Palestinian Advanced Physics School Condensed Matter

Palestinian Advanced Physics School

Condensed Matter Physics

Professor David Tong Lecture 2: Fermi Surfaces Review of Lecture 1

In the last lecture, we looked at a single electron moving in a lattice. The energy spectrum forms a band structure with gaps.

Figure 4: Energy dispersion for the free electron model.

In this lecture,Our we real look interest at what is what happens happens closewhen to we the have edge of many the Brillouin electrons. zone when is

small compared to the gap Vn. In this case we can expand the square-root to give 2 2 2 2 2 2 2 ~ n ⇡ ~ 1 n ~ ⇡ 2 E + V0 Vn + 1 ± ⇡ 2m a2 ±| | 2m ± V ma2 ✓ | n| ◆ The ﬁrst collection of terms coincide with the energy at the edge of the Brillouin zone (1.15), as indeed it must. For us, the important new point is in the second term which tells us that as we approach the gaps, the energy is quadratic in the momentum .

Band Structure We now have all we need to sketch the rough form of the energy spectrum E(k). The original quadratic spectrum is deformed with a number of striking features: For small momenta, k ⇡/a, the spectrum remains roughly unchanged. • ⌧ The energy spectrum splits into distinct bands, with gaps arising at k = n⇡/a • with n Z. The size of these gaps is given by 2 V , where V is the appropriate 2 | n| n Fourier mode of the potential.

The region of momentum space corresponding to the nth energy band is called the nth Brillouin zone. However, we usually call the 1st Brillouin zone simply the Brillouin zone.

–13– p3 + q3 = r3 p3 + q3 = r3

Z = ¯ expZ = S[ , ¯] ¯ exp S[ , ¯] D D D D Z Z ¯ ¯ 4 ? µ⌫ ¯ Z ¯ exp S[4 , ]+? iaµ↵⌫ d xFµ⌫ F Z exp! SD[ ,D ]+ia↵ d xFµ⌫ F ! D D Z ✓ Z ◆ p3 + q3 = r3Z ✓ Z ◆

Z = ¯ exp S[ , ¯] a = something D D a = something ! Z L R L R ! X X X X ¯ ¯ 4 ? µ⌫ G = U(1) Sp(r) SU(N) Z exp S[ , ]+ia↵ d xFµ⌫ F ⇥ ⇥ ! D D G = U(1) Sp(r) SU(N) Z ✓ Z ⇥◆ ⇥ G = SU(5) with 5 and 10 a = something G = SU(5) with 5 and 10 L R ! X X G = U(1) Sp(r) SU(N) G = SU(N)with(N 4) anti-fundamentals 5 and 10 ⇥ ⇥ 3. Fermi SurfacesG = SU(N)with(N 4) anti-fundamentals 5 and 10 3. Fermi Surfaces ikx InIn theG the= previous previousSU(5) chapter chapter with 5 we weand have have10 seen seen how how the the single-electron single-electron energy energy (x)= states statese form form abandstructureinthepresenceofalattice.Ourgoalnowistounderstandtheabandstructureinthepresenceofalattice.Ourgoalnowistounderstandthe ikx consequencesconsequences of of this, this, so so that that we we can can start start to to get get a a( feel feelx)= for for somee some of of the the basic basic properties properties ofof materials. materials. ⇡ ⇡ G = SU(N)with(N 4) anti-fundamentals 5 and 10 k< ThereThere is is one one feature feature in in particular particular that that will will be be important: important: materials materials a don’t don’t just just have havea ⇡ ⇡ oneone electron electron sitting sitting in in them. them. They They have have lots. lots. A A large large part partk< of of condensed condensed matter matter physics physics isis concerned concerned with with in in understanding understanding the the collective collective behavioura behaviour of ofa this this swarm swarm of of electrons. electrons.2 2 2 ikx 1 2 p ~ k This can often involve(x)= thee interactions between electronsE giving= risemv to= subtle and= This can often involve the interactions between electrons giving2 rise to subtle2m and 2m surprisingsurprising e e↵↵ects.ects. However, However, for for our our initial initial foray foray into into this this problem, problem, we we will will make make a a fairly fairly 2 2 2 brutalbrutal simpliﬁcation: simpliﬁcation: we we will will ignore ignore the the interactions interactions1 2 between betweenp electrons. electrons.~ k Ultimately, Ultimately, much of the basic⇡ physics that⇡ we describeE below= ismv unchanged= if= we turn on interactions, much of the basic physicsk< that we describe below2 is unchanged2m if we turn2pm= onmv interactions,= k althoughalthough the the reason reasona  for for this thisa turns turns out out to to be be rather rather deep. deep. ~

