magnetores

Handout 11 Magnetoresistance in three-dimensional systems 11.1 Introduction Magnetoresistance is a general term for the changes in the components of the resistivity and conductivity tensors of materials caused by the application of magnetic field. We are going to treat the magnetore- sistance of metals in a quite general and simple manner. First, however, the Hall effect in a system with more than one type of carrier will be described, as it helps to illuminate the more general discussion of metals that will follow, and gives a clue as to the origins of magnetoresistance. 11.2 Hall effect with more than one type of carrier 11.2.1 General considerations We consider the Hall effect with two or more carrier types present ( e.g. electrons and holes). The geometry of a Hall effect measurement is shown in Figure 11.1; the magnetic field Bis applied parallel to thezdirection ( i.e.B= (0,0,B)), whilst the current Iis driven through the sample in the x direction. The electric field Eis assumed to be E= (Ex,Ey,0); we assume that any effect is going to occur in the plane perpendicular to Bbecause of the nature of the Lorentz force (see Equation 11.1 below). Voltage measuring contacts are provided on the sample so that ExandEycan be deduced (see Figure 11.1). We assume that the drift velocity vof each species of carrier can be treated using the Relaxation Time Approximation, i.e. m∗{dv dt+v τ}=qE+qv×B, (11.1) whereqis the charge of the carrier, m∗is its effective mass (assumed isotropic and energy-independent) andτ−1is its relaxation (scattering) rate. Note that all changes in voccur in the plane perpendicular toB; therefore it is sufficient to split Equation 11.1 intoxandycomponents to give m∗{dvx dt+vx τ}=qEx+qvyB (11.2) and m∗{dvy dt+vy τ}=qEy−qvxB. (11.3) The Hall effect represents a steady state of the system, i.e. dvx/dt= dvy/dt= 0. Substituting Equation 11.2 into Equation 11.3 with dvx/dt= dvy/dt= 0 gives m∗vy τ=qEy−qBτ m∗{qEx+qvyB}, (11.4) 101 102 HANDOUT 11. MAGNETORESISTANCE IN THREE-DIMENSIONAL SYSTEMS Figure 11.1: Geometry of a Hall effect measurement on a sample of thickness sand widthw. On entering the sample, the current Ibecomes a current density Jof average magnitude I/ws . The magnetic field (flux density) is uniform within the sample. The positions of voltmeters for measuring Ex=Vx/dand Ey=Vy/ware shown symbolically. which can be rearranged to give vy{m∗ τ+q2B2τ m∗}=qEy−q2Bτ m∗Ex. (11.5) Dividing through by m∗/τand making the identification eB/m∗≡ωc(i.e.the cyclotron frequency) gives vy{1 +ω2 cτ2}=qτ m∗{Ey−ωcτEx}. (11.6) Hall effect experiments are usually quite deliberately carried out at low magnetic fields, such that ωcτ/lessmuch1, implying that terms ∼ω2 cτ2can be neglected. Therefore Equation 11.6 becomes vy=qτ m∗{Ey−ωcτEx}=qτ m∗{Ey−qBτ m∗Ex}. (11.7) We now consider an arbitrary number of carrier types, with each type being labelled by the integer j; thejth carrier type has effective mass m∗ j, chargeqj, scattering rate τjand number density nj. For each carrier type, Equation 11.7 therefore becomes vy,j=qjτj m∗ j{Ey−qjBτj m∗ jEx}. (11.8) Now the net transverse current must be zero, as there is nowhere for it to go (see Figure 11.1). Therefore /summationdisplay jnjvy,jqj= 0. (11.9) Equations 11.8 and11.9 can be used to derive the Hall coefficient for an arbitrary number of carrier types. We shall use them to treat the simple case of electrons and heavy holes in a semiconductor; as usual, the light holes, with their relatively feeble density of states compared to that of the heavy holes, will be ignored. 11.2.2 Hall effect in the presence of electrons and holes In the case of electrons in the conduction band (with effective mass m∗ c, scattering rate τ−1 c, chargeqc, densityn) and heavy holes in the valence band (with effective mass m∗ hh, scattering rate τ−1 hh, charge qhh, densityp), Equations 11.8 and11.9 combine to give nq2 cτc m∗c{Ey−qcBτc m∗cEx}+pq2 hhτhh m∗ hh{Ey−qhhBτhh m∗ hhEx}= 0. (11.10) 11.3. MAGNETORESISTANCE IN METALS 103 Equation 11.10 can be rearranged to give Ey{nµc+pµhh}=Ex{pµ2 hh−nµ2 c}B, (11.11) whereµhh=|qhhτhh/m∗ hh|is the heavy hole mobility and where µc=|qcτc/m∗ c|is the electron mobility, and I have substituted qhh≡+eandqc≡ −e. Now Ex=Jx σ=Jx |e|(nµc+pµhh), (11.12) whereJxis the current density