Cc says thanks so much for answering my question. That makes sense. I'm hoping I'm a happy that help Uh Andrea writes question What if a guy gives you his number, but then is too busy with work to talk much? It's like a contradiction. Here's my number. Don't call me Probably in the moment Andrea He did feel some sort of connection with you but not enough to actually go anything further This is the hard thing about dating these days. I think you know, that's phrase. He's just not that into you I think humans are so deprived in their lack of of look at most humans Biggest wound is I'm not good enough.  You didn't want to bother. You know, I'm just speculating here. Okay The reality is these days, you know, people are looking for so we're almost addicted to the need to have it perfect So perfect that if it doesn't feel perfect in that moment They might temporarily want to give you their number or get your number, but they're not willing to invest This is the dysfunctional dynamic of dating today dating You know dating means dysfunctional way of getting to know someone That's what dating means. It's a dysfunctional way to get to know someone because these days were meeting total strangers And and I say dysfunctional because humans aren't radically honest with one another They're just their highlight of their communication is how's your day going? Did you have a good day? I hope you had a good day Many of you ladies are afraid to ask deeper questions because you're willing you're afraid to scare a guy away and meant Oops, I got pit stains there And men are rather clueless in the dating process. So look at I know it's a mess out there My job is to help you make sense of it all so you can have the kind of results I get all the time Jonathan, I met a great guy. Jonathan. I met a great guy. Jonathan. I met a great guy And these are women that know the difference. So anyways, Andrea, thank you so much Um Hey, one of my members from my facebook group says I thoroughly enjoyed your book great read. Thank you so much All right And we're gonna end on this note here Stay well strong says when a man says you are the only woman who talks with me all the rest All of the rest claws come out and while he's yelling at me. Is that a common thing for men these days? Okay, I'm gonna finish up on this note. Okay Most humans have Weak emotional maturity and or excuse me poor emotional maturity and weak relationship skills. Look at the vast majority 75% of singles who are actively in the dating marketplace over 45 years old their divorce Now if women initiate two thirds of those divorces because the men they were with Had weak emotional maturity and weak relationship skills. And by the way, you women are no picnic either coming back to that Marilyn moan row quote. I can tell you women can be just equally as dysfunctional as men Okay But with that said men by the time they hit 40 They're on some level yearning to Become they're yearning to be loved and appreciated. They just don't know how we were never taught how to Dive into our emotions So what we need is someone who accepts that we may not our love language may not be yours as an example Or what I shared earlier and yet we want someone to guide us The problem is a lot of unhealed human beings are in the dating marketplace and what they do Many women alike use each other as therapists and these are unprofessional people Giving advice seeking that way of just connecting one in that female energy And this happens frequently and yet they're not capable of actually leaning into a healthy happy relationship This is happening habitually And this is one of the reasons why dating is so darn dysfunctional these days so um So coming back to what you said Unhealed human beings with weak emotional maturity can operate in the way that you just described So I hope that gives you some insight and then again what's most important is for you to To operate from your sovereignty your self-worth your self-esteem your self-respect your self-confidence And do a better job in what I teach in my private coaching. By the way, there's a link here Radical honesty laying your cards on the table and the rules of engagement Which I talk about when I work with my clients, okay? If you need some support check out all the links below to a free discovery call with me to join my group to follow Me on instagram and to get my book as an example All right, I think this will be a great place to wrap up today I hope you got value from this if you did please to co post a comment below Please like this video Please share this with your friends and please subscribe to my channel And i'm going to wrap up this video as I always do first off give myself a big gigantic johnathan mirag self love I'm going to reach into the camera excuse those pit stains give you a hug of love I'm going to ask you to turn to a friend to pet the teddy bear pillow and give itter them a hug of love Because hugs are a great source of love and let's face it. We could all use more love in our lives I want to thank michelle and sharon and andrea and jojica and jennifer and uh, I said michelle and andrea and stay well strong and cc and la And one of our uh members in facebook group cc Shah Priya, thank you all so much wishing you a fab day. Bye now


You can follow along with this presentation by going to nanohub.org and downloading the corresponding slides. Enjoy the show. Okay, so I'm going to be giving my last lecture now, and then I'm going to be looking forward to hearing Professor Alam talk about solar cells this afternoon. So this is a lecture, it's really not intended to be about graphene. I know that that's a very hot research topic right now, a lot of people are working on it. My intent here is really not to talk about graphene and why it's of interest and what it might be useful for, but it's really just to use it as an example to show when a new research topic emerges, you have to figure out how to understand what's going on here. You might read about graphene and you hear a lot of talk about relativistic Dirac particles and you might think that this is a very complicated material and it's difficult to understand. But what I want to show you is that most of what you need to know about electronic transport for building devices in graphene can be understood very simply with the concepts that we've introduced. It's just a different material with a different dispersion, you work everything out, we get all of the answers that we need. So let's talk about how this works. Just by way of review, these are our fundamental equations. So we're back with electrons now, not phonons. These are the expressions we've developed for the flow of electrical current and the flow of heat due to electrons. We have all these conventional expressions and we could express them as diffusive transport equations too just by having long samples. We can express them in one of two ways depending on what we want to make the independent variables gradient and quasi-firmly level and temperature or current and