Transport in SolidsIntroduction

Peter M LevyNew York University

A general review of the history of GMR can be found in:

http://wiki.nsdl.org/index.php/PALE:ClassicArticles/GMR

Material I cover can be found in

General:

Solid State Physics, N.W. Ashcroft and N.D. Mermin (Holt, Rinehardt and Winston, 1976)

Electronic Transport in Mesoscopic Systems, S. Datta (CambridgeUniversity Press, 1995).

Transport Phenomena, H. Smith and H.H. Jensen ( Clarendon Press,Oxford, 1989).

J. Rammer and H. Smith, Rev. Mod. Phys. 58, 323 (1986).

Ab-initio theories of electric transport in solid systems with reduceddimensions, P. Weinberger, Phys. Reports 377, 281-387 (2003).

Electrical conduction in magnetic media

How we got from 19th century concepts to applications in computer storage and memories.

1897- The electron is discovered by J.J. Thomson

~1900 Drude model of conductionbased on kinetic theory of gases {PV=RT}

~1928 Sommerfeld model of conduction in metals

€

l ~ 100 Ao

n⇒ neff ~ N (ε F )×δε

N(ε F ) ≅ nε F

;

δε = kBT or eEl

€

V = IR = RI Ohm's law

j = IA ; E =V

L ; ρ = ARL

⇒ E = ρj

r j = ner v

r v avg = er E τm

; τ is the time between collisions

r j = ne2τ

m ⎛ ⎝ ⎜

⎞ ⎠ ⎟r E =σ

r E

ρ = 1σ

l = vτ = mean free path ~ 10 -100A

Phenomena

While each atom scatters electrons, when they form a periodic array the atomic background only electrons from one state k to another with k+K.

This is called Bragg scattering; it is responsible for dividing the continuous energy vs. momentumcurve into bands.

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n⇒ neff ~ N (ε F )×δε ; N(ε F ) ≅ nε F

; δε = kBT or eEl

Provides explanation for negligible contribution of conduction electrons to specific heatof metals.

What distinguishes a metal from an insulator

Magnetoresistance

Lorentz force acting on trajectory of electron;longitudinal magnetoresistance (MR).

A.D. Kent et alJ. Phys. Cond.Mat. 13, R461(2001)

Anisotropic MR

A.D. Kent et alJ. Phys. Cond.Mat. 13, R461(2001)

Role of spin-orbit coupling on electron scattering

Domain walls

ReferencesSpin transport: Transport properties of dilute alloys, I. Mertig, Rep. Prog. Phys. 62, 123-142 (1999).

Spin Dependent Transport in Magnetic Nanostructures, edited byS. Maekawa and T. Shinjo ( Taylor and Francis, 2002).

GMR:Giant Magnetoresistance in Magnetic Layered and Granular Materials, by P.M. Levy, in Solid State Physics Vol. 47, eds. H. Ehrenreich and D. Turnbull (Academic Press, Cambridge, MA, 1994) pp. 367-462.

Giant Magnetoresistance in Magnetic Multilayers, by A. Barthélémy,A.Fert and F. Petroff, Handbook of Ferromagnetic Materials, Vol.12,ed. K.H.J. Buschow (Elsevier Science, Amsterdam, The Netherlands,1999) Chap. 1.

Perspectives of Giant Magnetoresistance, by E.Y. Tsymbal and D,G.Pettifor, in Solid State Physics Vol. 56, eds. H. Ehrenreich and F. Spaepen (Academic Press, Cambridge, MA, 2001) pp. 113-237.

CPP-MR:M.A.M. Gijs and G.E.W. Bauer, Adv. in Phys. 46, 285 (1997).

J. Bass, W.P. Pratt and P.A. Schroeder, Comments Cond. Mater. Phys.18, 223 (1998).

J. Bass and W.P. Pratt Jr., J.Mag. Mag. Mater. 200, 274 (1999).

Spin transfer:

A. Brataas, G.E.W. Bauer and P. Kelly, Physics Reports 427,157 (2006).

Spintronics- control of current through spin of electron

The two current model of conduction in ferromagnetic metals

Parallel configuration Antiparallel configuration

1988 Giant magnetoresistanceAlbert Fert & Peter Grünberg

Two current model in magnetic multilayers

Data on GMR

M.N. Baibich et al., Phys. Rev. Lett. 61, 2472 (1988).