3.13.1 Fermi Fermi Surfaces Surfaces 1 p2 2k2 p = mv = ~k 2 3 EvenEven in in the the absence absence2 of of any anyFermi interactions, interactions,~ Surfaces electrons electrons still still are are still still a a↵↵ectedected by by the~ the presence presence E = mv = = E = k2 ofof others. others. This This2 is is because because2m electrons electrons2m are are fermions, fermions, and and so so subject subject to to the thePauliPauli2m exclusion exclusioni principleprinciple.. This This is is the the statement statement that that only only one one electron electron can can sit sit in in any any given given state. state.i=1 As As 3 X Let’s firstwewe willthink will seeabout see below, below, a Fermi the the surface Pauli Pauli exclusionwithout exclusion a lattice principle, principle, coupled coupled2 with with the the general general features features of of ~ 2 2⇡n band structure,p goes= mv some= ~ wayk towards explainingE = the main propertiesk of materials.i Considerband a single structure, electron goes in a some box of way size towards L. The energy explaining is the2m main propertiesi ki of= materials. i=1 L FreeFree Electrons Electrons X 2 3 AsAs a a simple simple example, example,~ suppose suppose2 that that we we have have no no lattice. lattice.2⇡ We Weni E = k with k = n kkzzZ taketake a a cubic cubic box, box, with with2m sides sides ofi of length lengthLL,, and and throw throwi in in some some i i=1 L 2 largelarge number number of of electrons. electrons.X What What is is the the lowest lowest energy energy state state of of thisthis system? system? Free Free electrons electrons sit sit in in eigenstates eigenstates with with momentum momentum 22⇡n2 i ~~kkandand energy energyEkE===~~2kk2//22mm.. Because Because we we have have a a system system of of Make some assumptionsi L finitefinite size, size, momenta momenta are are quantised quantised as askki i=2=2⇡⇡nni/Li/L.. Further, Further, 2 theythey also also• carry carryElectron one one ofhas of two twotwo spin spinspin states, states: states, andoror . . 2 |"i|"i |#i|#i kkxx kkyy TheThe first first• Electrons electron electron cando can not sit sit interact in in the the statewith state eachkk== other 0 0 with, with, say, say, spin spin .. The The second second electron electron2 can can also also have havekk=0,butmusthave=0,butmusthave FigureFigure 18: 18: TheThe Fermi Fermi |"i|"i spinspinWe, also, opposite opposite need toa to key the the principle first. first. Neither Neither of physics: of of these these the electronsPauli electrons exclusion costs costs principlesurfacesurface |#i|#i anyany energy. energy. However, However, the the next next electron electron is is not not so so lucky. lucky. The The • No two electrons can occupy the same state minimumminimum energy energy state state it it can can sit sit in in has hasnni i=(1=(1, ,00, ,0).0). Including Including spin spin and and momentum momentum therethere are are a a total total of of six six electrons electrons which which can can carry carry momentum momentum kk =2=2⇡⇡/L/L.. As As we we go go on, on, | | | | wewe fill fill out out a a ball ball in in momentum momentum space. space. This This ball ball is is called called the theFermiFermi sea seaandand the the boundary boundary