gradient and temperature. So we get these standard expressions that are really very, very widely applicable. So we want to apply them to a very different material now. So in order to compute conductivity in Pelletier coefficient and all of these parameters, we just need to know two things if we have a new material. We need to know what is its density of channels and what is its transmission or its mean free path. Then everything works out. And we've been discussing when we worked out the density of channels for a two-dimensional conductor, we worked out this expression a few days ago, it's got an effective mass in it. The thing to remember, first of all, is that we assumed parabolic bands when we did this and if you work it out in 1D or 3D and still assume parabolic bands, you get different answers, but there's no need to assume parabolic bands. We can work out the answer for the number of channels if we have a dispersion. So what happens then? Graphene is a really good example because it has a very simple dispersion so that we can work out things analytically, but it's very different from the parabolic band. So that's primarily why I'm doing it. And of course, a lot of people are interested in it. As you know, the Nobel Prize in Physics this year was awarded to the pioneers in graphene. A lot of people are interested not only in graphene science but in potential graphene applications. Several of you have talked to me or tell me you're working on your thesis topics involve graphene. So it's worth knowing something about this material. So the objective is just to basically to try to understand the conductance in graphene. See what people measure. So I'll remind you, graphene is just a one atom thick planar sheet of carbon with this hexagonal honeycomb lattice. You can work out its band structure. If you run this little tool in the nano hub, you can compute its dispersion and E of K in density of states and things like that. But it has a really unusual band structure. So if you do a simple tight binding model and compute E of K in the two-dimensional Brillouin zone for graphene and you get an answer that looks like this on the left. Now, if I look at that KXKY plane, that two-dimensional Brillouin zone, it turns out that there are only two distinct points at the corners of that Brillouin zone. The other ones can all be reached by a reciprocal lattice vector so they're not distinct. So the red ones and the blue ones are distinct. And I can reach the other red ones or the other blue ones just by translating by a reciprocal lattice vector so they're not distinct. So when we talk about valley degeneracy, G will be two for graphene. The states that I'm going to be interested in, the Fermi level lies at zero for a pure piece of graphene in equilibrium, charge neutral. So I'm going to be interested at the states where that top and bottom layers are intersecting because that's about E equals zero. Most of the action is going to occur near the Fermi energy. So it's going to occur near those points. There are six of those points, but only two of them are distinct. So if I look near those points, the dispersion looks very simple. The EK looks like this. It's a cone. So there's a plane, they have a Kx and a Ky coming out of the page. The E of K looks like that cone. The Fermi level ends up being right at the intersection of those two cones. I can describe that very simply. It's just plus or minus h bar of velocity. We'll call the Fermi velocity times the magnitude of K. Magnitude of K is just square root of Kx squared plus Ky squared. So the velocity, I just take the slope of the EK. The velocity of an electron is going to be Vf. And what's interesting here is that an unusual in graphene is that the slope is constant everywhere. Not if I get way, you saw in the earlier picture, if I get very far away from this point, the velocity starts to change. But under practical conditions, I'm not going to get very far away from this point. So this linear dispersion is very good. The velocity is constant. It's about 1 times 10 to the 8 centimeters per second. That's very high. That's about 10 times higher than the average thermal velocity of electrons in silicon, so that's one of the reasons that people are very interested in it. It has a valid degeneracy of 2, which I'll have to put in when I do these things. And this point, if I locate the Fermi level at the intersection of those two cones, then everything is neutral. That's called the neutral point. It's also sometimes called the Dirac point. I'll think about the states above that point as the conduction band and the states below that point as the valence band, but there's no band gap. That's also what makes it unusual. Now, so people would call this, this would be called a semi-metal. In a metal, the Fermi level is deep inside the band and there are lots of states. In a semiconductor, there's a band gap. Here, we have states very close to the Fermi level, but exactly zero if I locate them at that point. All right. Okay, so I'll mention something that I'm not going to do very much with here. If I have a normal parabolic band structure, I'll have a band gap and I'll have a conduction band and a valence band and energy will go parabolicly with k. And when I need to do things like calculate densities of states and scattering rates and things, I'll need to know the electron wave function. And we usually think of the electron as a plane wave. We would think of it as a plane wave propagating in this 2D plane. So my electron wave function would have a form like this. The square root of area is to normalize it so that when I integrate psi star psi over an area, I get 1. And but now one of the things that happens is that when the band gap gets very small that I really have to think of a multi-component wave function. The wave function has contributions from both the conduction band and the valence band. In silicon, there's light holes, heavy holes, and a split off band. So it's really a four-component wave function in which all of those components are coupled. Sometimes we have to deal with that, especially when we deal with small band gap semiconductors when the conduction band is close to the valence band and everything is coupled. Frequently in a large band gap material like silicon, there isn't a tight coupling between the conduction and valence bands and I can just use this wave function for the electrons in the conduction band. But graphene is different. The conduction and valence bands are very close together. So you would think that you always have to think about a coupled wave function that has a character due to the conduction band and a character due to the valence band. So everything is symmetrical here. That's one of the things that makes graphene so nice. Conduction band and the valence bands are mirror images of each other. The wave function, you can see that it's a two-component vector. The first one just looks like a plane wave. But the second one depends on the angle that the electron is propagating at in