400 H (Oe)-40

400

110

H (kOe)-40 H / / [ 011]

spin-valve

multi-layer GMR -metallic spacer between magnetic layers-current flows in-plane of layers

Co95Fe5/Cu[110]

R/R~110% at RTField ~10,000 Oe

Py/Co/Cu/Co/Py

R/R~8-17% at RTField ~1 Oe NiFe + Co

nanolayer

NiFeCo nanolayerCuCo nanolayerNiFeFeMn

H(Oe)

H(kOe)[011]

S.S.P. Parkin

GMR in Multilayers and Spin-Valves

Current in the plane (CIP)-MR

vs

Current perpendicular to the plane (CPP)-MR

1995 GMR heads

From IBM website; 1.swf 2.swf

Tunneling-MR

Two magnetic metallic electrodes separated by an insulator; transport controlled by tunneling phenomena not by characteristics of conductionin metallic electrodes

2000 magnetic tunnel junctions used in magnetic random access memory

From IBM website;

http://www.research.ibm.

com/research/gmr.html

PHYSICAL REVIEW LETTERS VOLUME 84, 3149 (2000)Current-Driven Magnetization Reversal and Spin-Wave Excitations in CoCuCo PillarsJ. A. Katine, F. J. Albert, and R. A. BuhrmanSchool of Applied and Engineering Physics, Cornell University, Ithaca, New York 14853E. B. Myers and D. C. RalphLaboratory of Atomic and Solid State Physics, Cornell University, Ithaca, New York 14853

δm(z)

ML

MR

j

z

Spin Accumulation from left layer

δm(z)

ML

MR

j

z

Spin Accumulation-left layer-current reversed

How reversal in current directions changes alignment of layers

How can one rotate a magnetic layer with a spin polarized current?

By spin torques:Slonczewski-1996Berger -1996Waintal et al-2000Brataas et al-2000

By current induced interlayer coupling:Heide- 2001

Current induced switching of magnetic layers by spin polarized currents can be divided in two parts:

Creation of torque on background by the electric current, and

reaction of background to torque.

Latter epitomized by Landau-Lifschitz equation; micromagnetics

Former is current focus article in PRL:Mechanisms of spin-polarized current-driven magnetization switchingby S. Zhang, P.M. Levy and A. Fert. Phys. Rev. Lett. 88, 236601 (2002).

Extension of Valet-Fert to noncollinear multilayers

Methodology

Structures

•Metallic multilayers

•Magnetic tunnel junctions•Insulating barriers•Semiconducting barriers•Half-metallic electrodes•Semiconducting electrodes

To discuss transport two calculations are necessary:

•Electronic structure, and

•Transport equations; out of equilibrium collective electron phenomena.

different length scales

Prepared by Carsten Heide

Lexicon of transport parameters

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εF = Fermi energy

vF = Fermi velocity = 1h

∂ε∂k

kF = Fermi momentumτ mfp = Mean time between collisions

λ mfp = Distance travelled between collisions

⇒ G(r − r',ε F )∝ ei(kF + iλ ) r−r '

≈ vFτ mfp

Spin independent transport

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τ s = Spin dependent relaxation time s =↑,↓ / M,mτ sf = Time between spin flips

λ sdl ≅ λ sf λ mfp = Spin diffusion length

dJ = hvFJ = Spin coherence length

due to temporal precession;J = exchange constantλ tr = Transverse spin coherence length

≅ λ J ≅ dJ λ mfp = transverse spin diffusion length

lc = 1kF↑ − kF↓

= Transverse spin coherence length

due to spatial precession.

Spin dependent transport parameters

Spin and charge accumulation in metallic systems

Derivation of Landauer formula (see Datta)

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I = nevFor a conductor of length L (one dimensional electron gas)the electron density for each k state is 1/L. Thus the currentcarried by k states travelling in one direction is

I+ = eL

vf +

k∑ (ε)M(ε) = e

L1h

∂ε∂k

f +

k∑ (ε)M(ε)

⇒ 2(spin)k

∑ L2π

dk∫

I+ = 2eh

dεf +∫ (ε)M(ε)

I = I+(μ1) − I−(μ2) = 2eh

M(μ1 − μ2) = 2e2

hMV

Gc = 2e2

hM

Therefore the contact resistance of a ballistic conductor is

Gc−1 = 12.9kΩ

M

Landauer reasoned that when the conductor is not perfectlyballistic, i.e., has a transmission probability T<1 that

€

G = 2e2

hMT

so that

G−1 = h2e2M

1T

= h2e2M

+ h2e2M

1− TT

= Gc−1 + Gs

−1

In other words when T < 1 in addition to the contact resistancethere is a reistance due to the scattering in the conductor.

While the latter is independent of the length L of the conductor,it can be directly related to "ohmic" resistance as follows.

