MSci Project Report: Magnetoresistance in a quasi two-dimensional metal Spring 2011 Lucy Whalley [email protected]Supervisor Professor Andy Schofield Abstract In January 2010 it was claimed by Figarova and Figarov in [4] that the in-plane transverse magnetore- sistance (TMR) of a quasi two-dimensional metal is negative in weak fields. This theoretical result was calculated using the Boltzmann transport equation in the relaxation time approximation. We present in this paper a calculation which shows magnetoresistance to be positive within such limits. Firstly, we intro- duce the analytic and mathematical framework used: The Boltzmann equation, the Abrikosov-Chambers method and Jacobi elliptic functions. We then, following work by Schofield and Cooper in [7], reach an expression for TMR. The TMR is calculated numerically for a range of fields and it is seen that there is no region of negative magnetoresistance.
Transcript
MSci Project Report:Magnetoresistance in a quasi two-dimensional metal
In January 2010 it was claimed by Figarova and Figarov in [4] that the in-plane transverse magnetore-sistance (TMR) of a quasi two-dimensional metal is negative in weak fields. This theoretical result wascalculated using the Boltzmann transport equation in the relaxation time approximation. We present inthis paper a calculation which shows magnetoresistance to be positive within such limits. Firstly, we intro-duce the analytic and mathematical framework used: The Boltzmann equation, the Abrikosov-Chambersmethod and Jacobi elliptic functions. We then, following work by Schofield and Cooper in [7], reach anexpression for TMR. The TMR is calculated numerically for a range of fields and it is seen that there isno region of negative magnetoresistance.
Figarova and Figarov claim that negative TMR exists in an electron gas:
“We show that the transverse magnetoresistance of a degenerate quasi three-dimensional 1 elec-tron gas in the fields parallel to the layer plane is negative in weak fields.”[4, p.1]
Although negative TMR has beem found in a number of compounds (eg. [9]) this is commonly attributed tostrong scattering by inhomogenities or quantum effects such as weak localisation. Provision for such effectsare not made by Figarova and Figarov and so their claim goes against our physical understanding of thesystem under consideration.
The aim of this paper is to use the Abrikosov-Chambers method of coordinate substitution to demonstratethat there is no region of in-plane negative TMR for a quasi two-dimensional metal. The TMR will be dervied,results will be generated numerically and will then be compared to those found by Figarova and Figarov.
1.2 Previous work
In 1879 E. H. Hall applied an electric field and a transverse magnetic field to a conductor. He found thatthe resistance was independant of the magnetic field. This result was later derived for an isotropic metal byDrude2.
Cuprate metals are among the many compounds which consist of weakly coupled layers, forming a quasitwo-dimensional (anisotropic) metal. These compounds have a resistance which is dependant upon theapplied magnetic field. The semi-classical Boltzmann equation can be solved to produce this behaviour fornon-quantizing fields, eg. [8].
This paper follows an approach developed by Schofield and Cooper in 2000 [7]. In this paper we considerin-plane TMR whereas the electric field is orientated through the plane in [7].
1Quasi three-dimensional can be read to mean a quasi two-dimensional metal which consists of closed orbits only.2Annalen der Physik 306, Vol. 3, 566 (1900)
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Figure 1: The field geometry which we willconsider. Shown is applied electric field E =Ea, applied magnetic field H = Hb and theresulting hall field EH = −Ec.
1.3 Geometry
We use the same geometry as that used by Figarova and Figarov in [4]: see figure (1). We look at the in-planemagnetoresistance, meaning that both the electric and magnetic fields lie along the free electron plane of themetal. Transverse magnetoresistance (TMR) means that the electric and magnetic fields are at right anglesto one another.
4
2 Background
In this section we will introduce the physics used throughout the rest of the paper. First we look at theHall effect which arises under applied electric and magnetic fields and see how this affects the resistance of amaterial. Magnetoresistance is found by taking the inverse of the conductivity tensor σ, given by:
j = σE
=2e
(2π)3
∫ +∞
−∞d3knv (1)
where j is the current density, E the applied electric field and v the electron velocity. The electron distributionn(r,k, t) is given by the Boltzmann equation, which is introduced in the second subsection. In the case of ananisotropic metal the Boltzmann equation is best tackled using the Abrikosov-Chambers method of coordinatesubstitution, which is introduced in the final subsection.
2.1 Magnetoresistance and the Hall effect
The Hall effect is the production of an electrical potential differance when an electric and magnetic field areapplied to a conductor.
The applied electric field E produces a current (figure 2a). We then apply a magnetic field H and the resultingLorentz force q(v×H) deflects the charge carriers (figure 2b). This produces a Hall field EH which opposesthe charge carrier deflection (figure 2c). In equilibrium qEH balances the Lorentz force and current flowsalong only the a-axis 3 with current density ja.
The magnetoresistance ρ(H) is
ρ(H) =Eaja
(2)
Hall found this to be independant of the magnetic field, a result which was later derived by Drude for the caseof a non-interacting electron gas moving between immobile atoms. In the case of a quasi two-dimensionalmetal the electrons are not ’free’ to move in all directions - it requires energy to move between the planes asshown in figure 3. This anisotropy results in a non-zero magnetoresistance.
(a) (b) (c)
Figure 2: Charge carrier response to applied electric and magnetic fields
3We will label direction in real space with a,b,c. Directions in recipricol space will be labelled x,y,z.
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Figure 3: The structure of StrontiumRuthenate, Sr2RuO4. Electrons are ’free’particles on the ab-planes but their move-ment is restricted in the c-direction, throughthe planes. This anisotropy gives rise tonon-zero magnetoresistance.
2.2 The Boltzmann transport equation
The Boltzmann transport equation is used to calculate the non-equilibrium distribution of electrons in statek at position r and time t:
dn
dt=∂n
∂t+∂k
∂t· ∇kn+
∂r
∂t· ∇rn
with ∇kn =
(∂n
∂kx,∂n
∂ky,∂n
∂kz
)∇rn =
(∂n
∂x,∂n
∂y,∂n
∂z
)(3)
The LHS gives the rate of change of the electron distribution. The first term on the RHS accounts forany explicit time dependance, the second term accounts for how external fields change the wavevector of anelectron and the third term is a diffusion term (electron diffusion due to a thermal gradient for example).
We assume that any electric or magnetic fields are independant of space and time and that there are no otherexternal forces or thermal gradients which may cause electron diffusion. As a result ∂n
∂t and ∇rn are equal tozero. We also know that in the presence of electric and magnetic fields the force on an electron is
∂p
∂t= h
∂k
∂t
= e
(E +
1
cv ×H
)(4)
Equation 3 then becomes:
dn
dt= e
(E +
1
cv ×H
)· 1
h∇kn (5)
The Boltzmann equation is semi-classical. We consider the electrons to be electron quasi-particles4 whosemass depends upon the periodic potential created by the lattice they move through. Although their interactionwith the lattice is quantum mechanical, they respond to the Lorentz force as a classical particle would.
2.2.1 Relaxation time approximation
The equilibrium state distribution n0 when there are no external fields, thermal gradients or other terms whichwould affect electron transport is given by the Fermi-Dirac distribution. To solve equation (3) we assumethat the electron distribution does not deviate far from it’s equilibrium state and that the electron’s energy is
4We will refer to the charge carriers as electrons although they are considered to be electron quasi-particles throughout.
6
Figure 4:Main picture: theFermi surfacefor a quasi two-dimensional metal.Inset: Cross sec-tions along theplane py = con-stant show thepossible electrontrajectories ink-space.
unaffected by scattering. This is the relaxation time approximation:
dn
dt= −n− n0
τ= −g
τ(6)
with n0 =1
exp(E−µkT
)+ 1
(7)
where τ is the characteristic time it would take for an electron to return to it’s equilibrium state after allexternal fields have been turned off.