the x, y plane. And if you look, if the angle is zero and I'm propagating in the x direction, then those two components are both one. If the angle is pi and it's propagating in the minus x direction, then the second one has a minus one. Now, those two wave functions are orthogonal. If I take integral of psi one times psi two and integrate them, I'll get zero because they're orthogonal. And that leads to some interesting things, that sometimes people call this climb tunneling. Backscattering is suppressed because it's not possible for an electron to flip around because a scattering potential can't turn an electron exactly around because those two wave functions have no overlap. That happens in carbon nanotubes too. So that promotes higher mobilities, right? Now it's not quite as good as that and maybe people make a little bigger deal out of it than we really should for device work. Because they're only orthogonal if you exactly backscatter. But you have to take this into account when you're computing mobilities and it ends up I think if I remember right when you take this two component nature into effect, you get something like a factor of two higher mobility than you would if you ignored it. Because backscattering is suppressed completely for one particular angle but for the other ones it's reduced. Okay, now I had earlier there pointed you to a couple of lectures on Professor Dada on the nanohub. So if you'd like to see a careful discussion of band structure of graphene and how those plots are made I'd refer you to those. But that's about all we need for this lecture. So we need the number of modes, but the number of modes is the density of states times velocity. So the first thing we can talk about is the density of states. Okay, so this is going to be, this is a very simple dispersion now, but it's distinctly non parabolic. Doesn't matter, right? Because we can do a density of states. Now remind you the way that we would do that is we would look in the XY plane at a constant energy surface that would be a constant K. And we would ask how many states are there in that shaded ring? How many between NFK, DK in that shaded ring? Well the area of that shaded ring is the circumference 2 pi K times the VK. And if you remember when you impose boundary conditions on the wave function and count states, each state takes up a space 2 pi over length in the x direction and 2 pi over length in the y direction. So that's how much space each state takes up in the XY plane. So we divide that into the area, we get the total number of states. We multiply by 2 for spin and then we multiply by the valley degeneracy, which happens to be 2 for graphene also. So we get this expression for the number of states in that shaded ring. It's got a KDK. If I'd like to ask the question how many states are there between energy E and E plus DE, then I just have to do a change of variables from K to energy. And there I'll use the dispersion, E is equal to H bar VF times K. So I can do that change of variables. And if I do that change of variables, now what I have is the density of states. I'm multiplying by an A because it's the density of states per unit area, per joule, for graphene and it's EDE. So that's my answer. So if I plot the density of states above the drag point, it increases linearly with energy. If I plot the density of states below the drag point, it increases linearly with energy. Remember, for a 2D parabolic band, the density of states is constant and independent of energy. But that depends on that particular dispersion. For graphene, the density of states increases linearly with energy. All right, so density of states depends both on dimensionality and dispersion. Now how do we get the carrier density? We just integrate the density of states times the Fermi function over all the energies. Now one of the things that's nice about graphene is under most conditions, it's strongly degenerate. Once I get the Fermi level up away from that direct point, I'm deep inside the conduction band or deep inside the valence band. So the T equals 0 approximation actually works very well even at room temperature. So that simplifies everything. I don't have to deal with Fermi direct integrals very much. So I can use T equals 0, and I can ask, what's the electron density if my Fermi energy is up there? And we just do that integral. And now F is 1 between 0 and the Fermi energy, and it's 0 above that. So I only have to integrate up to EF. And the density of states goes linearly with energy. So I'm integrating EDE, and that just gives me an E squared over 2. So I get a simple expression that relates the sheet carrier density per square centimeter to the Fermi energy. It just goes as a square of the Fermi energy. And that works well even at room temperature. Okay, all right. So how do I get the number of modes, and then I can get the conductance? So remember, the number of modes, we have this general expression that works for any dispersion. It's the density of states. And we now know that times the average velocity in the direction of transport times H over 4, and then it's proportional to the width of the sample. So for graphene, we have the density of states. The velocity is independent of energy in graphene. That's because of its unusual dispersion, the linear dispersion. But I still have to average over angle, and we've done that a few times. So when I do that average over angle, just to get the x directed velocity, it's 2 over pi times the magnitude of the velocity. And since the magnitude of the velocity is independent of energy, when I multiply out and find what the number of channels is at a particular energy, it is also going to increase linearly with energy. So I have these two expressions for density of states. And for a number of channels, they both increase linearly with energy. That's a little different than when I did it for parabolic bands, I had different energy dependencies of the density of states and the number of modes because the velocity was energy dependent. In this case, I don't. Things are even simpler. The graphene is just like silicon, but a lot simpler. You can use t equals 0, calculations, things are easy. Okay, so we want conductance. There's our general expression. The second one is for near equilibrium. We know the Fermi function, transmission is going to be the same. We have to talk a little bit about what the mean free path might be for graphene. We know the number of channels, so it's easy to compute this. When I do the computation, I'm going to assume t equals 0, even at room temperature. So that derivative minus the FDE is just a delta function at the Fermi energy. So I can get simple expressions that work pretty well. So the expression is just, technically, I made the 0k assumption, but in practice it works pretty good at room temperature as well. 2q squared over h times the transmission at the Fermi energy times the number of channels at the Fermi energy. So if I'm able to, say, vary the Fermi energy up or down, I could modulate the conductance. So