–36––36– 3. Fermi Surfaces

3. FermiIn Surfaces the previous chapter we have seen how the single-electron energy states form abandstructureinthepresenceofalattice.Ourgoalnowistounderstandthe In theconsequences previous chapter of this, we so have that seen we can how start the to single-electron get a feel for some energy of thestates basic form properties abandstructureinthepresenceofalattice.Ourgoalnowistounderstandtheof materials. consequences of this, so that we can start to get a feel for some of the basic properties of materials.There is one feature in particular that will be important: materials don’t just have one electron sitting in them. They have lots. A large part of condensed matter physics Thereis is concerned one feature with in particularin understanding that will the be collective important: behaviour materials of this don’t swarm just have of electrons. one electronThis sitting can often in them. involve They the have interactions lots. A large between part of electrons condensed giving matter rise physics to subtle and is concernedsurprising with in e↵ understandingects. However, the for our collective initial behaviour foray into ofthis this problem, swarm we of electrons. will make a fairly 3. FermiThis Surfaces canbrutal often simplification: involve the interactions we will ignore between the interactions electrons giving between rise electrons. to subtle and Ultimately, surprisingmuch e↵ects. of the However, basic physics for our that initial we describe foray into below this is problem, unchanged we willif we make turn on a fairly interactions, In the previousbrutal simplification:although chapter the we reason we have will for seen ignore this how turns the the interactions out single-electron to be rather between deep. energy electrons. states Ultimately, form abandstructureinthepresenceofalattice.Ourgoalnowistounderstandthemuch of the basic physics that we describe below is unchanged if we turn on interactions, consequencesalthough of this,3.1 the so Fermi reason that Surfaces we for thiscan turns start out to get to be a feel rather for deep. some of the basic properties of materials. 3.1 FermiEven Surfaces in the absence of any interactions, electrons still are still a↵ected by the presence of others. This is because electrons are fermions, and so subject to the Pauli exclusion There is one feature in particular that will be important: materials don’t just have Even inprinciple the absence. This of any is the interactions, statement electrons that only still one are electron still a↵ canected sit by in the any presence given state. As one electronof sitting others.we in This will them. is see because below, They electrons have the Pauli lots. are exclusion A fermions, large part principle, and of so condensed subject coupled to with the matterPauli the physicsgeneral exclusion features of is concernedprinciple withband in. This understanding structure, is the statement goes the some collective that way only towards behaviour one electron explaining of can this the sit swarm main in any properties of given electrons. state. of materials. As This can oftenwe will involve see below, the the interactions Pauli exclusion between principle, electrons coupled giving with rise the general to subtle features and of surprising eband↵ects. structure,Free However, Electrons goes for some our way initial towards foray explaining into this problem, the main properties we will make of materials. a fairly brutal simplification:As a we simple will example, ignore the suppose interactions that we have between no lattice. electrons. We Ultimately, Free Electrons kz much of the basic physicstake a cubic that webox, describe with sidesFermi below of length is Surfaces unchangedL, and throw if we turnin some on interactions, As a simple example, suppose that we have no lattice. We although the reasonlarge for number this turns of electrons. out to be What rather is the deep. lowest energy state of kz take a cubicthis system? box, with Free sides electrons of length sit inL, eigenstates and throw with in some momentum large number of electrons. What2 2 is the lowest energy state of 3.1 Fermi Surfaces~k and energy E = ~ k /2m. Because we23 have a system of thisNow system? we throw Free N electrons into sit in the eigenstates box. (With withN=10 momentumroughly). What is the ground state? finite size, momenta are quantised as ki =2⇡ni/L. Further, 2 2 Even in the~k absenceand• they energyThe of first also anyE electron carry= interactions,~ k one has/2m of k=0. two Because electronsand, spin say, states, we spin still have are aor system still. a↵ ofected by the presence of others. Thisfinite is size, because momenta electrons are quantised are fermions, as ki =2 and⇡n|i so"i/L subject. Further,|#i to the Paulikx exclusion ky they also• The carryThe second first one electronofelectron two spin canhas states, k=0 sit in and the spin stateor k. = 0 with, say, spin principle. This is the statement that only one|"i electron|#i can sit in any given state. As . The second electron can also have k =0,butmusthavekx Figure 18: kThey Fermi we will see below,The first|"i the electron Pauli can exclusion sit in the principle, state k = coupled 0 with, withsay, spin the general features of • spinThe next, oppositeelectron must to the have first. non Neither-zero momentum. of these electrons We can costsset k=2surfacep/L. band structure,. The goes second• someThere|#i electron way are six towards states can also with explaining have this kenergy=0,butmusthave the (three main directions; properties Figurespin ofup materials. or 18: down)The Fermi |"i any energy. However, the next electron is not so lucky. The spin , opposite to the first. Neither of these electrons costs surface |#i• minimumNow we keep energy going. state As we it can throw sit in in more has electrons,ni =(1, 0 they, 0). have Including to have spin higher and momentum Free Electronsany energy. However, the next electron is not so lucky. The theremomentum are a total and ofhigher six electronsenergy. which can carry momentum k =2⇡/L. As we go on, minimum energy state it can sit in has n =(1, 0, 0). Including spin| and| momentum As a simple example,we fill suppose out a ball that in momentum we have space. noi lattice. This ball We is called the Fermi sea and the boundary there are a total of six electrons which can carry momentum k =2⇡/Lk.z As we go on, take a cubic box, with sides of length L, and throw in some | | we fill out a ball in momentum space. This ball is called the Fermi sea and the boundary large number of electrons. What is the lowest energy state of this system?The Free electrons electrons fill out sit a in ball eigenstates in momentum with space momentum–36– ~k and energy E = ~2k2/2m. Because we have a system of –36– finite size, momenta are quantised as ki =2⇡ni/L. Further, they also carry one of two spin states, or . |"i |#i kx ky The first electron can sit in the state k = 0 with, say, spin . The second electron can also have k =0,butmusthave Figure 18: The Fermi |"i spin , opposite to the first. Neither of these electrons costs surface |#i any energy. However, the next electron is not so lucky. The minimum energy state it can sit in has ni =(1, 0, 0). Including spin and momentum there are a total of six electrons which can carry momentum k =2⇡/L. As we go on, | | we fill out a ball in momentum space. This ball is called the Fermi sea and the boundary