2.2.2 Other approximations
We assume that the applied fields are small. The magnitude of the magnetic field is not so large as to allowlandau quantization ωcτ << 1 and so the electron dispersion relation remains fixed, independant of magneticfield. The electric field is small enough to ignore joule heating and all calculations can be done to first orderin E.
We also assume that that the metal is at fixed temperature T=0 throughout. All of these assumptions werealso used by Figarova and Figarov in [4].
2.3 The Abrikosov-Chambers method
Solving the Boltzmann equation for an isotropic metal can be done by expanding g(k) in terms of E and Bthen solving at each order. However for an anisotropic metal it is easier to solve using the Abrikosov-Chambersmethod of coordinate substitution. Instead of using variables kx, ky and kz to give n(k) we use energy ε,time of motion along the trajectory t1 and wavevector ky to give n(ε, t1, ky).
Note that the variables ky and ε are conserved quantities for the individual elecron orbits around the Fermisurface. The Fermi surface for a quasi two-dimensional metal is shown in figure (4). The electrons which canparticipate in conduction are on the Fermi surface. The magnetic field, acting normal to the electrons velocityvector, produces no change in energy and so the electrons remain on the Fermi surface. If the magneticfield is acting along the b-direction then ky also remains unchanged. Possible trajectories in k-space are thusintersections of the Fermi surface by planes with ky = constant. The electric field is assumed small enoughto have no effect on electron velocity.
7
2.3.1 The Boltzmann equation in new coordinates
To write the Boltzmann equation in terms of these new variables we must first show that these variables areenough to calculate the area S =
∫dpxdpz when py is constant, see figure (5) for a visualisation of this in
the isotropic case. Note that the integral over dl is along a contour of constant energy.
S =
∮dl
∫dε
∣∣∣∣∂p⊥∂ε∣∣∣∣
=
∫dε
∮dl
v⊥(8)
We now calculate the time of motion along a trajectory. We assume that eE <<e
cv ×H so that the force
on an electron is given by equation (9).
dp
dt= e
(E +
1
cv ×H
)≈ e
c(v ×H) (9)
Write equation (9) as projections upon the x-z plane.
dpxdt
= −ecvzH and
dpzdt
=e
cvxH (10)
Use equations (10) to get an expression for t1:
dp2x + dp2zdt2
=e2
c2H2(v2x + v2z)
dl
dt=e
cv⊥
=⇒ t1 =c
eH
∫dl
v⊥(11)
Substitute this into equation (8) and integrate over py also to get a volume in k-space:
V =
∫dkxdkydkz =
eH
hc
∫dpydεdt1 (12)
Knowing the integral volume V allows us to calculate the current density as given in equation (1). It alsomeans that the Boltzmann equation for a metal in the presence of constant electric and magnetic fields maybe written as:
dn
dt=∂n
∂t1t1 +
∂n
∂εε+
∂n
∂pypy (13)
We can calculate the time period T for a closed trajectory by integrating equation (11) over the entirecontour:
T =c
eH
∮dl
v⊥(14)
Using equation (8) this gives
T =c
eH
∂s
∂ε
=2πcm∗
eH(15)
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Figure 5: Area S includes theshaded and striped regions. Thestriped ring has a width of magni-tude dp⊥ in the direction dε.
where m∗ =1
2π
∂s
∂εis the cyclotron mass. In the isotropic case this is identically equal to the effective mass
m.
We now find expressions for ε, py and t.
ε =∂ε
∂p
∂p
∂t
= v∂p
∂t(16)
with
∂p
∂t= e
(E +
1
cv ×H
)=⇒ ε = ev ·E (17)
=⇒ py = eEz (18)
Substituting equations (17), (18) anddt1dt≈ 1 into the Boltzmann equation gives
dn
dt=∂n
∂t1+
∂n
∂pyeEz +
∂n
∂εev ·E (19)
Note that there are no terms containing H; the Chambers method allows us to solve to all orders in H.
2.3.2 Finding an expression for the electron distribution n
We will now find an expression for the electron distribution n by solving the Boltzmann equation. This is
done by the substitution of n = n0 −∂n0∂ε
φ into equation (19). All terms of order E2 or above are ignored
(E is assumed small) and n0 is independant of t1.
dn
dt=∂n0∂t1
+∂n0∂py
eEz +∂n0∂ε
v · eE− ∂2n0∂t1∂ε
φ− ∂n0∂ε
∂φ
∂t1− ∂2n0∂py∂ε
φeEz
− ∂φ
∂py
∂n0∂ε
eEz −∂2n0∂ε2
φveE− ∂φ
∂ε
∂n0∂ε
φveE
=∂n0∂py
eEz +∂n0∂ε
v · eE− ∂n0∂ε
∂φ
∂t1
=∂n0∂ε
eE ·V − ∂n0∂ε
∂φ
∂t1(20)
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Use the relaxation time approximation to set the LHS of equation (19) asdn
dt=n0 − nτ
=∂n0∂ε
φ
τ. Equation
20 becomes
φ
τ= ev ·E− ∂φ
∂t1(21)
Solve equation (21) using the integrating factor method to give
φ(t1) =
∫ t1
c
ev(t2) ·Ee−t1−t2τ dt2 (22)
where periodic boundary conditions v(t1) = v(t1 +T ) require c = −∞ for an electron in an orbit with periodT .
2.3.3 Contribution to current density j
We now substitute our expression for the electron distribution into the equation for current density (1),adapted for the new coordinate system by equation (12). This will eventually give us a conductivity tensorwhich we invert to find the magnetoresistance.
jα =2e
(2π)3h3
∫nvαd
3p
=2e2H
(2πh)3c
∫dpydt1dε (23)
with α = x or z. Substitute into (23) an expression for n using equation 22:
jα = σαβEβ
=2He3
(2πh)3c
∫ +p0
−p0dpy
∫ T
0
dt1vα(t1)
∫ t1
−∞vβ(t2)Eβe
− (t1−t2)τ dt2 (24)
with β = x or y. The triple integral gives us an expression for the conductivity. To continue we needan expression for the electron velocities. In the isotropic case electron velocities are easily found and theconductivity can be calculated analytically; this will be the focus of the next subsection. We will then moveon to the anisotropic case where velocities are in the form of Jacobi elliptic functions and the conductivity iscalculated numerically.
2.3.4 Magnetoresistance in the isotropic case
Equation (10) can be solved to give
vx = v⊥ cos(ωct1) and vz = −v⊥ sin(ωct1) (25)
with ωc denoting angular cyclotron frequency ωc =eH
mc. Substitute this into equation (24):
(jxjz
)=
2He3
(2πh)3c
∫ p0
−p0v2⊥dpy
∫ T
0
dt1
(cos(ωct1)− sin(ωct1)
)∫ t1
−∞(Ex cos(ωct2)− Ez sin(ωct2))e−
t1−t2τ dt2 (26)
10
We can calculate the integral over dt2 by considering∫ t1
Use symmetry to calculate the integral over dt1:∫ T
0
cos(ωct1) sin(ωct1)dt1 = 0 and
∫ T
0
cos2(ωct1)dt1 = 0
=⇒(jxjz
)=
2He3
(2πh)3c
T
2(ω2c + τ−2)
(τ−1Ex + ωcEz−ωcEx + τ−1Ez
)∫ p0
−p0v⊥
2dpy (29)
The final integral can be calculated using v2⊥ = v2 − v2y =p20 − p2ym
. Using equation (15) and n0 =p30
3π2h3we calculate the current density and conductance tensor σc:(
jxjz
)=
He3
(2πh)3c
T
(ω2c + τ−2)
(τ−1 −ωcωc τ−1
)(ExEz
)=
(σ0
1+(ωcτ)2− ωcτσ0
1+(ωcτ)2ωcτσ0
1+(ωcτ)2σ0
1+(ωcτ)2
)(ExEz
)= σc
(ExEz
)(30)
If there is no magnetic field applied this reduces to well-known form j =ne2τ
mE. Inverting the matrix gives
us the magnetoresistance tensor ρ:
ρ =
( 1σ0
ωcτσ0−ωcτ
σ0
1σ0
)(31)
We see that the TMR is zero.