when I push the Fermi level up in the conduction band, I'll get a conductance that just increases linearly with the Fermi energy, assuming that the transmission doesn't change with energy too much. And if I go down, I'll also have states there. So I'll get a conductance that increases linearly with Fermi energy also. Now, in practice, the way you control the Fermi energy is by controlling the electron density. So usually people have a gate and they can control n and sub s, not the Fermi energy. So the plot will usually be g versus n sub s. So what should that look like? Well, if I plot it versus n sub s, we already determined that n sub s goes as the square of the Fermi energy. So that means that the number of channels, which is proportional to Fermi energy, goes as a square root of n sub s. So if I were to plot the expected conductance versus carrier density, which is what I can do in the lab, I would get a characteristic that looks nonlinear, looks like that. Assuming that the transmission is about constant, if it's got some strong energy dependence, it'll change this. So these are our expected results. If I go to a finite temperature, I'm going to have some spread of the Fermi function. Once I get away from that Dirac point, that spread really isn't going to matter because it's relatively small and things are not changing abruptly there. It could matter a lot right near the Dirac point. But as I'll mention a little bit later, we need to stay away from the Dirac point because things get very messy and complicated there. Near the Dirac point, if I put the Fermi level at the Dirac point, my t equals 0 assumption will say there are no states there, so there is no conductance there. Well, there's a little bit of thermal spread, so I'll get some states above the Dirac point contributing in some states below. But in practice, something else happens that is even more important. So this is not too much of a detail to worry about. All right, so at finite temperature in an ideal system, the red line is what I would expect to get. Okay, so these are the key equations. The first one describes the conductance. Whether the Fermi level is above or below the conduction band, we can, as long as we're away from the Dirac point, we can describe the conductance with that expression. Technically, for t greater than 0, I would have to work out a Fermi-Dirac integral and I have contributions from both bands. I'm going to have to talk about what the mean free path might be. People usually like to express this in terms of a sheet conductance. So you just express the conductance as sheet conductance times W over L. And what's usually reported in experiments is the sheet conductance. So the sheet conductance, if we work it out like we've done before, would be given by this expression. Where lambda apparent, remember, is the actual mean free path if I have a diffusive sample. But if I have a sample that may be between the ballistic and diffusive regime, it's the smaller of the length of the channel and the physical mean free path. This is actually a little bit important in graphene, because it has very high mean, long mean free paths. And you can see these ballistic effects. Okay, so let me talk a little bit about scattering. So if we want to understand this in more detail, we have to understand something about the mean free path. Remember, it's a mean free path for back scattering. So we have to understand something about scattering and graphene. So as I mentioned earlier, but for many scattering mechanisms, the way the scattering works is an electron comes in at some energy E. It gets scattered to some final energy. If the scattering is elastic, an acoustic phonons, the ones that scatter electrons most strongly are near the center of the Brillouin zone and they don't carry a lot of energy. So it's almost elastic. So we would expect the probability that you'll scatter per unit time one over tau to be proportional to the number of states that you can scatter to. So it should be proportional to the density of states. The density of states is proportional to energy. That means the scattering time for a mechanism like this, like acoustic phonon scattering, would be proportional to one over energy. So as you get higher and higher away from the direct point and you have more and more states, the time, the scattering time decreases because there's more and more states to scatter to. So our energy mean free path would be proportional to one over, our energy dependent mean free path would be proportional to one over energy. Now it's interesting to ask what would this do, what kind of conductance would we measure if this were the dominant scattering mechanism? Because again, it's straightforward to calculate, but it's a little surprising when you see the answer. But it just comes from this unusual dispersion of silicon. So this is our expression for, let's say, diffusive transport. That's our expression for the sheet conductance. 2q squared over h times mean free path times a number of channels at the Fermi energy. The mean free path for this type of scattering is proportional to one over Fermi energy. The number of channels is proportional to Fermi energy. So the conductance is constant. So this is not usually what we expect. Usually you expect if you have more electrons in the conduction band, you'll have lower conductance. What we have for graphene is it's independent of the number of electrons. If acoustic phonon scattering is dominated, you put twice as many electrons in, you get exactly the same conductance. That happens because you have twice as much scattering. You put in twice as many, you get twice as much scattering. Your mean free path is halved and you get the same conductance. So that's a very unusual thing that was pointed out several years ago. So you could ask yourself, and I'll refer you to some notes later, you can work out these scattering times for acoustic phonon scattering. And you can put in actual numbers if you know things like deformation potentials, and you could compute what is the conductance of graphene if you just have acoustic phonon scattering. And it turns out that putting in numbers at room temperature, it's about 30 ohms per square. So I think there was a question yesterday about transparent conductors for solar cells or something like that. And people who design solar cells, they like a sheet resistance on the order of 10 ohms per square so that they get good performance. So you can see that this is sort of an intrinsic sheet conductance of graphene because you're always going to have some acoustic phonon scattering at your temperature, the best you can do is 30 ohms per square. So you'd have to do multiple layers of graphene to get its sheet resistance down to a place that you could use it for an application like that. Now in many of these samples, the resistivity is much higher because there are things like charged impurities that can scatter electrons. So let's say I'm looking down on the top. I put a sheet of graphene on a silicon