in the xdirection. Combining Equations 11.11 and11.12 gives RH≡Ey JxB=1 |e|(pµ2 hh−nµ2 c) (nµc+pµhh)2. (11.13) This treatment is explored in more depth in the Problems.1 11.2.3 A clue about the origins of magnetoresistance Equations 11.9 and11.10 show that, although no netcurrent flows in the ydirection, the currents carried in the ydirection by a particular type of carrier may ( i.e.probably will) be non-zero. Carriers flowing in the ydirection will experience a Lorentz force caused by Bin the negativexdirection (you can satisfy yourself that this will always be the case). This backflow of carriers will act to change the apparent resistivity Ex/Jx,i.e.cause magnetoresistance.2 In the following Section we shall explore this idea more formally and in a very general manner. We shall see that the presence of more than one “carrier type” (and here the term is used very imprecisely) is necessary for magnetoresistance to be observed. In order to treat all of the different contributions to the conductivity and resistivity in a sanitary fashion, we shall introduce the idea of conductivity and resistivity tensors. 11.3 Magnetoresistance in metals 11.3.1 The absence of magnetoresistance in the Sommerfeld model of metals We consider first of all a metal with a simple spherical Fermi surface and isotropic, energy-independent effective mass. As in Section 11.2.1 , the magnetic field Bwill be parallel to z(see Figure 11.1), and we shall use the same symbols (effective mass m∗, scattering rate τ−1and electronic charge −e). Applying Equation 11.1, we have m∗{dv dt+v τ}=−eE−ev×B. (11.14) Two things may be deduced from this equation. •The motion of the electrons in the direction parallel to Bis unaffected. Therefore there will be no longitudinal magnetoresistance ; in this context, the longitudinal resistivity is measured in the direction parallel to the field B(i.e.both the applied current density and the measured electric field are parallel to B). •There may well be transverse magnetoresistance . Here transverse resistivity means that measured in the direction perpendicular to the field B(i.e. both the applied current density and the measured electric field are in the plane perpendicular to B). 1Some excellent illustrative data are shown in Figure 4.3 of Semiconductor Physics , by K. Seeger (Springer, Berlin 1991). 2Those who are unconvinced by this hand-waving argument should go back to Equation 11.6 and repeat the above derivation for ωcτ/greatermuch1. They will find that (( p/µhh) + (n/µc))Ey= (p−n)ExB(i.ethe Hall field is zero if n=p). Putting n=pyields Jx= ((p/µhh)+(n/µc))eEx/B2,i.e.ρ∝B2. This is the reason for the very large magnetoresistance in compensated semimetals (equal number of holes and electrons at Fermi surface) such as Bi. 104 HANDOUT 11. MAGNETORESISTANCE IN THREE-DIMENSIONAL SYSTEMS Jx ExJy J Figure 11.2: Geometrical interpretation of the components of current density Jx,Jyand caused by electric field component Exand magnetic field (0 ,0,B);Jis the total current density. Let us look at the second point in more detail. To simplify matters, we shall initially consider an electric field directed only along the xdirection ( i.e.E= (Ex,0,0)); our tactic will be to deduce the components of the current density J= (Jx,Jy,0) that flow in response to BandE. As in the previous Section, we are dealing with a steady state of the system, i.e.dvx/dt= dvy/dt= 0. Taking B= (0,0,B) and E= (Ex,0,0) as defined above, we rewrite Equation 11.2and Equation 11.3 in the form vd,x=−eτ m∗{Ex+vd,yB} (11.15) and vd,y=eτ m∗vd,xB, (11.16) where the subscript “d” emphasises the fact that we are dealing with a drift velocity. Equations 11.15 and11.16 show that the magnetic field has made the conductivity anisotropic; it has become a tensor, rather than a scalar. In order to work out the components of the conductivity tensor , we look at the current densities Jx=−nevd,xandJy=−nevd,y. Substituting these into Equations 11.15 and11.16 yields, after some rearrangement Jx=σxxExandJy=σyxEx, where σxx=σ0 1 +ω2cτ2(11.17) and σyx=σ0ωcτ 1 +ω2cτ2. (11.18) Hereσ0=ne2τ/m∗is the zero-field conductivity in the Sommerfeld model and ωc=eB/m∗is the cyclotron frequency. The conductivity tensor shows that, in a magnetic field, the total current density Jno longer flows parallel to the