This can be done analytically, which is not possible for the anisotropic case. Before we find an expression forvelocity we must first consider the dispersion relation for our problem; this is the subject of the next section.
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3 The Dispersion Relation
All results reached so far (excluding those in section 2.3.4) are valid for a range of dispersion relations. Wenow introduce the dispersion relation used by Figarova and Figarov in [4] and then in the following sectionfind the magnetoresistance for this particular problem.
The dispersion relation for a quasi two-dimensional metal is
ε(k) =h2
2m(k2x + k2y) + 2t⊥(1− cos(ckz)) (32)
where x,y,z label directions in recipricol space and a,b,c label directions in real space; the direction perpendic-ular to the layer plane is c. It describes free electrons in the ab-plane with mass = m0. Electrons travellingalong the c-axis have an effective mass m∗ given by
m∗ =h2
2t⊥c2(33)
The transfer integral 2t⊥ accounts for the cost in energy when electrons are able to occupy sites on planesparallel to the ab-plane. The larger the value of t⊥, the more energy is required for electrons to move betweenplanes.
Figarova and Figarov consider the dispersion relation to remain fixed and independant of magnetic field. Thisis a fair assumption in the weak field limit (ωcτ << 1) where the uncertainty in the energy of an electron isof order hωc and Landau level quantisation can be ignored.
By Taylor expanding the dispersion relation we see that the dispersion relation is ellipsoidal at small kz:
ε =h2k2⊥2m
+ 2t⊥(1− cos kzc)
≈ h2k2⊥2m
+ 2t⊥(1− (1− k2zc2))
≈ h2k2⊥2m
+ 2t⊥k2za
2 (34)
At this limit electrons will move around closed ellipses in k-space.
In the case where t⊥ is zero the the Fermi surface becomes a cylinder with its axis in the z-direction. Betweenthese two extremes the electron orbits look like distorted ellipses as shown in figure (6). Note that the fermisurface must always intercept the Brillouin zone boundary at right angles.
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Figure 6: The delta parameter,controlled by the transfer integralt⊥, determines if electron orbitsin k-space will be closed or a mix-ture of closed and open.
3.1 The delta parameter
We now introduce an important dimensionless parameter delta, which is controlled by the transfer integralt⊥:
δ =4t⊥εf
(35)
Delta measures the anisotropy of the system. When delta is large all electron orbits are closed whilst whendelta is small some electron orbits are open and some are closed, depending upon their value of kz. At thepoint k = (0, 0, πc ) and at energy εf the dispersion relation gives εf = 4π and so δ = 1. This is where wemove from the closed to open orbits shown in figure (6).
We now move on to solve the triple integral given in equation (24) to give us the conductivity tensor for thisdispersion relation.
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4 The Integral
To solve the triple integral given for the current density in (24) we have to obtain expressions for the electronvelocity. We find that these take the form of Jacobi elliptic functions: Well behaved, periodic functionswhich can be expanded into summations of cosine and sine. The integral over time period T results in manyterms disappearing due to orthogonality and we will eventually be left with a summation term and an integralover py. This will then be calculated numerically to give us an expression for the conductivity tensor andmagnetoresistance.
4.1 Electron velocity: Jacobi elliptic functions
We begin by rearranging the expression for the Lorentz force to find an expression for kz(t):
hdk
dt= −ev ×H
dt = − he
dkzvaHb
(36)
The velocity va5 from dispersion relation (32) is
va =dεfdpy
=hkym
=h
m
√
2mεf + 4t⊥m cos(kzc)− 4mt⊥ − h2k2yh
(37)
Substitute this into the Lorentz force expression (36) and use cos(kzc)− 1 = −2 sin2
(kzc
2
)to give
dt = −mheH
dkz√2mεf − h2k2y + 4mt⊥(cos(kzc)− 1)
= −mheH
dkz√2mεf − h2k2y − 8mt⊥ sin2
(kzc2
)= −√αc
2ωc
dkz√1− α sin2
(kzc2
) (38)
where we have defined a new cyclotron frequency
Ωc =eHc
h
√2t⊥m
(39)
and α is a new dimensionless parameter
α =8t⊥m
2mεf − h2k2y(40)
Integrate equation (38) to get an expression for t
Ωct√α
= − c2
∫ kz(t)
0
dkz√1− α sin2
(kzc2
) (41)
5When we talk of velocity it is assumed that this is the velocity of an electron on the Fermi surface with energy εf
14
The integral is an incomplete elliptic integral of the first kind. The elliptic functions are inverses of this typeof integral. The upper limit kz(t) is the jacobi amplitude (’amp’) and can be written as
kz(t) = −2
camp
[Ωct√α, α
](42)
The three elliptic functions SN
[Ωct√α, α
],CN
[Ωct√α, α
]and DN
[Ωct√α, α
]are given by
SN
[Ωct√α, α
]= sin
(amp
[Ωct√α, α
])= sin
(− c
2kz(t)
)(43)
CN
[Ωct√α, α
]= cos
(amp
[Ωct√α, α
])= cos
(− c
2kz(t)
)(44)
DN
[Ωct√α, α
]=
√1− α sin2
(amp
[Ωct√α, α
])=
√1− α sin2
( c2kz(t)
)(45)
We will look at some of the useful properties of these functions in the next subsection. We will now deriveexpressions for vc(t) and va(t) in terms of Jacobi elliptic functions.
vc(t) =dε
dpz
=2t⊥c
hsin(kzc)
=2h
cm⊥cos
(ckz(t)
2
)sin
(ckz(t)
2
)= − 2h
cm⊥cos
(amp
[Ωct√α, α
])sin
(amp
[Ωct√α, α
])= − 2h
cm⊥CN
[Ωct√α, α
]SN
[Ωct√α, α
](46)
va(t) =dε
dpx
=hkxm
=1
m
√2mε+ 4t⊥(cos(kzc)− 1)− h2k2y
=2h
c√mm⊥α
√1− α sin2
(ckz2
)
=2h
c√mm⊥α
√1− α sin2
(amp
[Ωct√α, α
])=
2h
c√mm⊥α
DN
[Ωct√α, α
](47)
We now have the velocities needed to calculate current density j. In the next section we look at the Jacobielliptic functions further and introduce some identities which will make calculations easier.