dioxide layer, which is sitting on a wafer. I've made a resistor here, but there are stray charges around. They may be charges in the oxide. They may be charges in the ambient or due to the processing that has been left over. And those stray charges can scatter electrons. So that's the charged impurity scattering that we've talked about a couple of times. So that's going to fluctuate the direct point up and down. It's going to introduce this scattering potential. And again, whenever I have a scattering potential like this, the high energy carriers aren't going to see those fluctuations in potential. They're not going to be so important. So the high energy carriers will scatter less. So this would give me a mean free path that would increase as the energy increases. And actually, if you work out this scattering rate and you do it carefully, you'll get some complicated expressions. But you'll find that in the end that the mean free path is approximately proportional to the energy. The higher the energy, the longer the mean free path. When you're dominated by this type of scattering. So if I go back and ask what would that do to the measured sheet conductance, I start with 2q squared over h times a mean free path times a number of channels that are occupied at the Fermi energy. I now have a mean free path that's proportional to the Fermi energy. That means my conductance is proportional to the Fermi energy squared. Remember that the carrier density was proportional to the Fermi energy squared. For this particular scattering mechanism, the conductance is proportional to the carrier density, which is what you would expect. So you'd expect to see a constant mobility if you do a calculation like this. Okay, so let's talk a little bit about how you do these experiments and look at some data. So in practice, people do near equilibrium transport measurements of graphene to try to understand the properties of graphene and how good their graphene is. The way they'll do it is that they will use a gate to move the Fermi energy up and down. And typically in the experiment, you have a layer of graphene on a silicon dioxide layer, you attach two contacts to it, or maybe four, so you can do four probe measurements, you use the electrode on the bottom as a gate to modulate the position of the Fermi level. And what you really control there is the electron density, which is related to the Fermi energy. So this is what it looks like from a side view. There's a layer of graphene on top of a silicon dioxide layer. It's usually something like 300 nanometers thick because if you put one monolayer of graphene on a 300 nanometer thick layer of SiO2 and illuminate it with typical illumination, the interference patterns will be set up in a way that you can see one monolayer of graphene and you can tell you've got it there. So 90 nanometers, I think you can also see it. So you tend to see the layers in one of those two dimensions. Okay, all right, so the idea is we're going to measure the voltage. This would be a two probe measurement that might suffer from contacts, but you could do four probe measurements as well. We use the bottom contact as a gate and we gate it from the bottom. And because of that, I'm not actually showing. We don't actually move the Fermi level. The Fermi level stays constant. It's fixed by the contacts. But we actually move the Dirac point up and down. So if we apply a positive voltage, we would lower the Dirac point and that would put the Fermi level in the conduction band. If we would apply a negative gate voltage, that would pull the Dirac point up and the Fermi level would be in the valence band. So we can sweep the valence band through the Dirac point and measure the conductance. So if you did the measurement, you would get something like this. The minimum, you could identify the minimum and say, well, that's where the Dirac point is at that voltage. It won't typically occur at zero voltage because there will be some work function difference between the gate and the graphene. And there will be some straight charges that are fixed amount that are there that will shift it. So people will typically then report, they'll shift the zero to the origin and they'll report it that way. So it's easy to relate the gate voltage to the sheet carrier density because charge is just capacitance times voltage. Now you have to be a little more careful than that. The graphene has capacitance two. It's really two capacitors in series and the graphene capacitance is something people sometimes call a quantum capacitance. And you can do that calculation more carefully. What you find is that whenever the SiO2 layer is bigger than a few nanometers or so, that this is a very good approximation. The only capacitance you have to worry about is the capacitance of the insulator. So this gives people a very simple way experimentally to control the density of electrons in the graphene. And then you can study how the conductance varies. So here are some measured results. So we can see linear characteristic. We're plotting it versus gate voltage. And N sub s is capacitance times gate voltage. We see a linear characteristic. So right away we think we must be dominated by charged impurity scattering in this particular sample. Now I could go through at any point and I could take my expression. I've measured the sheet conductance and that's reported here. I know what the carrier density is because my gate voltage gives that to me. And I know the insulator capacitance. So I could deduce what the mean free path is at any point. And if you do that up here at a gate voltage of 100 and you take the measured conductance, you deduce what the sheet carrier density is. Plug numbers into that expression. That sheet carrier density will correspond to a Fermi energy of 3 tenths of an EV. So what we find is that at that particular point, the mean free path is 130 nanometers. It's much less, let's see, the length of that resistor is 5,000 nanometers. So it's much less than the length. So I'm in the diffusive regime. I don't have to worry about ballistic transport. If I go down here to a lower voltage, I can pick off the sheet conductance there. I've got a lower carrier density that I can figure out. The Fermi energy is 0.2 EV above the direct point at this point. And I can solve that expression I showed earlier and find out what the mean free path is. And it's about 90 nanometers, so it's shorter. So the ratio of the mean free path is about 69%. The ratio of the energies is about 67%. So it looks like the mean free path is increasing linearly with energy. That's what you expect for charged impurity scattering. And that's likely what's going on here. Since the conductance is proportional to the carrier density in this regime, I can deduce a mobility. If I were to deduce a mobility from the slope of that line, it's about 12,000 or so. Nice high mobility. Another reason that people are interested in graphene. Now, this region near the bottom, I mentioned that we usually try to stay away from that