Chapter 2. Spinelektronik: Grundlagen und Anwendung spinabhängiger Transportphänomene. Winter 05/06

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1 Winter 05/06 : Grundlagen und Anwendung spinabhängiger Transportphänomene Chapter 2 : Grundlagen und Anwendung spinabhängiger Transportphänomene 1

2 Winter 05/ Scattering of charges (electrons) In order to establish a steady current under an applied magnetic field, the electrons must be scattered inelastically. Main scattering mechanisms in (non-magnetic) metals Viewgraph 1 phonons (temperature dependent) defects (temperature independent) Viewgraph 2 Defects are responsible for residual resistivity at low temperature. 2.1 General Expression for j Current density r j = e 4π 3 dk r r ν ( k r ) f ( k r ) = σ E introduce expression for f ( r k ) = f o ( r k ) τ υ ee( f o E ) r j = ε 4π 3 d r k ( r υ ( r k ) f o (k)) r υ ( k r ) o r υ k r τ ( k r ) ee r f o E first term vanishes symm. Integration, asymm. Integrand σ = e2 4π 3 dk r τ ( k r ) r υ ( k r ) o r υ ( k r ) f o E Metals: Behavior determined by Fermi function f o changes only within kτ f o E = δ(e E F ) only states at Ε F contribute to the transport : Grundlagen und Anwendung spinabhängiger Transportphänomene 2

3 Winter 05/06 σ = e2 4π 3 h E=E F ds F τ ( k r r υ ( k r F ) F ) o r υ ( k r F ) r υ ( k r F ) Integration only over Fermi surface Actual shape of the Fermi surface is important. Viewgraph 3 For a cubic crystal (e.g. Cu) σ = σ xx = σ yy = σ zz e2 σ = 12π 3 h FS ds F υ F τ ( k r F ) for a free electron gas this again yields the Drude conductivity σ met = e2 n τ (k F ) m For semiconductors, the situation is more complicated as EF lies within the gap. There are contributions from both electrons (conduction band) and defect electrons holes (valence band). Drude term: Important: σ sc = e2 nτ N m n τ n ( r k ) τ p ( r k )! + e2 pτ P m p 2.2 Influence of a Magnetic Field Assuming only weak magnetic fields, the electron in the solid are subject to the Lorentz force r F = e( E r + r υ B r ) Boltzmann equation must include an extra term : Grundlagen und Anwendung spinabhängiger Transportphänomene 3

4 Winter 05/06 f t magn = e h ( r υ B r ) f k r k r Problem: f r k cannot be simply replaced by f o ( r k ), as the influence of r B on the equilibrium distribution f o ( r k ) is zero. next order in the expansion is needed. This yields for the deviation g( r k ) from f o (k) g( k r ) = τ [ r υ f o r e r E f o h k r e h ( r υ B r ) g k r ] despite g k r << f o k r, the last term is important, because the magnetic field B can cause strong equivalent electric fields. in metals E~1 V/m B = 100mT υ F B = 10 6 m s 1 Vs m 2 = 106 V m A simple solution for g can only be given for the NFE approximation assuming r B r E. A solution to this problem g = τ ee r r υ f o + e ε m ( v υ B v ) g υ ] g = eτ f o 1 E 1+ ω 2 0 τ (E 2 x [ E y ω 0 τ )υ x + (E y + E x ω o τ )υ y ] : Grundlagen und Anwendung spinabhängiger Transportphänomene 4

5 Winter 05/06 with the cyclotron frequency ω o = eb m (valid for electrons, ω o for holes!) This situation describes the Hall effect j x = M 1 E x M 2 E y r j = σe r Viewgraph 4 j y = M 1 E x + M 1 E y (2 x 2) tensor For metals, one obtains M 1 = e2 n m τ (E F ) 1+ ω 0 2 τ 2 (E F ) B=0 M 1 = e2 nτ m M 2 = e2 n m τ 2 (E F ) ω 0 1+ ω 0 2 τ 2 (E F ) B=0 M 2 = 0 The Hall field Ey is measured with j y = 0 Viewgraph 5 E y = j x M 2 M M 2 2 : Grundlagen und Anwendung spinabhängiger Transportphänomene 5

6 Winter 05/06 The Hall coefficient R H is defined as j x = 1 R H E y B R H = 1 en Hall effect depends linearly on the magnetic field B and on the electron density. Hall coefficient is field independent. Expression for j can be rewritten in a tensor form σ (B) = e2 nτ 1 1 +ω 0 τ m 1+ ω 2 0 τ 2 ω 0 τ 1 σ xx are independent of B in this model. This is a consequence of the effective single band model σ xy = σ yx Side remark: for higher fields there is a transition to quantum Hall effect. Viewgraph Normal Magnetoresistance Experiment shows that normally σ xx = f (B)! single band model is insufficient! : Grundlagen und Anwendung spinabhängiger Transportphänomene 6