–36– 3. Fermi Surfaces

In the previous chapter we have seen how the single-electron energy states form abandstructureinthepresenceofalattice.Ourgoalnowistounderstandthe consequences of this, so that we can start to get a feel for some of the basic properties of materials.

There is one feature in particular that will be important: materials don’t just have one electron sitting in them. They have lots. A large part of condensed matter physics is concerned with in understanding the collective behaviour of this swarm of electrons. This can often involve the interactions between electrons giving2 rise3 to subtle and ~ E = k2 surprising e↵ects. However, for our initial foray into this problem,2m we will makei a fairly i=1 brutal simpliﬁcation: we will ignore the interactions between electrons.X Ultimately, much of the basic physics that we describeFermi below is Surfaces unchanged if we turn on interactions, although the reason for this turns out to be rather deep. 2⇡ni k = i L 3.1 FermiEverything Surfaces in the Fermi surface is called the Fermi-something

Even in the absence• This inside of any is interactions,the Fermi sea electrons still are still a↵ected by the presence of others. This is because electrons are fermions, and so subjectn toi theZ Pauli exclusion 2 principle. This• The is the edge statement is the Fermi that surface only one electron can sit in any given state. As we will see below,• The theelectrons Pauli on exclusion the edge have principle, Fermi momentum coupled with, kF the general features of band structure, goes some way towards explaining the main properties2 2 of materials. ~ kF • The energy of this last electron is the Fermi energy E = F 2m Free Electrons As a simple example, supposeAcknowledgements that we have no lattice. We An important fact: kz take a cubic box, with sides of length L, and throw in some large numberOnly ofthe electrons. electronsWe near What are the supported Fermi is the surface lowest by can energy the contribute European state to of Research Council under the European Union’s this system?any Freedynamical electrons processSeventh sit. in Framework eigenstates Programme with momentum (FP7/2007-2013), ERC grant agreement STG 279943, 2 2 ~k and energyAll othersE are= ~ trapped.k“Strongly/2m . Because Coupled we Systems”. have a system of finite size, momenta are quantised as ki =2⇡ni/L. Further, they also carry one of twoReferences spin states, or . |"i |#i kx ky The first electron can sit[1] in M. the E. Peskin,state k “=Mandelstam 0 with, say, ’t spin Hooft Duality in Abelian Lattice Models,” Annals Phys. . The second electron can also have k =0,butmusthave Figure 18: The Fermi |"i 113, 122 (1978). spin , opposite to the first. Neither of these electrons costs surface |#i any energy. However, the next electron is not so lucky. The minimum energy state it can sit in has ni =(1, 0, 0). Including spin and momentum there are a total of six electrons which can carry momentum k =2⇡/L. As we go on, | | we fill out a ball in momentum space. This ball is called the Fermi sea and the boundary

–36–

3 Fermi Surfaces with a Lattice

How does this change in a solid where there is a lattice?

• The energy spectrum for a single electron forms bands.

• The bands sit within Brillouin zone

Let’s consider electrons moving in two-dimensions. Throw in electronsFigure and 4: Energy they dispersion form for a the Fermi free electron surface model.

Our real interest is what happens close to the edge of the Brillouin zone when is

The region of momentum space corresponding to the nth energy band is called the nth Brillouin zone. However, we usually call the 1st Brillouin zone simply the of the ball is called the Fermi surface. The states onBrillouin the Fermi zone. surface are said to have 2 2 Fermi momentum ~kF and Fermi energy EF = ~ kF /2m. Various properties of the free Fermi sea are explored in the lectures on Statistical Physics. –13–

3.2 Metals vs Insulators Here we would like to understand what becomes of the Fermi sea and, more importantly, the Fermi surface in the presence of a lattice. Let’s recapitulate some important facts that we’ll need to proceed:

A lattice causes the energy spectrum to splits into bands. We saw in Section • 2.6 that a Bravais lattice with N sites results in each band having N momentum states. These are either labelled by momenta in the ﬁrst Brillouin zone (in the reduced zone scheme) or by momentum in successive Brillouin zones (in the extended zone scheme).