regime. The reason is that if you look, this is I guess a scanning tunneling electron. Using some scanning probe measurements to try to determine what the electron density is versus position in graphene. And what you find is that the graphene isn't perfect. It's got charges and variations that are moving the direct point up and down. So when you get near the direct point, you see these fluctuations. You can't assume when I gate this structure with a voltage that should put the Fermi level at the direct point, it's not going to be at the direct point everywhere. So you have to average over all of these fluctuations, the electron hole puddles. And that's why the description near the direct point gets more complicated. If I get the Fermi energy way above those potential fluctuations, then they're not so important. And the theory that I discussed works just fine. Okay, now these are some experiments where they controllably put in potassium dopants. So they introduced charged scattering centers in a controllable way. And you can see the first line there on the right is the initial sample where it wasn't exposed to phosphorous at all. And you get a characteristic that's slightly linear. And as you expose the characteristics to more and more phosphorous, the characteristics get more and more linear, and the slope gets less and less. So that's what we would expect then for charged impurities scattering is that we take an initial sample that doesn't have a lot of charged impurities. And as we increase charged impurities, they just get more and more, we just get more and more scattering and we get a lower and lower slope. Notice that the neutral point also shifts because the background charge that there is shifts the electrostatics as well. So I could look at that first one and I could say that, well, I know that way back when I did it ballistically, I told you that the ballistic characteristic should go as the square root of gate voltage or carrier density. So I could look at that initial one and I could ask how close is it to the ballistic limit? Am I dealing with a diffusive sample or a ballistic sample? So again, we had this expression here which relates to measured conductance and the carrier density that we've induced with a gate voltage to the mean free path. So I could just go there and read off data and estimate the mean free path. And it's about 164 microns. And I forget in this particular sample, I think the sample length is over a micron. So we're in a diffusive regime. So that nonlinear characteristic is not coming from ballistic transport. It's coming from some other scattering mechanisms. Now this is another experiment. This is done at Columbia. This is an initial sample. You see the blue line where the graphene was sitting on the substrate. And then they could etch out the underlying substrate so that the graphene was just suspended and they could anneal it and drive off all of the stray charges. So they had a much more pure sample. And what they see is higher conductance and the initial linear characteristic is converted into a characteristic that's nonlinear. Now you could ask on this, is it near the ballistic limit or is it near the diffusive limit? Well if you look at that length scale up there, the length of the resistor that they're measuring, you can see a four probe pattern there to do four probe measurements and Hall Effect measurements. You can see that the length is on the order of a couple of microns and the mean free path that they extract is on the order of a micron. So we're probably getting close to a ballistic transport. If we just use our expression in there, we can deduce an apparent mean free path. Notice the one that we get is a little bit longer than the one that they quoted in the paper because ours is this mean free path for back scattering, this landauer mean free path that's got the pi over 2 in it. So it's a little bit longer. But you can see that we're getting close to the dimensions of the sample. So at this regime, we're starting to approach ballistic transport. So it's possible with very clean graphene to get close to the ballistic limit. Now these are some other measurements, these are temperature dependent measurements. And now this isn't conductance, it's one over conductance. It's resistivity. And what you're seeing here on the right are plots of resistivity versus temperature for two different graphene samples. And the main point I want you to notice there is just the two different regimes. There's an initial regime where the resistivity increases linearly with temperature, and then it starts to increase much more rapidly with temperature. So for low temperatures below 100K, the resistivity or resistance is proportional to T. But above that, the resistivity starts to go exponentially with temperature. Now there you start to suspect scattering by optical phonons. Because the population of optical phonons depends on their energy, and it goes as something like H bar omega over KT. So that could be optical phonons in graphene. It turns out that people really think that what's happening is that those are optical phonons in the SiO2 underneath the graphene. Because they have the energies that seem to explain when this starts to kick up. So you can understand the general characteristics very simply. The resistance is one over the conductance. So the resistance is proportional to one over the mean free path. That means the resistance is proportional to one over tower, the scattering rate. The scattering rate should be proportional to the number of phonons that are there to scatter from. The number of phonons is given by that Bose-Einstein factor. So the resistance should be proportional to that Bose-Einstein factor. Acoustic phonons have small energy. So H bar omega tends to be small compared to KT. That means that exponential, I can expand it for small argument. And that little n and capital N are the same thing. But the n naught then becomes KT over H bar omega. That's very intuitive. Thermal energy is KT. H bar omega is the energy of the acoustic phonon. I just divide the two and I get the number. So if acoustic phonons were dominating the scattering process, I would expect the resistance to be proportional to temperature. That's what we saw initially. Optical phonons have much bigger energies. They're comparable to KT. So I can't do that Taylor series expansion of the exponential. I'm just going to expect the resistance to have that exponential characteristic. Once the thermal energy gets comparable to H bar omega, I should start to increase the population of optical phonons and I get more scattering. That's why it increases. So these are the kind of, as I mentioned, a lot of what people use near equilibrium transport measurements for is when you have a new material that you're working with. You do these kinds of measurements, you vary the temperature and you try to deduce what's going on. What is the band structure? Is it what I think it is? What scattering mechanisms are controlling the performance? And