4.2 Useful identities and relationships
For simplification we will denote the Jacobi elliptic function as SN[u,m], CN[u,m] and DN[u,m] where
u =Ωct√α
and m = α. There is a restriction that m ≤ 1, which will not always be true in our case . However
15
we can introduce the recipricol parameter µ = m−1 and the following identities:
DN[u,m] = CN[um12 , µ] (48)
CN[u,m] = DN[um12 , µ] (49)
SN[u,m] = µ12 SN[um
12 , µ] (50)
This means we will have to split the integral into two, one for the case α < 1 and one for the case α ≥ 1.We will also make use of relationships between Jacobi elliptic derivatives:
d
duDN[u,m] = −mSN[u,m]CN[u,m] (51)
d
duCN[u,m] = −SN[u,m]DN[u,m] (52)
4.3 Fourier expansions of the Jacobi elliptic functions
SN[u,m], CN[u,m] and DN[u,m] can be expanded into summations of cosine and sine as shown below
SN[u|m] =2π
m12K
∞∑n=o
qn+12
1− q2n+1sin
((2n+ 1)πu
2k
)(53)
CN[u|m] =2π
m12K
∞∑n=o
qn+12
1 + q2n+1cos
((2n+ 1)πu
2k
)(54)
DN[u|m] =π
2k+
2π
K
∞∑n=1
qn
1 + q2ncos
((2nπu
2k
)(55)
Where K is known as the complete integral of the first kind:
k =
∫ π2
0
dθ√1−m sin2 θ
(56)
and the nome q is given by
q = exp
(−π(1− k)
k
)(57)
Plotted in figure (7) are SN[u, 0.5], CN[u, 0.5] and DN[u, 0.5] along with the first term of their expansions.This first term of the expansion is already very close to the exact results - the sums are quickly convergent andand so computing them numerically for a finite n is a possiblity. Also note that the complete elliptic integralK is a quarter period of SN[u,m],CN[u,m] and a half period of DN[u,m] - this will later enable terms of sineand cosine to disappear due to the orthogonality relations below:∫ T
0
sin
(nπt
T
)sin
(mπt
T
)dt =
T
2δnm (58)∫ T
0
cos
(nπt
T
)cos
(mπt
T
)dt =
T
2δnm (59)∫ T
0
sin
(nπt
T
)cos
(mπt
T
)dt = 0 (60)
4.4 Integral limits
In the equation for current density (24) there are three variables to integrate over. The first two (over dt1and dt2) can be done analytically with the variables and limits given. For the third one over py we change to
16
Figure 7: Plots of SN[u, 0.5], CN[u, 0.5] and DN[u, 0.5] along with the first term of their expansions.
the new variable α given in (40). As mentioned in section 4.1 if orbits can be open and closed (δ < 1) wemust split the integral into two: one for the case when α < 1 and one for the case when α ≥ 1 where we will
introduce the recipricol parameter β =1
α. Note that the limits
∫ pb0−pb0≡ 2
∫ pb00
as the integrand is symmetric
in vy.
When py = 0
α =8t⊥m0
2m0εf − h2k2y=
4t⊥εf
= δ (61)
=⇒ β =1
δ(62)
When py = max =√
2m0εf
α =8t⊥m0
2m0εf − h2k2y=
8t⊥m0
0= undefined (63)
=⇒ β = 0 (64)
17
When kx = 0 and kz = πc the electron is moving on the trajectory which marks the transition from closed to
open orbits:
When kx = 0, kz =π
c(65)
=⇒ py =√
2m0εf − 8t⊥m0
α =8t⊥m0
2m0εf − h2k2y= 1 (66)
=⇒ β = 1 (67)
We have now shown that the limits for the integral over dpy become:
2
∫ 0
min(1,δ−1)
dβ In the case δ ≥ 1, all orbits closed (68)
2
[∫ 0
1
dβ +
∫ 1
δ
dα
]In the case δ < 1, orbits a mixture of open and closed (69)
4.5 Calculating conductivity σcc in the case of closed orbits
We now have the velocities, identities, expansions and limits needed to derive an expression for conductivity.We will calculate the term σcc in the case of closed orbits (α ≥ 1); we will not show the calculations for theother terms in the tensor as these are of a very similar nature.
An expression for vc in the case of closed orbits is given by taking the expression for vc in equation (46) andusing the recipricol parameter identities (49)-(50):
vc = −2h√β
cm⊥DN[Ωct, β]SN[Ωct, β] (70)
We then use the identity for the derivative CN (52) and expand using equation (54)
vc =2h√β
cm⊥
d
d(Ωct)CN[Ωct, β]
=2h√β
cm⊥
d
d(Ωct)
[2π
k√β
∞∑n=0
qn+12
1 + q2n+1cos
((2n+ 1)πΩct
2k
)]
=−4hπ2
cm⊥k2
∞∑n= 1
2
nqn
1 + q2nsin
(nπΩct
k
)(71)
Substitute this expression for vc into equation (24) for the current density
jc =4e3Hπ
hc2m2⊥
∫ pb0
−pb0
1
k4dpb
∫ T
0
∞∑n= 1
2
nqn
1 + q2nsin
(nπΩct
k
)e−
tτ dt
∫ t
−∞
∞∑m= 1
2
mqm
1 + q2msin
(mπΩct2
k
)et2τ dt2Ec (72)
The integral over dt2 can be done by applying integration by parts twice to give
jc =4e3Hπ
hc2m2⊥
∫ pb0
−pb0
1
k4dpb
∫ T
0
∞∑n= 1
2
nqn
1 + q2nsin
(nπΩct
k
)e−
tτ
τ2
1 +(mπΩcτ)2
k2[etτ
τsin
(mπΩct
k
)− mπΩce
tτ
k√α
cos
(mπΩct
k
)]dtEc (73)
18
The next integral is over time period T, which is equal to the time period of vc ∝ CN[Ωct, β]:
Ωct =
∫ φ dθ√1− β2 sin2 θ
ΩcT
4=
∫ π2 dθ√
1− β2 sin2 θ
ΩcT
4= K
=⇒ T =4K
Ωc(74)
We see that the sines and cosines in equation (73) take the form sin nπtT and cos nπtT . We can then use
orthogonality equations (58) and (60), to leave only the terms which have sin2(..).
jc =4e3Hπ
hc2m2⊥
∫ pb0
−pb0
1
k4dpb
∞∑n= 1
2
(nqn
1 + q2n
)2τ2
1 +(nπΩcτ)2
k2
1
τ
2k
ΩcEc
=2πσ0
√δm0
m⊥
∫ min(1,δ−1)
0
∞∑n= 1
2
(nqn
1 + q2n
)21
1 +(nπΩcτ)2
k2
1
k3√
1− βδdβEc
=⇒ σcc =2πσ0
√δm0
m⊥
∫ min(1,δ−1)
0
∞∑n= 1
2
(nqn
1 + q2n
)21
1 +(nπΩcτ)2
k2
1
k3√
1− βδdβ (75)
Where we have used the substitutions
β =1
α=
2m0εf − h2k2y8t⊥m0
(76)
σ0 =2e2τ
c3m⊥(77)
Similar methods to that outlined above were used to give σcc in the case of open orbits and σaa, σac in thecase of open and closed orbits:
σcc for the case of open orbits, α < 1 :
σcc =8πσ0
√δm0
m⊥
∫ 1
δ
∞∑n=1
(nqn
1 + q2n
)21
1 +(nπΩcτ)2
αk2
1
k3α3√α− δ
dα (78)
σaa for the case of closed orbits, α ≥ 1 :
σaa =8σ0√δ
π
∫ min(1,δ−1)
0
∞∑n= 1
2
(qn
1 + q2n
)21
1 +(nπΩcτ)2
k2
1
k√
1− βδdβ (79)
σaa for the case of open orbits, α < 1 :
σaa =σ0√δ
π
∫ 1
δ
1 + 8
∞∑n=1
(qn
1 + q2n
)21
1 +(nπΩcτ)2
αk2
1
α2k√α− δ
dα (80)
σac for the case of open and closed orbits :
σac =eHτ
m0σcc (81)
19
The Onsager symmetry principle for kinetic coefficients means that
σac = −σca (82)
The Lorentz force cross product v ×H results in zero electron velocity along the axis of the magnetic fieldunless the electric field also lies along that axis, giving:
σab = σba = σbc = σcb = 0 (83)
The complete elliptic integral k and the nome q are given by equations (56) and (57) respectively. They takeα as there argument when α < 1 and β as there argument when α ≥ 1.