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For a wide conductor W with many channels or modes of conductionM ~ W/(π/kF), so that the conductance is

G = e2W (m /πh2)(vFT(L) /π )

How does T depend on L?If we neglect treat the quantum interference between electrons, the transmission probability through a conductor of length L which containsscatterers is :

T(L) = L0

L + L0

where L 0 is the average distance travelled between scatterings. Thisis derived as follows :

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When one has a sequence of two scatterings with probabilities of transmissionT1 and T2 , then the joint transmission probability is not simply the product of them;rather one have to take into account the multiple reflections :

T1T2R1R2 ,T1T2R12R2

2 ,..... so one arrive at,

T12 = T1T2

1− R1R2

where Ti =1− Ri . This can be rewritten as

1-T12

T12

= 1- T1

T1

+1- T2

T2

,

which for N scatterings in series can be extended to

1-T (N )T (N )

= N 1-TT

⇒ T (N ) = TN(1−T )+T

.

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If we denote the linear density of scatters in a sample as υ ; then the number of scatterers in a conductor of length L is N = υL. Placing this in the preceeding equation we find

T(L) = L0

L + L0

where L0 = Tυ (1− T)

.

To identify L0 we note that the mean free path,Lm , is the average distance an electron travels before being scattered; as the probability of scattering is (1- T) we canwrite

(1− T)υLm ~ 1 ⇒ Lm = 1υ (1− T)

~ L0,

when T ~ 1.

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By placing this result

T(L) = L0

L + L0

into the conductance

G = e2W (m /πh2)(vFT(L) /π )we find

G = e2(m /πh2)(vF L0 /π ) WL + L0

⇒ G−1 = L + L0

σW.

We thereby arrive at resistance made up of the combination of an actual (Ohmic), and a contact resistance :

Gs−1 = L

σW, and Gc

−1 = L0

σW.

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In general we can always write

G−1 = h2e2m

1T

= h2e2m

+ h2e2m

1− TT

= Gc−1 + actual resistance.

Conclusion

The contact resistance is also known as the Sharvin resistance.

Ballistic transport: see S. Datta Electronic Transport in Mesoscopic Systems (Cambridge Univ. Press, 1995).

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Collisionless regime; transport conditions set by reservoirs at boundaries. Conductance measured by transmission

through states on Fermi surface Tkσ →k 'σ ' ∝ tk 'σ ',kσ2

in units of the quantum of conduction 2e2 /h =12.9kΩ−1

G = 2e2

hMT , where M is the number of channels.

Critique of the “mantra” of Landauer’s formula; see M.P. Das and F. Green, cond-mat/0304573 v1 25Apr 2003.

Application to magnetic multilayers

Semi-classical approaches to electron dynamics

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External fields are treated classically, while potential of periodic background is not.

r•

= vn(k) =1 h ∂εn (k) ∂k( )

hk•

= −e E(r, t)+ 1c

vn (k)× H (r, t) ⎡ ⎣ ⎢

⎤ ⎦ ⎥

f (ε n(k)) = 1e εn (k )−μ( ) /kBT +1

Validity

€

As long as one does not try to localize electron on length scale of unit cell, and wavelength of applied fields long compared to lattice constant.

eEa,hωc <<ε gap2 (k) ε F .

Diffusive transport

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Collisions assure local equilibrium of current; howevera << λ mfp << L, where a is lattice constant. Also,

λ mfp << phase coherence length of wavefunctions.

In the diffusive regime processes that occur on a length scale long with repsect to the mean free path have to be averaged,e.g., the distance traversed by an electron undergoing randomscattering is

L2 ~τ

τ mfp

(λ mfp cosϑ )2 ⇒ 1/3 vF2ττ mfp .

By definning a diffusion constant D ≡1/3vF2τ mfp we find

L2 = Dτ .

In this regime one can usually describe transport by semi - classical Boltzmann equation. This is an equationof motion for an electron distribution function, f (r,k, t).

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∂f ∂t + v ⋅∇f − eE ⋅vδ(ε −ε F )

= −1 τ f − f{ }

€

df (k,r, t)dt

= dkdt

• ∇ k f + drdt

• ∇ r f + ∂f (k,r, t)∂t

From semiclassical electron dynamics :

r•

= vn (k) =1 h ∂εn (k) ∂k( )

hk•

= −e E(r, t) + 1c

vn (k) × H(r, t) ⎡ ⎣ ⎢

⎤ ⎦ ⎥

and from :

∇ k f = ∂εn (k) ∂k∗∂f 0 ∂εn ⇒ hvn (k)δ(εn −εF ) at T = 0K.Thus we find :

Simple derivation