there's been a lot of work over the past three or four years where people have done a lot of these near equilibrium measurements and tried to sort this all out. And the general picture is something like this. We said, if you have a ballistic device, we worked it out and we said the conductance should go as a square root of voltage, square root of carrier density. If we're dominated by acoustic phonons scattering, let's say we have a pure sample. We're at a low enough temperature that we're not exciting optical phonons. We have no charged impurities. Then we said that the conductance is independent of gate voltage. So it's just that horizontal line. And if the conductance is dominated by charged impurity scattering, it gives a linear characteristic. And in practice, we might have some contribution to charged impurities. And we might have some contribution due to acoustic phonons. So we could expect to see some kind of nonlinear characteristic, that is sort of the sum of those two. So that initial, remember when I talked to you about that experiment with phosphorus doping, the initial one before any charged impurities were intentionally put in was a little bit nonlinear. It was probably a little bit nonlinear because we were seeing a combination of some background unintentional charged impurity scattering. But it was weak enough that we were beginning to see the limitations due to acoustic phonons scattering as well. And when we intentionally introduced more scattering centers, then we were just totally dominated by the charged impurity scattering and the other one was so higher that we didn't see it. Okay, now I should mention, if you do these two probe measurements, you'll tend to get a characteristic that looks like this. It won't be symmetrical. The conduction and valence band of graphene are perfectly symmetrical. And when people first saw this, it wasn't obvious what the difference might be. What could be causing it? There are a number of things. One of the things is if you look at how an electron scatters off of a positive charge and how an electron scatters off of a negative charge, it's different. So on one branch, we have electrons scattering and on the other branch, we have holes scattering. So the sign of the interaction is changing. If you calculate the scattering rate by Fermi's golden rule, it doesn't make any difference. But if you do the scattering calculation a little more carefully, when there's an attractive potential between the two, it scatters a little more strongly than when there's a repulsive one. So that was hypothesized and that may be part of the answer. But the main answer seems to be the asymmetries are coming from the contacts. That there's some nonlinear characteristic and the IV characteristic that comes from the contacts. So you have a different contact resistance on the N branch than you do on the P branch. You can eliminate that. You can test that hypothesis by doing a four probe measurement. And when they did a four probe measurement, they found that the characteristics were symmetrical. So most of it seems to be coming from the contacts. And that's something that people worry about here now. Okay, so all right, there's something a little interesting here. I want to just mention it because it's kind of cute. This is our expression for sheet conductance, okay? And I can always equate my sheet conductance to Nq mu. And then I can derive a theoretical expression for mobility. So let's do that. We have an expression for the number of channels. We have an expression for the mean free path. We have an expression for the carrier density. I can just solve the top equation for the mobility. And if I solve the top equation for the mobility, this is the expression I'll get. Q tau over EF squared over Fermi velocity squared. That looks like Q tau over M. It looks like the effective mass of graphene is Fermi level over BF squared. Now that's actually kind of cute. That sort of looks like a relativistic expression. E is equal to MC squared. And this has a dispersion that's like light. So that may not be unexpected. Remember, people do Hall effect measurements in graphene and they worry about the cyclotron resonance frequency. For a parabolic band, the cyclotron resonance frequency is QB over M. What is the cyclotron resonance frequency in graphene? It turns out it's QB over this M. You get the right answer for that too. So it's actually a little more than fortuitous. If you look at Professor Dada's notes, probably in the ones that he put on the flash drive. He has an argument that a better way to define effective mass for transport is that it's the ratio of momentum to velocity or crystal momentum to velocity. If you use that definition and the dispersion of graphene, you'll get this answer for the effective mass of graphene. If you use that definition of effective mass and a parabolic dispersion, you'll get the normal effective mass. So people say graphene has no effective mass, but there are cases where if you're careful about thinking about it, you can assign an appropriate effective mass to it. All right, if you're looking for a challenge, you can think about working out the Hall effect in graphene. The answer is in the notes that I put on the flash drive, but it takes a little bit of effort. Also one of the interesting features of graphene is that it has an unusually high thermal conductivity. So you could try to think about why it has a high thermal conductivity. See if you could figure that out. Okay, so that's basically it. And as I said, the point was just to show you how if you encounter a new material and you start reading papers, it can frequently look very strange. But if it's a problem in electrical transport, the first thing you want to do is to go back and say, well, can I understand these ideas in terms of the concepts that I know from other materials? And it turns out you can. So graphene has some really interesting features that come from this dispersion. But in terms of understanding all of these, all we have to do is to explain the same concepts that start with a different dispersion, then everything falls out pretty naturally. There are a lot of interesting things, applications that people are exploring these days, it's a little hard to see which ones are going to end up being important, it's likely that some of them will be. But it also makes a wonderful set of homework problems because the dispersion is so simple, it's so easy once you learn things with parabolic bands to give a homework exercise, work it out for graphene and it's easy to do analytically. Okay, if you want a lot more information about this, we actually had one of the previous summer schools. We had a set of four or five lectures on graphene, so I can refer you to those. We have a set of notes where we work out a lot of these things like densities, escapes, states, and scattering times. And my student Dionysus was a co-author on this, the lead author on this, so you can find that if you're interested in it too. Of course, there's a