applied E-field,Ex; instead, it now contains both xandycomponents. Figure 11.2gives a geometrical interpretation of Jand the components of current density Jx,Jycaused by electric field component Exand magnetic field (0 ,0,B). Equation 11.17 shows that as B→ ∞ ,σxx∝B−2. We might therefore expect to see some magnetoresistance. However, most experiments (see Figure 11.1) measure voltages dropped in the x andydirections between pairs of contacts, rather than measuring the xandycomponents of the current density. In such experiments the current is forced to go along the xdirection, so that J≡Jxe1; in contrast, the electric field will have components in both xandydirections (see Figure 11.3). Therefore we want the components ρxx≡Ex Jxandρyx≡Ey Jx(11.19) of the resistivity tensor, rather than the conductivity. The general conductivity tensor is σ=/parenleftbiggσxxσyx σxyσyy/parenrightbigg . (11.20) 11.3. MAGNETORESISTANCE IN METALS 105 JxEx EyE Figure 11.3: Geometrical interpretation of the components of electric field Ex,Eyand the total field E caused by current density component Jxand magnetic field (0 ,0,B). The derivations which start at Equations 11.15 and11.16 can be repeated with E= (0,Ey,0) to yield σxy=−σyxandσyy=σxx,3so that we have σ=/parenleftbiggσxx−σxy σxyσxx/parenrightbigg =σ0 1 +ω2cτ2/parenleftbigg1ωcτ −ωcτ 1/parenrightbigg . (11.21) This tensor can then be inverted using standard methods to give the resistivity tensor ρ=/parenleftbiggρxxρyx ρxyρyy/parenrightbigg =1 σ0/parenleftbigg1−ωcτ ωcτ 1/parenrightbigg . (11.22) The components of interest in the experimental arrangement shown in Figure 11.1 are ρxx=ρ0andρyx=−ρ0ωcτ=−B ne, (11.23) whereρ0= 1/σ0(see Figure 11.3). Therefore we get no magnetoresistance in the diagonal components of the resistivity tensor and the familiar Hall effect for one carrier in the off-diagonal components. 11.3.2 The presence of magnetoresistance in real metals Almost all real metals exhibit some form of magnetoresistance, and so we must try to find out what is wrong with the approach above. In the above derivation we assumed that all carriers had the same value ofm∗andτ. However in a real metal we could have •electrons with different values of m∗(e.g.from anisotropic bands); •electrons with different values of τ(e.g.some parts of the Fermi surface may have higher scattering probabilities than others); •a combination of both. We therefore split the current density Jinto several components Jj=σxx,jExe1+σyx,jEye2 (11.24) 3This fact can also be deduced using symmetry considerations. 106 HANDOUT 11. MAGNETORESISTANCE IN THREE-DIMENSIONAL SYSTEMS ExJ J2 J1 Figure 11.4: Geometrical interpretation of the components of current density J1andJ2due to two different species of carrier caused by electric field component Exand magnetic field (0 ,0,B). where the index jindicates a contribution from the jth type of carrier. Each type of carrier will have a different density njand/or effective mass m∗ jand/or scattering rate τ−1 j. Hence, the components of the conductivity tensor σxx,jandσyx,jwill differ for each type of carrier, resulting in Jjs which do not in general point in the same direction as each other. The total current Jis just the sum of all of the components J=/summationdisplay jJj. (11.25) In order to see what happens, we take a very simple case of just two carrier types, j= 1,2. Equa- tion11.24 shows that, barring some very unlikely coincidence, J1andJ2will be in different directions. This situation is illustrated in Figure 11.4; the application of the magnetic field means that J1andJ2 are no longer parallel, so that |J| ≤ |J1+J2|, (11.26) i.e.the resistivity increases with increasing magnetic field. We therefore have magnetoresistance. We note in passing that, as above, σxx∝B−2asB→ ∞ (see Equation 11.17 and the paragraph following it). This will be important in the discussion of the following section. 