4.6 Expression for the magnetoresistance
We invert our conductance tensor to get the resistance tensor ρ:
j =1
det(σ)adj(σ) (84)
ρaa ρba ρcaρab ρbb ρcbρac ρbc ρcc
=1
σaaσbbσcc − σacσbbσca
σbbσcc 0 −σbbσac0 σcaσac − σaaσcc 0
−σbbσca 0 σbbσaa
(85)
=1
σaaσbbσcc − σacσbbσca
σbbσcc 0 −σbbσac0 −(σ2
ac + σaaσcc) 0σbbσac 0 σbbσaa
(86)
The in-plane TMR is given by
ρaa =σbbσcc
σaaσbbσcc − σacσbbσca=
σccσaaσcc + σ2
ac
(87)
The quantity we are interested in calculating is
∆ρ
ρ=ρ(H)− ρ(0)
ρ(0)
This allows cancellation of the parameters σ0 and mom⊥
so that the only variables left to consider are δ
(anisotropy) and Ωcτ (magnetic field strength).
4.7 A programme for numerical integration
See Appendix A for the programme written to calculate σaa for any value of δ. We chose to use the Pythonprogramming language with the open-source ’SciPy’ and ’NumPy’ scientific computing libraries. These aredesigned to easily handle arrays of numbers and have built in integration routines.
The user enters the values of Ωcτ and δ of which they would like to calculate TMR. They also choose howthey want to present the information graphically: TMR vs δ, TMR vs Ωcτ , each data set on a separate plotor all datasets on one plot. All plots and the unprocessed information are automatically saved.
We sum over 100 terms of the sine/cosine expansions. We then check that the 100th and 101st term areidentical (to machine precision), if they are not then the programme stops running6.
6We have not seen this happen yet.
20
In this section we have shown how to calculate the in-plane TMR for a quasi two-dimensional metal. Thedispersion relation is important as it determines the electron velocity. In our case we could expand the electronvelocity into summations of sine and cosine; orthogonality then allowed a much simpler expression for theconductivity. This approach could be extended for use with other dispersion relations where the velocity isperiodic. In the next section we look at what the existing literature would expect our TMR to be.
21
Figure 8:Main: Fermi surface for a 2dmetalInset: Cross sections along theky = constant and kz = con-stant planes. The rate of pre-cession is independant of k
5 Expected Results
There has been a lot of work done in predicting and finding the magnetoresistance of metals in a varietyof conditions. Many authors consider magnetoresistance in the classical case, as the bending of electrontrajectories by the Lorentz force. Approached in this way, “the effect of the magnetic field is almost alwaysto increase the resistance” [10, p. 490].
We will now justify this claim using a physically intuitive approach. Although the dotted path in figure (8) islarger than the dashed path, the acceleration (∝ vH sinφ) is also larger; in the case of an isotropic metal therate of precession is independant of k. This result was reached in equation (15) where the time period for acomplete orbit in an isotropic metal is shown to be independant of electron momentum. If all electrons aredeflected through the same angle, the Hall field is able to deflect them all back to the a-axis direction andthere is no change in resistance. This result is true for all ellipsoidal Fermi surfaces.
In the case of a quasi two-dimensional metal it is not true that the rate of precession is independant of k. Inequation (74) we see that the time period T depends upon k(α). If the electrons are deflected by varyingamounts, then the Hall field will be unable to deflect them all back to the a-axis direction and they will movethroughout the ab-plane. As there is no increase in their energy (and consequently magnitude of velocity)this must result in an increase of resistance.
A few other well-known general results are also worth mention. Kohler’s rule states that
∆ρ
ρ0= F
(H
ρ0
)(88)
where F is a function which depends only upon the metal and geometry of the fields. This rule depends uponthere being a single relaxation time τ which is inversely proportional to ρ0 for a given metal. Deviations fromKohler’s rule exist when relaxation times depend upon temperature or differ depending upon the cause ofscattering.
22
Figure 9: Two electron wavefunctions(dashed and dotted) constructively in-tefere. When a magnetic field is appliedthe dashed wavefunction changes phaseand inteference is no longer be construc-tive.
Ong et al. in [6] show that magnetoresistance is equal to the variance of the local hall angle θH , the anglethrough which the electron moves when a magnetic field is applied. This agrees with the isotropic case whereall electrons are deflected by the same amount and so the variation on this amount is zero. However thisresult is valid only when the magnetic field is out of plane.
There are also predictions for the TMR in the weak- and strong-field limits. In weak fields it can be shown[1, p82-83] that
∆ρ
ρ= (Ωcτ)2 (89)
which is due to the curving of electron trajectories. It is also shown [1, p.87-90] that in strong fields theresistance saturates in the direction of closed electron orbits (as in the case of in-plane TMR which we areinvestigating).
Finally, the method we have used follows that done by Schofield and Cooper in [7]. Our expressions forconductance match exactly except for a factor of α2 in expression (80).
5.1 Negative magnetoresistance
Negative magnetoresistance has been found experimentally and is often attributed to strong anisotropicscattering from impurities or quantum effects such as weak localisation. In [9] negative magnetoresistancewas found in a metal with nonplanar topography. Negative magnetoresistance has also be found in layers offerromagnetic and non-ferromagentic material in the form of Giant Magnetoresistance GMR.
Weak localisation is found in metals where electrons are scattered from impurities in a random walk. Quantummechanics tells us that to find the probability of an electron taking a particular path we must considerinteference. Electrons which travel in a closed loop can also travel around the same loop counter-clockwise,which leads to constructive inteference. The probability that the electron remains localised is increased andso resistivity increases.
When a magnetic field is applied we see the Aharonov-Bohm effect; the phase of the electron wavefunction ischanged in on direction but not the other. Inteference is no longer constructive, the probability that electronremains localised is reduced, and the resistivity decreases. See figure (9).
Adjusting the relaxation time τ so that it becomes field-dependant could account for weak localisation effects.
23
6 Results
In figure (10) we compare our results to that found by Figarova and Figarov. We have used the same Ωcτrange, the same value for δ and it is for the same geometry. These plots are for the case of closed orbits only,δ = 2. We see that they produce a region of negative magnetoresistance for small Ωcτ , whilst we do not.
(a) Our results
(b) Figarova and Figarov results
Figure 10: Comparison of our results against Figarova & Figarov’s.
24
Figure (11) displays our results for small (11a) and large (11b) values of Ωcτ . We see a clear quadraticdependance at small fields and saturation at large fields .
(a) Small Ωcτ
(b) Large Ωcτ
Figure 11: TMR at small and large fields.
25
We see in figure (12) that in the case of mixed orbits, with δ = 0.5, we have the same quadratic dependanceat small field, saturation at large field and positive TMR throughout.
(a)
(b)
Figure 12: TMR for a range of field strengths when δ = 0.5, the case of open and closed orbits.
26
TMR for a range of δ values is plotted in figure (13). As δ increases the TMR increases.
(a)
(b)
Figure 13: TMR for a range of δ values.
27
7 Discussion
7.1 Comparison with Figarova and Figarov
In Figarova and Figarov’s plot (10b) we clearly see a region of negative magnetoresistance in the limit of
small fields. Note that as Ωcτ → 0,∆ρ
ρdoes not tend towards zero - a sharp change in behaviour is required
for the weak to zero field transition. In comparison, the plot we have produced (10a) is a smooth functiontending towards zero at weak fields and shows no region of negative magnetoresistance.