lot of information on graphene available in many different places. Okay, so I'll stop there and see if you have any questions. Yes, skip. When you first started talking about a lot of purities, they thought, well, maybe it's going to be an SA2 or interstitial or something like that. But then when you run into a graphene and purity, I thought I could use it in a crystal graphene glass. Yeah. So it's in the crystal graphene glass that it's possible to shift the frontal lobes with these purities. And as soon as it wasn't there, as you know, we talked about changing the shape of the structure, which I've been talking about for a long time. Yeah. I'm going to confuse these purities in the crystal structure. Yeah, so the question is where are these, these imperities, we're not doping graphene with substitutional doping, right? So the imperities that I'm speaking about here, like the phosphorous, these are not in the graphene honeycomb structure. You're coating the graphene with something on top. And the charge then is affecting the Dirac point and moving things up down. So it might be analogous to modulation doping for semiconductor people. And the background imperities, when you take a sample and put it on an SA2 layer and then measure the characteristics and you see a linear characteristic, you know, there are impurities somewhere. A lot of them tend to be in the SIO2 that you did the processing on. A lot of them tend to be left over from the ambient when you did all of the processing on the materials. You can heat it up and try to drive some of them off. But it's only one monolayer thick, so it's very sensitive to whether there are any charges in the environment close to it. Yeah. When you look at the sheet, they show sheet tend to be on the order of 20 and 20, but there is no steps. There is no what? There is no quantization steps that we expect. There is no quantity. Oh. You mean there is no quantized conductance? Yeah. So all you're right that in the characteristics I showed there was no evidence of quantized conductance. And that's because these were all very wide structures for which there were many modes. And when I go to very wide structures, then the conductance is just proportional to the width of the resistor. Now, you know, when you experimentally early on when I showed you that experiment that showed these steps, that was a very narrow conductor. And I could electrically control the width of the conductor through a shot key barrier, so I could then deplete it and I could individually count the number of modes. Now, I don't think I've seen that in graphene. If you try to make a graphene ribbon, you know, a lot of research these days is devoted to how do we induce a band gap in graphene so that we can make a better transistor out of it and turn it off. And one of the ways is to make a very narrow strip. And then you could start to count modes. And the difficulties there is you have so much roughness and disorder at the edges that it's hard. The challenge is to try to figure out ways to make smooth edges that don't introduce so much scattering and randomness that it affects the transport. Any other questions? Right down here. We know graphene consists of a number of modes and I'm learning the experience when we change some of the levels of this atom by other atoms. For example, the volume of other atoms. Yeah. So where you would substitute other experiments, where you would substitute other atoms for carbon in them. Yes, it works. Yeah. Yeah. It doesn't mean that there aren't, but I'm not aware of any. These carbon-carbon bonds are very strong so it's hard to do substitutional doping the way that you do in silicon. The limits of force. Pardon me? We have other elements of force group, which have also force. Right, but you have to have enough thermal energy to break a bond and dope it or you have to grow it in that way. And maybe people have looked at that, I'm not aware of anything. There are a variety of other materials. The thing that was nice about graphene is you could take a chunk of graphite and then you could rub it across an SIL2 layer and the individual layers would peel off. And if you learn how to do that just right, you can get one monolayer. That's what the folks who got the Nobel Prize learned how to do. There are other layered materials like this where the bonding is very strong within a layer but very weak between layers. People are interested in other materials like this where you can produce single monolayers of various materials. So there may be other examples of materials that are one monolayer thick that are of interest for electronic applications. In fact, there are. But it's just amazing how quickly this Nassau-Love and Geim learned how to do this. Let's see, they got the layers, I think they use Scotch tape to put Scotch tape on graphene and pull off one monolayer and then transfer it to an SIL2 layer. Now you can get graphene layers over 12-inch silicon wafers. It's just amazing how quickly, people have done it in roll-to-roll processing now where you can have large sheets. It's amazing how quickly it has come in just a few years. Now those layers tend to have many more defects. They're polycrystalline, they have grain boundaries, it's scattered, but the transport of the individual grain is described by concepts like this. Yes. And I like that Tuskegee, we've experimented with graphene, actually making graphene product among forgathal clay. We also can challenge some impurities with silicon, some aluminum that we try to wash away with that phenol ethyl. But using the furnace, we've tried to actually graphene with cadmium sulfide. We have like quantum dots on top of this sheet. I was wondering, by using an acoustic wave through ultrasound with that, using graphene oxide, because it does have impurities. Because the graphene that we order is a graphene powder. It's just sort of a mesh, how small, 400 nanometers or something like that. But we can't look copper as the impurity. It won't be pretty on the TEM, we said that it was 97% pure. But it wasn't actually, we had our copper along with it. So this is, you're saying you purchased graphene or you obtained it from another source? So you don't? Purchase and we obtained the cheap graphene by furnace under high temperature. So we're able to produce graphene, but it's not pure graphene. We actually have pureness coming up. And I think one of the growth techniques for this large area of graphene now involves growing it on copper. The copper catalyzes it. But then it's maybe not surprising that you find some copper in the graphene. It's not surprising, but this is from the super market graphene, the super market that we purchased. It's taking that as none, so I'm just saying you have to be careful about what you do. And so what you would probably want to do is to do some transport characterization and see what kind of mobilities you're getting. See if the characteristics look linear and if you can identify this as charged impurity scattering or something else. Thank you.