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 δ 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 δ ± ⇡ 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 δ. 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.