Figures (11a), (11b) show behaviour at weak and strong fields respectively. We see that there is the expectedquadratic behaviour at weak fields and saturation at strong fields.
Figures (10b), (11) are for the case δ = 2, where all orbits are closed. Figure (12) plots the TMR whenδ = 0.5 and there are mixed orbits. We see that the behaviour is very similar to our previous result - quadraticat weak fields with saturation at high fields.
The TMR is smaller in the case of open orbits than it is in the case of closed orbits. This is not expected - aswe approach the ellipsoidal limit (as δ →∞ ) we would expect the TMR to approach zero, not deviate awayfrom it.
Figure (13) plots TMR against Ωcτ for a wide range of delta values. We see that as delta increases the TMRincreases. There appears to be two levels of saturation - one for δ > 1 and one for δ < 1. This is alsoperculiar behaviour and indicates that there may be a problem with the computer programme.
7.2 Figarova and Figarov’s method
To understand why we should reach different behaviour from that of Figarova and Figarov when the sameassumptions have been used we will need to understand the methodology of their paper [4]. Unfortunatelythe theory underpinning their work is not fleshed out in the paper itself and has been hard to find elsewhere.They appear to use a method proposed by B.M Askerov in [3] - unfortunately only part of this text could befound for consideration.
The Askerov method they employ appears to be valid for the following ellipsoidal dispersion relation:
B(ε) =h2
2
(k2xm1
+k2ym2
+k2zm3
)(90)
where B is a function of energy. This differs from that considered in [4] except in the limit of δ →∞ wherethe fermi surface becomes ellipsoidal. If this is the case then Figarova and Figarov have employed an equationwhich was designed for use with other dispersion relations and that is the reason they have found negativemagnetoresistance.
7.3 Further work
To produce a strong argument against Figarova and Figarov’s claims we must first make sense of our results.The fact that they do not agree with very well accepted behaviour - zero magnetoresistance for ellipsoidaldispersions - indicates that there is something wrong with either our expression for magnetoresistance or thecomputer programme.
To verify our expression for magnetoresistance we could calculate the TMR for the ellipsoidal case (which isthought can be done analytically using the Abrikosov Chambers method) and compare this with our result inthe limit δ →∞. There may also exist methods to find the saturation level in high fields.
28
We could also extend our method to incorporate the effects of weak localisation of anisotropic scattering. Inthe case of weak localisation we could introduce a dispersion relation which is dependant upon the magneticfield. For anisotropic scattering we could use a relaxation time tensor which is dependant upon the wavevectorof an electron.
29
8 Summary
The aim of this report is to persuade the reader that the region of negative TMR in the weak field limitas claimed by Figarova and Figarov in [4] is unjustified. We began by introducing magnetoresistance, theBoltzmann transport equation and the Abrikosov-Chambers method which was then applied to the isotropiccase. We then focused upon our quasi two-dimensional dispersion relation and derived expressions for electronvelocity in terms of Jacobi elliptic functions. After deriving one term of the conductivity tensor we thenpresented the expression for magnetoresistance.
Previous work in the field was discussed, with a focus on general results applicable to our problem. We thenpresented plots of TMR against Ωcτ which clearly show no region of negative magnetoresistance for any valueof δ. Although quadratic behaviour and saturation was found at the weak and strong fields respectively, wedid not find TMR tending towards zero in the ellipsoidal limit which contradicts existing well-accepted results.For this reason we can not make any strong statement as to the veracity of Figarova and Figarov’s claimsalthough we have highlighted a number of issues with their results.
30
References
[1] A. A. Abrikosov. Fundamentals of the theory of metals (North-Holland, 1988).
[2] N. W. Ashcroft and N. D. Mermin. Solid State Physics (Holt, Rinehart and Winston, 1976).
[3] B. M. Askerov. Electron transport phenomena in semiconductors (World Scientific, 1993).
[4] S. R. Figarova and V. R. Figarov. Transverse magnetoresistance in layered electron systems. EPL,89(37004) [2010].
[5] J. M. Harris et al. Violation of Kohler’s Rule in the Normal-State Magnetoresistance ofYBa2Cu3O7−δ and La2SrxCuO4. PRL, 75(7) [1995].
[6] N. P. Ong. Geometric interpretation of the weak-field Hall conductivity in two-dimensional metals witharbitrary Fermi surface. PRB, 43(1) [1991].
[7] A. J. Schofield and J. R. Cooper. Quasilinear magnetoresistance in an almost two dimensionalband structure. PRB, 69(16) [2000].
[8] M. F. Smith and R. H. McKenzie. Anisotropic scattering in angular-dependent magnetoresis-tance oscillations of quasi-2D and quasi-1D metals: beyond the relaxation-time approximation. PRB,77(235123) [2008].
[9] N. M. Sotomayor et al. Negative linear classical magnetoresistance in a corrugated two-dimensionalelectron gas. PRB, 70(235326) [2004].
[10] J. M. Ziman. Electrons and phonoms (Oxford University Press, 1960).
[11] J. M. Ziman. Principles of the theory of solids (University printing house, Cambridge, 1972).
31
A Q2D TMR.py
1 #! / u s r / b in / env python
3 import s c i p y . i n t e g r a t e as sp#s c i e n t i f i c l i b r a r y , i n t e g r a t i o n module .
5 import s c i p y . s p e c i a l as s p s#s c i e n t i f i c l i b r a r y , s p e c i a l f u n c t i o n s module .
7 import numpy as np# mathemat i ca l l i b r a r y
9 import m a t p l o t l i bimport m a t p l o t l i b . p y p l o t as p l t
35 ####### User I n pu t s ####################################################
37 o t = np . l i n s p a c e ( 0 . 0 0 1 , 0 . 1 , 1 0 )d e l = np . a r r a y ( [ 0 . 5 ] )
39 # ar r a y o f v a l u e s f o r omega tau and d e l t a . Can e n t e r e x p l i c i t l y# ( np . a r r a y ( [ a , b . . ] ) ) o r c r e a t e a l i n e a r l y spaced a r r a y
41 # (np . l i n s p a c e ( s t a r t , stop , s t e p ) ) .p r o c e s s = 2
43 #1: mu l t i p l e p l o t s f o r tmr v omega c#2 : s i n g l e p l o t f o r tmr v omega c
45 #3: mu l t i p l e p l o t s f o r tmr v d e l t a#4 : s i n g l e p l o t f o r tmr v d e l t a
51 l e n g t h d e l = np . s i z e ( d e l )# np . s i z e ( a r r a y ) r e t u r n s l e n g t h o f the a r r a y .
53 TMR= np . empty ( shape =( l e n g t h o t , l e n g t h d e l ) )# Create a two−d imen s i o n a l empty a r r a y to which v a l u e s o f TMR w i l l be
55 # added . omega tau l a b e l s rows , d e l t a l a b e l s columns . The c r e a t i o n o f an# empty a r r a y i s an improvement upon i n e f f i c i e n t ’ append ’ methods which
57 # c r e a t e mu l t i p l e c o p i e s o f the same a r r a y .