Because each electron carries one of two spin states, each band can accommodate • 2N electrons.

Each atom of the lattice provides an integer number of electrons, Z, which are • free to roam the material. These are called valence electrons and the atom is said to have valence Z.

From this, we can piece the rest of the story together. We’ll discuss the situation for two-dimensional square lattices because it’s simple to draw the Brillouin zones. But everything we say carries over for more complicated lattices in three-dimensions.

Suppose that our atoms have valence Z = 1. There are then N electrons, which can be comfortably housed inside the ﬁrst Brillouin zone. In the left-hand of Figure 19 we have drawn the Fermi surface for free electrons inside the ﬁrst Brillouin zone. However, we know that the e↵ect of the lattice is to reduce the energy at the edges

–37– Fermi Surfaces with a Lattice

How does this change in a solid where there is a lattice?

• The energy spectrum for a single electron forms bands.

• The bands sit within Brillouin zone

Let’s consider electrons moving in two-dimensions. Throw in electronsFigure and 4: Energy they dispersion form for a the Fermi free electron surface model.

Our real interest is what happens close to the edge of the Brillouin zone when is

Because each electron carries one of two spin states, each band can accommodate • 2N electrons.

Each atom of the lattice provides an integer number of electrons, Z, which are • free to roam the material. These are called valence electrons and the atom is said to have valence Z.

–37– Fermi Surfaces with a Lattice

How does this change in a solid where there is a lattice?

• The energy spectrum for a single electron forms bands.

• The bands sit within Brillouin zone

Let’s consider electrons moving in two-dimensions. Throw in electronsFigure and 4: Energy they dispersion form for a the Fermi free electron surface model.

Our real interest is what happens close to the edge of the Brillouin zone when is

Because each electron carries one of two spin states, each band can accommodate • 2N electrons.

Each atom of the lattice provides an integer number of electrons, Z, which are • free to roam the material. These are called valence electrons and the atom is said to have valence Z.

–37– Figure 20: Lithium. Figure 21: Copper. of the Brillouin zone. We expect, therefore, that the Fermi surface — which is the equipotential EF —willbedistortedasshowninthemiddleﬁgure,withstatescloser to the edge of the Brillouin zone ﬁlled preferentially. Note that the area inside the Fermi surface remains the same.

If the e↵ects of the lattice get very strong, it may that the Fermi surface touches the edge of the Brillouin zone as shown in the right-hand drawing in Figure 19. Because the Brillouin zone is a torus, if the Fermi surface is to be smooth then it must hit the edge of the Brillouin zone atReal right-angles. Fermi Surfaces

This same physics can be seen in real Fermi surfaces. Lithium has valence Z =1. It forms a BCC lattice, and so the Brillouin zone is FCC. Its Fermi surface is shown above, plotted within its Brillouin zone2. Copper also has valency Z =1,withaFCC lattice and hence BCC Brillouin zone. Here the e↵ects of the lattice are somewhat stronger, and the Fermi surface touches the Brillouin zone.

In all of these cases, there are unoccupied states with arbitrarily small energy above E . (Strictly speaking, this statement holds only in the limit L of an inﬁnitely F !1 large lattice.) This means that if we perturb the system in any way, the electrons will easily be able to respond. Note, however, that only those electrons close to the Fermi surface can respond;Figure those 20: thatLithium. lie deep within the FermiFigure sea 21: areCopper. locked there by the Pauli exclusion principleLithium and require much larger amountsCopper of energy if they wish to escape.of the Brillouin zone. We expect, therefore, that the Fermi surface — which is the

equipotential EF —willbedistortedasshowninthemiddlefigure,withstatescloser Thisto is an the important edge of the point, Brillouin so I’ll zone say filled it again. preferentially. In most Notesituations, that the only area those inside electrons the which lieFermi on surface the Fermi remains surface the same. can actually do anything. This is why Fermi surfaces play such a crucial role in our understanding of materials. If the e↵ects of the lattice get very strong, it may that the Fermi surface touches the 2This,edge and of other the pictures Brillouin of zone Fermi as surfaces, shown in are the taken right-hand from http://www.phys.ufl.edu/fermisurface/ drawing in Figure 19. Because . the Brillouin zone is a torus, if the Fermi surface is to be smooth then it must hit the edge of the Brillouin zone at right-angles.

