Journal Club

New Spin Mechanism forNegative Magnetoresistance in

Hopping RegimePRB 89, 100201(R) 2014

Thuong T. Nguyen

April 16, 2014

Outline

1 Overview of Magnetoresistance (MR) in Hopping Regime

♠ Orbital Mechanism

♠ Spin Mechanism

2 New Spin Mechanism

♠ Qualitative picture

♠ Simple Model

Overview of Magnetoresistance

in Hopping Regime

Hopping ConductionStrongly localized systems at low temperatureThe main contribution to electrical conductivity comes from

electrons hopping between impurities (tunneling).

MR in hopping regime NOT well understood.

Orbital-related mechanisms Spin-related mechanisms

Orbital Mechanism

Metallic regime: Negative MR is explained by weaklocalization theory (back-scattering).

∝ 1kF l

Orbital MechanismHopping regime: anomalously large negative MR1.

∝ B4/5

Characteristics• field: H0 ∼ Φ0/S = ~c/eS.

• ANISOTROPY in two dimension.

1Ioffe & Spivak, JETP 117, 551 (2013)

Absence of interference effects for free spins2

(more on Shumilin and Kozub, PRB 85,115203 (2012))

2Shklovskii & Spivak, in Hopping Transport in Solid, 1991

Spin Mechanism

{ISOTROPIC + POSITIVE} Magnetoresistance 3

3Kamimura et. al, 1985

Spin Mechanism

{ISOTROPIC + POSITIVE} Magnetoresistance 3

3Kamimura et. al, 1985

Spin Mechanism

{ISOTROPIC + POSITIVE} Magnetoresistance 3

3Kamimura et. al, 1985

Summary

Orbital Mechanism

• Negative

∝ B4/5

• Anisotropy

Spin Mechanism

• Positive

∝ B2

• Isotropy

New Spin Mechanism

New Spin Mechanism

• Characters: NEGATIVE + ISOTROPIC

• Ingredients: fluctuation of g factor in space; long memory of

non-equilibrium spin correlation.

Qualitative PictureHopping rate of an electron i→ j depends on the relative spin

configuration of hoping electron and a spin nearby.

Qualitative Picture1 Spin-memory ←− nonequilibrium electric current: decreases

conductivity for H = 0

2 Strong disordered system: random g factor: increases

conductivity.

Qualitative Picture1 Spin-memory ←− nonequilibrium electric current: decreases

conductivity for H = 0

2 Strong disordered system: random g factor: increases

conductivity.

Qualitative Picture

1 Spin-memory ←− nonequilibrium electric current

2 Strong disordered system: random g factor

Characteristic field:

δgµBH∗τ ∼ 1

with τ � τsEstimation: R ∼ 10−9Ω, δg ∼ 0.01 −→ µBnH∗/T ∼ 10−4, i.e

field of order of gauss at T ∼ 1K .

Simple model

• Main assumptions:� indirect transition rate: γij � 1 - perturbation parameter.� link spins are rare.

• Parametrize states: P 0i + TrP̂ 1i = 1

ni = TrP̂1i , Si = Tr(σP̂

1i )

Simple model

d 〈ni〉dt

= −∑j

[〈ni〉+ γij (〈ni〉 − 〈Si · sij〉)

τi→j− (i↔ j)

]d 〈Si〉dt

= hi × 〈Si〉 −∑j

[〈Si〉+ γij (〈Si〉 − 〈nisij〉)

τi→j− (i↔ j)

]d 〈sij〉dt

= hij × 〈sij〉

Simple modelNon-equilibrium variables:

〈ni〉 → neqi (1 + ψi) , Si → neqi S̃i, sij → sij

=⇒

neqidψidt

=∑j

1

τij

[ψj − ψi − γij

〈(S̃j − S̃i

)· sij

〉]neql

[(d

dt+

1

τs

)Cαβl;ij − �αγδh

γl C

αβl;ij − �βγδh

γijC

αδl;ij

]= −

∑k 6=l

Cαβl;ij − Cαβk;ij

τlk+ γijδαβ (δil − δjl)

ψi − ψjτij

γij

〈(S̃j − S̃i

)· sij

〉= Qij (H) (ψj − ψi)

Gij =e2

Tτij[1−Qij (H)]

Simple model: Resultσ(H →∞)− σ(0)

σ(0)∼ A = ργ2ij !Small

δσ(H)

Aσ(0)∼ −Γ

(−ds

2

) 1∑l=−1

[(ilH

H∗+τ

τs

)ds/2+

(τ

τs

)ds/2]

Conclusion

Negative Magnetoresistance

Spin memory effect Strongly disordered systems

Thanks for Your Attention!

Overview of Magnetoresistance (MR) in Hopping Regime Orbital Mechanism Spin Mechanism

New Spin Mechanism Qualitative picture Simple Model