32
59def c a l c u l a t e ( omega tau , d e l t a ) :
61 #Ca l c u l a t e TMR a long a−a x i s f o r a g i v en v a l u e o f d e l t a and omega tau
63 s i g m a a a = GetSigma aa ( omega tau , d e l t a )#c a l c u l a t e c o n d u c t i v i t y a l ong a a x i s
65s i g m a c c = GetSigma cc ( omega tau , d e l t a )
67 #c a l c u l a t e c o n d u c t i v i t y a l ong c a x i s
69 s i g m a a c = GetSigma ac ( omega tau , s i g m a c c )#c a l c u l a t e Ha l l component
71s i g m a c a = GetSigma ca ( s i g m a a c )
73 #c a l c u l a t e Ha l l component
75 TMR = s i g m a c c / ( ( 1 . / np . p i ) ∗ s i g m a a a ∗ s i g m a c c − (2∗ np . p i ) ∗ s i g m a a c ∗s i g m a c a )
r e t u r n TMR77 #Use th e s e c o n d u c t i v i t e s to c a l c u l a t e TMR a long a a x i s .
79 def n o r m a l i s e (TMR, TMR zero ) :p r i n t ’TMR’ , TMR
81 p r i n t ’TMR z e r o ’ , TMR zero#This removes unneeded con s t a n t s
83 TMR normal = (TMR − TMR zero ) /TMR zeror e t u r n TMR normal
85def GetSigma aa ( omega tau , d e l t a ) :
87 #c a l c u l a t e c o n d u c t i v i t y a l ong a a x i si f d e l t a > 1 :
89 #I f de l t a>= 1 then a l l o r b i t s a r e c l o s e ds i g m a a a = 8∗ I n t e g r a l ( i n t e g r a n d 1 , 0 , 1 . / d e l t a , omega tau , d e l t a )
91 r e t u r n s i g m a a ai f d e l t a <= 1 :
93 #I f d e l t a < 1 then t h e r e i s a m ix tu r e o f open and c l o s e d o r b i t ss i g m a a a = 8∗ I n t e g r a l ( i n t e g r a n d 1 , 0 , 1 , omega tau , d e l t a ) + I n t e g r a l (
i n t e g r a n d 2 , d e l t a , 1 , omega tau , d e l t a )95 r e t u r n s i g m a a a
97 def GetSigma cc ( omega tau , d e l t a ) :#c a l c u l a t e c o n d u c t i v i t y a l ong c a x i s
99 i f d e l t a > 1 :#I f de l t a>= 1 then a l l o r b i t s a r e c l o s e d
101 s i g m a c c = I n t e g r a l ( i n t e g r a n d 3 , 0 , 1 . / d e l t a , omega tau , d e l t a )r e t u r n s i g m a c c
103 i f d e l t a <= 1 :#I f d e l t a < 1 then t h e r e i s a m ix tu r e o f open and c l o s e d o r b i t s
105 s i g m a c c = I n t e g r a l ( i n t e g r a n d 3 , 0 , 1 , omega tau , d e l t a ) + 4∗ I n t e g r a l (i n t e g r a n d 4 , d e l t a , 1 , omega tau , d e l t a )
r e t u r n s i g m a c c107
def GetSigma ac ( omega tau , s i g m a c c ) :109 #c a l c u l a t e Ha l l component
s i g m a a c = omega tau∗ s i g m a c c111 r e t u r n s i g m a a c
113 def GetSigma ca ( s i g m a a c ) :#c a l c u l a t e Ha l l component .
115 s i g m a c a = − s i g m a a c# Use symmetry o f geometry .
33
117 r e t u r n s i g m a c a
119 def I n t e g r a l ( i n t e g r a n d , lower , upper , omega tau , d e l t a ) :I n t e g r a l = sp . quad ( i n t e g r a n d , lower , upper , a r g s =(omega tau , d e l t a ) )
121 r e t u r n I n t e g r a l [ 0 ]# Sc ipy f u n c t i o n used to compute the d e f i n i t e i n t e g r a l o f
123 #’ i n t e g r a nd ’ from ’ l owe r ’ to#’ upper ’ . Adapted from Fo r t r an l i b r a r y QUADPACK. I f ’ f u n c t i o n ’ t a k e s
125 # more than one argument i t i s i n t e g r a t e d a long the a x i s o f the# f i r s t argument
127 # and use keyword a r g s to pas s i n o t h e r arguments .
129 def i n t e g r a n d 1 ( beta , omega tau , d e l t a ) :i n t e g r a n d 1 = s1 ( beta , omega tau ) /( s p s . e l l i p k ( b e t a )
131 ∗np . s q r t (1−b e ta ∗ d e l t a ) )# ’ s1 ’ i s a summation d e f i n e d below . ’ sp s . e l l i p k ’ i s a s c i p y
133 # fun c t i o n which c a l c u l a t e s# the qu a r t e r p e r i o d o f a j a o b i e l l i p t i c f u n c t i o n o f the f i r s t k i nd
135 # with modulus# passed i n as the argument .
137 r e t u r n i n t e g r a n d 1
139 def i n t e g r a n d 2 ( a lpha , omega tau , d e l t a ) :i n t e g r a n d 2 = (1 + 8∗ s2 ( a lpha , omega tau ) ) /( s p s . e l l i p k ( a l p h a )
141 ∗ a l p h a ∗∗2∗np . s q r t ( a lpha−d e l t a ) )# ’ s2 ’ i s a summation d e f i n e d below .
143 r e t u r n i n t e g r a n d 2
145 def i n t e g r a n d 3 ( beta , omega tau , d e l t a ) :i n t e g r a n d 3 = s3 ( beta , omega tau ) /( s p s . e l l i p k ( b e t a ) ∗∗3
147 ∗np . s q r t (1−b e ta ∗ d e l t a ) )# ’ s3 ’ i s a summation d e f i n e d below .
149 r e t u r n i n t e g r a n d 3
151 def i n t e g r a n d 4 ( a lpha , omega tau , d e l t a ) :i n t e g r a n d 4 = s4 ( a lpha , omega tau ) /( s p s . e l l i p k ( a l p h a ) ∗∗3∗ a l p h a ∗∗3
153 ∗np . s q r t ( a lpha−d e l t a ) )# ’ s4 ’ i s a summation d e f i n e d below .
155 r e t u r n i n t e g r a n d 4
157 def s1 ( beta , omega tau ) :sum=0
159 f o r n i n r a ng e (100 ,0 ,−1) :n = n−1./2.
161 sum += nome ( b e t a ) ∗∗(2∗n ) /(1+nome ( b e t a ) ∗∗(2∗n ) ) ∗∗2∗(1/(1 + ( n∗np .p i ∗omega tau / s p s . e l l i p k ( b e t a ) ) ∗∗2) )
163 n = 1 0 0 . 5sum next = sum + nome ( b e t a ) ∗∗(2∗n ) /(1+nome ( b e t a ) ∗∗(2∗n ) ) ∗∗2∗(1/(1 + ( n∗np
. p i ∗omega tau / s p s . e l l i p k ( b e t a ) ) ∗∗2) )165
r e t u r n CheckConvergence ( sum , sum next )167
def s2 ( a lpha , omega tau ) :169 sum=0
f o r n i n r a ng e (100 ,0 ,−1) :171 sum += nome ( a l p h a ) ∗∗(2∗n ) /(1+nome ( a l p h a ) ∗∗(2∗n ) ) ∗∗2∗(1/(1 + ( n∗np
. p i ∗omega tau /( s p s . e l l i p k ( a l p h a ) ∗np . s q r t ( a l p h a ) ) ) ∗∗2) )
173 n = 101sum next = sum + nome ( a l p h a ) ∗∗(2∗n ) /(1+nome ( a l p h a ) ∗∗(2∗n ) ) ∗∗2∗(1/(1 + ( n∗
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np . p i ∗omega tau /( s p s . e l l i p k ( a l p h a ) ∗np . s q r t ( a l p h a ) ) ) ∗∗2) )175
r e t u r n CheckConvergence ( sum , sum next )177
def s3 ( beta , omega tau ) :179 sum=0
f o r n i n r a ng e (100 ,0 ,−1) :181 n = n−1./2.