This same physics can be seen in real Fermi surfaces. Lithium has valence Z =1. –38– It forms a BCC lattice, and so the Brillouin zone is FCC. Its Fermi surface is shown above, plotted within its Brillouin zone2. Copper also has valency Z =1,withaFCC lattice and hence BCC Brillouin zone. Here the e↵ects of the lattice are somewhat stronger, and the Fermi surface touches the Brillouin zone.

In all of these cases, there are unoccupied states with arbitrarily small energy above E . (Strictly speaking, this statement holds only in the limit L of an inﬁnitely F !1 large lattice.) This means that if we perturb the system in any way, the electrons will easily be able to respond. Note, however, that only those electrons close to the Fermi surface can respond; those that lie deep within the Fermi sea are locked there by the Pauli exclusion principle and require much larger amounts of energy if they wish to escape.

This is an important point, so I’ll say it again. In most situations, only those electrons which lie on the Fermi surface can actually do anything. This is why Fermi surfaces play such a crucial role in our understanding of materials.

2This, and other pictures of Fermi surfaces, are taken from http://www.phys.uﬂ.edu/fermisurface/.

–38– Fermi Surfaces with a Lattice

What happens if we keep adding electrons?

Figure 22: FermiWith surfaces no lattice, for valencethe electronsZ = spill 2 with over increasing into the next lattice Brillouin strength, zone moving from a metal to an insulator.

Materials with a Fermi surface are called metals. Suppose, for example, that we apply a small electric field to the sample. The electrons that lie at the Fermi surface can move to di↵erent available states in order to minimize their energy in the presence of the electric field. This results in a current that flows, the key characteristic of a metal. We’ll discuss more about how electrons in lattices respond to outside influences in Section 5

Before we more on, a couple of comments:

The Fermi energy of metals is huge, corresponding to a temperature of E /k • F B ⇠ 104 K, much higher than the melting temperature. For this reason, the zero temperature analysis is a good starting point for thinking about real materials.

Metals have a very large number of low-energy excitations, proportional to the • area of the Fermi surface. This makes metals a particularly interesting theoretical challenge.

Let’s now consider atoms with valency Z =2.These have 2N mobile electrons, exactly the right number to fill the first band. However, in the free electron picture, this is not what happens. Instead, they partially fill the first Brillouin zone and then spill over into the second Brillouin zone. The resulting Fermi surface, drawn in the extended zone scheme, is shown in left-hand picture of Figure 22

If the e↵ects of the lattice are weak, this will not be greatly changed. Both the ﬁrst and second Brillouin zones Figure 23: Beryllium will have available states close to the Fermi surface as shown in the middle picture. These materials remain metals. We sometimes talk

–39– Fermi Surfaces with a Lattice

What happens if we keep adding electrons?

Figure 22: FermiThe surfaces lattice fordistorts valence the energyZ = 2 spectrum, with increasing creating a lattice gap at strength,the BZ. If the moving gap from a metal to an insulator.is small, the electrons still spill over into the second BZ

Before we more on, a couple of comments:

Metals have a very large number of low-energy excitations, proportional to the • area of the Fermi surface. This makes metals a particularly interesting theoretical challenge.

–39– Fermi Surfaces with a Lattice

What happens if we keep adding electrons?

Figure 22: FermiBut surfaces if the lattice for valence is strong,Z then= 2 the with electrons increasing fill up latticeall of the strength, first BZ and moving from a metal to an insulator.none of the second BZ.

Before we more on, a couple of comments:

Metals have a very large number of low-energy excitations, proportional to the • area of the Fermi surface. This makes metals a particularly interesting theoretical challenge.

–39– Fermi Surfaces with a Lattice

Figure 22: Fermi surfaces for valence Z = 2 with increasing lattice strength, moving from a These two cases are similar. If we perturb This is very different. If we perturb metal to anthe insulator. system a little bit, the electrons can shift the system, the electrons can move. into new states at very little cost of energy, There is an energy gap to the next available state Materials with a Fermi surface are called metals. Suppose, for example, that we apply a small electric field to the sample. The electrons that lie at the Fermi surface can move to di↵erent available states in order to minimize their energy in the presence of the electric field. This results in a current that flows, the key characteristic of a metal. We’ll discuss more about how electrons in lattices respond to outside influences in Section 5

Before we more on, a couple of comments:

Metals have a very large number of low-energy excitations, proportional to the • area of the Fermi surface. This makes metals a particularly interesting theoretical challenge.

–39– Fermi Surfaces with a Lattice

These are metals This is an insulator

Before we more on, a couple of comments:

Metals have a very large number of low-energy excitations, proportional to the • area of the Fermi surface. This makes metals a particularly interesting theoretical challenge.