11.3.3 The use of magnetoresistance in finding the Fermi surface shape We consider first a closed section of Fermi surface, about which a carrier can perform closed orbits under the influence of a magnetic field (see Figure 11.5). AsB→ ∞ ,ωcτ→ ∞ , so that an electron will tend to make many circuits of the Fermi surface before scattering. Therefore, the velocity of the electron in the plane perpendicular to Bwill average to zero; this is the reason why σxxandσyyboth vary asB−2in very high fields. Using the conductivity tensor components, the current densities can be written Jx=σxxEx+1 RBEy (11.27) and Jy=−1 RBEx+σyyEy (11.28) whereRis the Hall coefficient, and the off-diagional tensor components have been written σxy= 1/RB andσyx=−1/RB. Eliminating Eygives Ex=1 σxx+ (R2B2σyy)−1Jx+RB 1 +R2B2σxxσyyJy. (11.29) As mentioned above, σxxandσyyboth vary as B−2in very high fields, so that as B→ ∞ , (B2R2σyy)−1/greatermuchσxx. Therefore, ρxx=Ex/Jxtends to a constant at high fields, i.e.itsaturates . 11.4. THE MAGNETOPHONON EFFECT 107 Figure 11.5: (a) Schematic of electron motion on a closed section of Fermi surface in a magnetic field. The arrows indicate the velocities of an electron following a closed orbit about the Fermi surface in a plane perpendicular to the magnetic field B. (b) An open orbit on the Fermi surface. In an in-plane magnetic field, electrons will be driven across the Fermi surface, so that their velocities (shown by arrows) will rock from side to side. We now turn to an open section of Fermi surface (see Figure 11.5) about which an electron cannot perform closed orbits under the influence of a magnetic field. In this case, even as B→ ∞ , the average value ofvyremains finite; therefore σyy→C, a constant. Substituting σyy=Candσxx=AB−2, whereAis another constant, into Equation 11.29 yields ρxx=Ex Jx=1 AB−2+ (R2B2C)−1∝B2, (11.30) i.e.ρxx∝B2asB→ ∞ . We therefore have two distinct results •closed orbits produce aρxxwhich saturates as B→ ∞ ; •open orbits produce aρxxproportional to B2asB→ ∞ . This has been used to great effect in elucidating the Fermi surface of metals such as Copper, where changing the orientation of the magnetic field can produce open or closed orbits about the Fermi surface (see Figure 11.6). 11.4 The magnetophonon effect Oscillations can be observed in the resistivity of both bulk and two-dimensional semiconductors at ele- vated temperatures ∼100 K; this is known as the magnetophonon effect ormagnetophonon resonance . The effect is caused by resonant inter-Landau-level scattering of electrons by long-wavelength longitu- dinal optic (LO) phonons; such phonons are very effective scatterers of electrons. (Why? think about the type of polarisation field that they produce.) As such phonons have virtually zero wavevector, the transition is “vertical”, i.e.between almost identical points in k-space in the initial and final Landau levels involved. By conservation of energy, the condition for the magnetophonon effect to occur is therefore jωc=ωLO, (11.31) 108 HANDOUT 11. MAGNETORESISTANCE IN THREE-DIMENSIONAL SYSTEMS Figure 11.6: Magnetoresistance of Copper at a temperature of 4.2 K and a fixed magnetic field of 1.8 T; the current has been applied in the [100] direction (perpendicular to the plane of the page) and the magnetic field has been rotated from the [001] direction to the [010] direction. The magnetoresistance has been plotted radially as ( ρ(B)−ρ(B= 0))/ρ(B= 0). (Data from J.R. Klauder and J.E. Kunzler, The Fermi Surface , edited by W. Harrison (Wiley, New York, 1960.) 11.5. READING 109 Figure 11.7: Magnetophonon resonances in the longitudinal and transverse resistvities of InSb at 90 K. whereωLOis the phonon frequency, i.e.an integer number jof Landau level spacings matches an LO phonon energy. This leads to oscillations in the resistivity periodic in 1 /B; if the phonon frequency is known, the effective mass can be deduced from Equation 11.31 . The conditions for magnetophonon resonance to be observed are •the temperature should be low enough for the Landau levels to be resolved; •the temperature should be high enough for a substantial population of LO phonons. In practice, 70-100 K seems to be a good compromise. Figure 11.7 shows magnetophonon resonances in InSb at 90 K. 11.5 Reading Some useful general reading on magnetoresistance is contained in Electrons in Metals and Semicon- ductors , by R.G. Chambers (Chapman and Hall, London 1990) Chapters 1, 2 and 11, Semiconductor Physics , by K. Seeger (Springer, Berlin 1991) Chapter 9 (hard), Solid State Physics , by N.W Ashcroft and N.D. Mermin (Holt, Rinehart and Winston, New York 1976) Chapters 12, 13 and 15 (hard). 110 HANDOUT 11. MAGNETORESISTANCE IN THREE-DIMENSIONAL SYSTEMS