sum += ( n∗∗2∗nome ( b et a ) ∗∗(2∗n ) ) /(1+nome ( b e t a ) ∗∗(2∗n ) ) ∗∗2∗(1/(1 +( n∗np . p i ∗omega tau / s p s . e l l i p k ( b e t a ) ) ∗∗2) )
183n = 1 0 0 . 5
185 sum next = sum + ( n∗∗2∗nome ( b et a ) ∗∗(2∗n ) ) /(1+nome ( b e t a ) ∗∗(2∗n ) ) ∗∗2∗(1/(1+ ( n∗np . p i ∗omega tau / s p s . e l l i p k ( b e t a ) ) ∗∗2) )
187 r e t u r n CheckConvergence ( sum , sum next )
189 def s4 ( a lpha , omega tau ) :
191 sum=0f o r n i n r a ng e (100 ,0 ,−1) :
193 sum += ( n∗∗2∗nome ( a l p h a ) ∗∗(2∗n ) ) /(1+nome ( a l p h a ) ∗∗(2∗n ) ) ∗∗2∗(1/(1+ ( n∗np . p i ∗omega tau /( s p s . e l l i p k ( a l p h a ) ∗np . s q r t ( a l p h a ) ) ) ∗∗2) )
195 n = 101sum next = sum + ( n∗∗2∗nome ( a l p h a ) ∗∗(2∗n ) ) /(1+nome ( a l p h a ) ∗∗(2∗n ) )
∗∗2∗(1/(1 + ( n∗np . p i ∗omega tau /( s p s . e l l i p k ( a l p h a ) ∗np . s q r t ( a l p h a ) ) ) ∗∗2))
197r e t u r n CheckConvergence ( sum , sum next )
199def nome ( b et a ) :
201 r e t u r n np . exp(−np . p i ∗ s p s . e l l i p k (1−b e ta ) / s p s . e l l i p k ( b et a ) )
203 def CheckConvergence ( sum , sum next ) :i f sum next == sum :
205 r e t u r n sume l s e :
207 p r i n t ’ The summation has not c o n v e r g e d ’r a i s e K e y b o a r d I n t e r r u p t
209def p r o c e s s ( n ) :
211 np . s a v e t x t ( ’TMR. t x t ’ , TMR)# Save a r r a y o f r e s u l t s to . t x t f i l e
213m a t p l o t l i b . r c ( ’ a x e s ’ , l a b e l s i z e =16)
215i f n==1:
217 c o u n t d e l =0f o r d e l t i n d e l :
219 #f o r each v a l u e o f d e l t a p l o t TMR vs omega c . Save thef i g u r e
#then c l o s e .221 p l t . f i g u r e ( )
p l t . p l o t ( ot , TMR [ : , c o u n t d e l ] , c o l o r= ’ g r e e n ’ , marker= ’ . ’ ,l a b e l= ’ d e l t a =%.3 f ’%d e l t )
223 p l t . x l a b e l ( r ’ $\omega c \ tau$ ’ )p l t . y l a b e l ( r ’ $\D e l t a \ rho$ / $\ rho ( 0 ) $ ’ )
225 p l t . t i t l e ( r ’TMR vs $\omega c\ tau$ ’ )p l t . l e g e n d ( )
227 p l t . s a v e f i g ( ’ d e l t a =%.3 f . png ’%d e l t )
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p l t . draw ( )229 p l t . show ( )
c o u n t d e l+=1231
i f n==2:233 p l t . f i g u r e ( )
c o u n t d e l =0235 f o r d e l t i n d e l :
#p l o t TMR vs omega c . Save the f i g u r e237 #then c l o s e .
p l t . p l o t ( ot , TMR [ : , c o u n t d e l ] , marker= ’ . ’ , l a b e l= ’ d e l t a=%.3 f ’%d e l t )
239 c o u n t d e l+=1p l t . x l a b e l ( r ’ $\omega c \ tau$ ’ )
241 p l t . y l a b e l ( r ’ $\D e l t a \ rho$ / $\ rho ( 0 ) $ ’ )p l t . t i t l e ( r ’TMR vs $\omega c\ tau$ ’ )
243 p l t . l e g e n d ( )p l t . s a v e f i g ( ’ ManyDelta . png ’ )
245 p l t . draw ( )p l t . show ( )
247i f n==3:
249 c o u n t o t = 0f o r ot i n o t :
251 #f o r each v a l u e o f omega c p l o t TMR vs d e l t a . Save thef i g u r e
#then c l o s e .253 p l t . f i g u r e ( )
p l t . p l o t ( d e l , TMR [ c o u n t o t , : ] , c o l o r= ’ g r e e n ’ , marker= ’ . ’ ,l a b e l=r ’ $\omega c \ tau$ = %.3 f ’%ot )
255 p l t . x l a b e l ( r ’ $\ d e l t a $ ’ )p l t . y l a b e l ( r ’ $\D e l t a $ $\ rho$ / $\ rho ( 0 ) $ ’ )
257 p l t . t i t l e ( r ’TMR vs $\ d e l t a $ ’ )p l t . l e g e n d ( )
259 p l t . draw ( )p l t . show ( )
261 p l t . s a v e f i g ( ’ omega−tau=%.3 f . png ’%ot )c o u n t o t+=1
263i f n==4:
265 p l t . f i g u r e ( )c o u n t o t = 0
267 f o r ot i n o t :#p l o t TMR vs d e l t a . Save the f i g u r e
269 #then c l o s e .p l t . p l o t ( d e l , TMR [ c o u n t o t , : ] , marker= ’ . ’ , l a b e l=r ’ $\
omega c \ tau$ = %.3 f ’%ot )271 c o u n t o t+=1
p l t . x l a b e l ( r ’ $\ d e l t a $ ’ )273 p l t . y l a b e l ( r ’ $\D e l t a $ $\ rho$ / $\ rho ( 0 ) $ ’ )
p l t . t i t l e ( r ’TMR vs $\ d e l t a $ ’ )275 p l t . l e g e n d ( )
p l t . draw ( )277 p l t . show ( )
p l t . s a v e f i g ( ’ManyOmega . png ’ )279
281 i f n a m e == ” m a i n ” :# This a l l ow s the code below to be ran as a s c r i p t ( ’ python
283 # MagnetoCode . py ’ on command l i n e ) . I f the module were impor ted t h i s# code would not e x e cu t e .
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285 c o u n t d e l =0f o r d e l t i n d e l :
287 p r i n t ’ d e l t= ’ , d e l t#f i x d e l t a
289 TMR zero = c a l c u l a t e ( 0 . , d e l t )#Ca l c u l a t e TMR when t h e r e i s no magnet ic f i e l d a p p l i e d
291 c o u n t o t=0f o r ot i n o t :
293 #f i x omega taup r i n t ’ omega tau= ’ , ot
295 TMR = c a l c u l a t e ( ot , d e l t )#Ca l c u l a t e TMR a long a−a x i s f o r a g i v en v a l u e o f d e l t a
and297 #omega tau
TMR [ ( c o u n t o t , c o u n t d e l ) ] = n o r m a l i s e (TMR, TMR zero )299 # Remove unneeded c on s t a n t s
c o u n t o t +=1301 #Move a long omega tau a x i s o f a r r a y
c o u n t d e l+=1303 #Move a long d e l t a a x i s o f a r r a y
305 p r o c e s s ( p r o c e s s )#Save r e s u l t s to f i l e and produce p l o t s .