–39– Fermi Surfaces with a Lattice

To make an insulator, we have to fill one Brillouin zone exactly, with no spillover.

Recall: Figure 22: Fermi surfaces for valence Z = 2 with increasing lattice strength, moving from a metal to an insulator. • Each Brillouin zone contains N states. (where N is the number of atoms) Materials with a Fermi surface are called metals. Suppose, for example, that we • Eachapply electron a small electrichas two field spin to the states, sample. up The and electrons down that lie at the Fermi surface can move to di↵erent available states in order to minimize their energy in the presence • So ofeach the electricBrillouin field. zone This can results accommodate in a current that 2N flows,electrons the key characteristic of a metal. We’ll discuss more about how electrons in lattices respond to outside influences in Section 5

Before we more on, a couple of comments:

Metals have a very large number of low-energy excitations, proportional to the • area of the Fermi surface. This makes metals a particularly interesting theoretical challenge.

–39– Metals vs Insulators

• Each atom donates some number Z of electrons which wander through the materia; with Z an integer. This is called the valence of the atom. The number of electrons in a solid is NZ.

• For Z=1, the first BZ must be exactly half-filled. Any element with Z=1 is a metal! • e.g. Lithium, Copper

FigureFigure 20: Lithium. 20: Lithium. FigureFigure 21: 21: Copper.Copper.

of the Brillouinof the Brillouin zone. zone. We expect, We expect, therefore, therefore, that that the the Fermi Fermi surface surface — — which which is is the the equipotential E —willbedistortedasshowninthemiddlefigure,withstatescloser equipotential E —willbedistortedasshowninthemiddlefigure,withstatescloserF to the edgeF of the Brillouin zone filled preferentially. Note that the area inside the to the edgeFermi of surface the Brillouin remains the zone same. filled preferentially. Note that the area inside the Fermi surface remains the same. If the e↵ects of the lattice get very strong, it may that the Fermi surface touches the If theedge e↵ects of the of the Brillouin lattice zone get as very shown strong, in the it right-hand may that drawing the Fermi in Figure surface19 touches. Because the edge ofthe the Brillouin Brillouin zone zone is a as torus, shown if the in Fermi the right-hand surface is to drawing be smooth in then Figure it must19. Becausehit the the Brillouinedge of zone the Brillouin is a torus, zone if at the right-angles. Fermi surface is to be smooth then it must hit the edge of theThis Brillouin same physics zone atcan right-angles. be seen in real Fermi surfaces. Lithium has valence Z =1. It forms a BCC lattice, and so the Brillouin zone is FCC. Its Fermi surface is shown This same physics can be seen in real Fermi surfaces. Lithium has valence Z =1. above, plotted within its Brillouin zone2. Copper also has valency Z =1,withaFCC It formslattice a BCC and lattice, hence BCC and soBrillouin the Brillouin zone. Here zone the is e FCC.↵ects of Its the Fermi lattice surface are somewhat is shown 2 above, plottedstronger, within and the its Fermi Brillouin surface zone touches. Copper the Brillouin also has zone. valency Z =1,withaFCC lattice and hence BCC Brillouin zone. Here the e↵ects of the lattice are somewhat stronger,In and all the of these Fermi cases, surface there touches are unoccupied the Brillouin states with zone. arbitrarily small energy above E . (Strictly speaking, this statement holds only in the limit L of an infinitely F !1 In alllarge of these lattice.) cases, This there means are that unoccupied if we perturb states the with system arbitrarily in any way, small the electrons energy above will easily be able to respond. Note, however, that only those electrons close to the Fermi EF . (Strictly speaking, this statement holds only in the limit L of an infinitely surface can respond; those that lie deep within the Fermi sea are!1 locked there by the large lattice.) This means that if we perturb the system in any way, the electrons will Pauli exclusion principle and require much larger amounts of energy if they wish to easily be able to respond. Note, however, that only those electrons close to the Fermi escape. surface can respond; those that lie deep within the Fermi sea are locked there by the Pauli exclusionThis is an principle important and point, require so I’ll much say it again. larger In amounts most situations, of energy only if those they electrons wish to escape. which lie on the Fermi surface can actually do anything. This is why Fermi surfaces play such a crucial role in our understanding of materials. This is an important point, so I’ll say it again. In most situations, only those electrons 2This, and other pictures of Fermi surfaces, are taken from http://www.phys.ufl.edu/fermisurface/. which lie on the Fermi surface can actually do anything. This is why Fermi surfaces play such a crucial role in our understanding of materials.