7 Winter 05/06 A two band model with m 1, m2 and τ 1,τ 2 yields an expression (e.g., Cu belly and neck, see above) σ xx = (σ 1 + σ 2 )2 + (σ 1 ω 2 τ 2 σ 2 ω 1 τ 1 ) 2 σ 1 + σ 2 + σ 1 ω 2 2 τ σ 2 ω 1 2 τ 1 2 = σ xx (B) with σ 1 = ne2 τ 1 σ * 2 = pe2 τ 2 * m 1 m 2 and ω 1 (B),ω 2 (B) cyclotron frequencies. This behavior is called normal magnetoresistance. σ xx (B) > σ xx (O) positive magnetoresistance Viewgraph 7 Definition of MR: ρ(b) ρ(o) ρ(o) = (ω c τ ) 2 = ( eb m τ )2 : Grundlagen und Anwendung spinabhängiger Transportphänomene 7

8 Winter 05/06 = ( R H ρ o ) 2 B 2 Normal MR can have large values at low T, if the remaining resistivity is small in very clean metals. Typical values are of the order of a few percent. Viewgraph 8 In very thin films, behavior may be different. Fuchs- Sondheimer model predicts a negative normal magnetoresitance. 2.4 Fuchs-Sondheimer Model Present derivation relies on f f ( r ), i.e., distribution function does not depend on space. For confined systems this assumption breaks down. For thin films, for example, f r k 0 at the surfaces/interfaces. The Fuchs-Sondheimer model rederives the Boltzmann equation for a spatially inhomogeneous situation Thin film with normal z g = df o de r hk τ υ ee + m r r g with the solution g = df 0 de τ υ e E + τ hk z m g z : Grundlagen und Anwendung spinabhängiger Transportphänomene 8

9 Winter 05/06 g(z,k) = υ x ee df 0 de 1+ exp m z hτk z with boundary conditions g(z = 0) = g(z = t) = 0 (t film thickness) on gets σ = e2 4π 3 h υ( k r )τ ( k r ) df 0 1+ exp mz drdk de hτk z ρ = 1+ 3λ ρ B 8t t >> λ mean free path Viewgraph 9 Sie können auch den anderen Grenzfall noch diskutieren. Initial motivation for the FS model was to determine λ experimentally. Can be done by measuring resistivity as function of film thickness. Why is the MR signal negative in some cases? Viewgraph 10 The FS model is important in context of the giant magnetoresistance (GMR). : Grundlagen und Anwendung spinabhängiger Transportphänomene 9

10 Chapter 2

11 Electron scattering in k-space constant current density j electrons are inelastically ( ka kb ) scattered from occupied (A) into unoccupied states (B) elastic scattering only expands Fermi sphere inter 05/06

12 Electron (charge) scattering mechanisms ~1/T natrium scattering at phonons depends on temperature T scattering at defects residual resistivity at low T inter 05/06

13 Fermi surface of Cu Fermi surface of Cu based on the 6th band (mainly pd-like states) states are different at belly and neck normal magnetoresistance can also be observed in Cu inter 05/06

14 Hall effect magnetoresistance = difference in δkx electric field Ex displaces Fermi surface magnetic field Bz rotates Fermi surface Hall field EH displaces Fermi surface (jy=0!) inter 05/06

15 Hall effect σ (B) = e2 nτ 1 1 +ω 0 τ m 1+ ω 2 0 τ 2 ω 0 τ 1 resistivity depends linearly on B Hall constant is independent of B inter 05/06

16 Quantum Hall effect Initial linear increase is replaced by steplike function in ρxy resonances in ρxx (closed orbits of charge carriers within the sample) inter 05/06

17 Normal magnetoresistance B j B j 100 nm Co film high field MR is positive peak at low fields is due to anisotropic MR inter 05/06

18 Normal magnetoresistance ρ(300k) (µω cm) Δρ/ρ Ni Fe Co a-fe80b ~0 amorphous alloy has higher resistivity due to disorder normal MR reaches ~ 1% inter 05/06

19 Fuchs-Sondheimer model ρ = 1+ 3λ ρ B 8t FS model considers inhomogeneous distribution function resistivity in thin films increases with respect to infinite bulk situation inter 05/06

20 Normal magnetoresistance B j B j 3 nm Co film high field MR is negative peak at low fields is due to anisotropic MR inter 05/06