2This, and other pictures of Fermi surfaces, are taken–38– from http://www.phys.uﬂ.edu/fermisurface/.

–38– Metals vs Insulators

• Each atom donates some number Z of electrons which wander through the materia; with Z an integer. This is called the valence of the atom. The number of electrons in a solid is NZ.

• For Z=1, the first BZ must be exactly half-filled. Any element with Z=1 is a metal! • e.g. Lithium, Copper

Figure 4: Energy dispersion for the free electron model.

Our real interest is what happens close to the edge of the Brillouin zone when is

The region of momentum space corresponding to the nth energy band is called the nth Brillouin zone. However, we usually call the 1st Brillouin zone simply the Brillouin zone.

–13– Figure 22: Fermi surfaces for valence Z = 2 with increasing lattice strength, moving from a metal to an insulator.

Before we more on, a couple of comments:

The Fermi energy of metals is huge, corresponding to a temperature of E /k • Metals vs InsulatorsF B ⇠ 104 K, much higher than the melting temperature. For this reason, the zero temperature analysis is a good starting point for thinking about real materials.

Metals have a very large• numberFor Z=2 of, low-energy with a weak excitations, lattice, there proportional will be some to thespillover. Some elements with Z=2 are metals. • • e.g. Beryllium area of the Fermi surface. This makes metals a particularly interesting theoretical challenge.

–39– Metals vs Insulators

• For Z=2, with a strong lattice, the first BZ will be exactly filled with no spillover. This is an insulator.

Figure 4: Energy dispersion for the free electron model.

Our real interest is what happens close to the edge of the Brillouin zone when is

The region of momentum space corresponding to the nth energy band is called the nth Brillouin zone. However, we usually call the 1st Brillouin zone simply the Brillouin zone.

–13– Metals vs Insulators

• For Z=3 we get a metal again.

conduction band

Figure 4: Energy dispersion for the free electronvalence model. band

Our real interest is what happens close to the edge of the Brillouin zone when is

The region of momentum space corresponding to the nth energy band is called the nth Brillouin zone. However, we usually call the 1st Brillouin zone simply the Brillouin zone.

–13– Electrons and Holes

Take an insulator, and inject enough energy to move an electron from the valence band to the conduction band.

This is an electron which is free to move around

This is an absence of an electron, which is also free to move around. We call it a Figure 4: Energy dispersion for the free electron model. hole. It acts like a particle with positive charge. Our real interest is what happens close to the edge of the Brillouin zone when is The hole is like an anti-particle! (Although it is not related to special relativity.) small compared to the gap Vn. In this case we can expand the square-root to give 2 2 2 2 2 2 2 ~ n ⇡ ~ 1 n ~ ⇡ 2 E + V0 Vn + 1 ± ⇡ 2m a2 ±| | 2m ± V ma2 ✓ | n| ◆ The ﬁrst collection of terms coincide with the energy at the edge of the Brillouin zone (1.15), as indeed it must. For us, the important new point is in the second term which tells us that as we approach the gaps, the energy is quadratic in the momentum .

The region of momentum space corresponding to the nth energy band is called the nth Brillouin zone. However, we usually call the 1st Brillouin zone simply the Brillouin zone.

–13– Semiconductors

• A semiconductor is an insulator, but with a small energy gap

Energy gap D

Figure 4: Energy dispersion for the free electron model. If the temperature T is comparable to the gap, kBT ~ D, then electrons get excited into the conduction band, leaving behind holes. This is a semi-conductor. Our real interest is what happens close to the edge of the Brillouin zone when is small compared to the gap Vn. In this case we can expand the square-root to give 2 2 2 2 2 2 2 ~ n ⇡ ~ 1 n ~ ⇡ 2 E + V0 Vn + 1 ± ⇡ 2m a2 ±| | 2m ± V ma2 ✓ | n| ◆ The ﬁrst collection of terms coincide with the energy at the edge of the Brillouin zone (1.15), as indeed it must. For us, the important new point is in the second term which tells us that as we approach the gaps, the energy is quadratic in the momentum .

The region of momentum space corresponding to the nth energy band is called the nth Brillouin zone. However, we usually call the 1st Brillouin zone simply the Brillouin zone.

–13– Summary

• Lattice + Quantum Mechanics gives band structure

• Pauli exclusion principle gives metals and insulators

Lots of very interesting things still to discuss, and even more still to discover Bose-Einstein Condensates Superconductors

• Conventional superconductors • High temperature superconductors Graphene Topological Insulators Quantum Hall Effect