Chapter 9. Spintransport in Semiconductors. Spinelektronik: Grundlagen und Anwendung spinabhängiger Transportphänomene 1

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1 Chapte 9 Spintanspot in Semiconductos : Gundlagen und Anwendung spinabhängige Tanspotphänomene 1

2 Winte 05/06 Why ae semiconductos of inteest in spintonics? They povide a contol of the chage as in conventional micoelectonic devices but also of the spin, as we will see in the following. 9.0 Motivation "Simple" device in semiconducto physics: Field effect tansisto (FET). Thee-teminal device with souce (S), gate (G) and dain (D). Viewgaph 2 "electic valve": cuent between souce and dain contolled by gate voltage V g. Onoff atio may be < 10 2 much lage than in spin valves: ΔR/R < 100 % facto of 2 Essential ingedient in a FET: two-dimensional electon gas (2-DEG) below the gate electode. Tansfe to magnetic systems: Spin tansisto Viewgaph 3 : Gundlagen und Anwendung spinabhängige Tanspotphänomene 2

3 Winte 05/06 poposed by Datta and Das in 1990 (in a diffeent context). Idea: modulate a spin-polaized cuent by an electical voltage, not only by affecting the chage distibution, but also diectly the spin polaization P of the cuent. This is possible via the Rashba effect (see below). This idea has stimulated a temendous amount of wok ove the last 15 yeas, which evealed the numeous difficulties that must be solved. Thee majo poblems have to be addessed: spin injection into the semiconducto spin tanspot though the semiconducto channel spin detection of the electons at the end of the semiconducto channel 9.1 Semiconducto Popeties Reminde Semiconductos ae insulatos with a small band gap (ΔE 1.5 ev). Fo undoped semiconductos, the Femi levels usually lies mid-gap. Intinsic conductivity of semiconductos is vey low and stongly tempeatue dependent (themal activation of fee chage caies) Themal activation of an intinsic semiconducto ceates electons n o and holes p o at the same ate n o = p o = n; chage neutality $ n 2 i = N c " N v " exp # Eg ' & ) % kt ( Viewgaph 4 N c, N v : density of states in conduction and valence band. The caie density in semiconductos is much lowe (<10 19 cm -3 in a nondegeneate semiconducto) than in metals (~10 23 cm -3 ). This low caie density has a significant consequence fo the electostatics: sceening length is much lage than in metals electostatically induced caie pofiles may extend ove lage distances (~ 100 nm) Chage caie density may be inceased by doping: Donos Acceptos n conductivity p conductivity Viewgaph 5,6 Usually the dopant electonic levels ae close to the valence band E v (acceptos) o conduction band edge E c (donos) and ae ionized aleady at low tempeatues. : Gundlagen und Anwendung spinabhängige Tanspotphänomene 3

4 Winte 05/06 E F is the electochemical potential and shifts with tempeatue E F = " c + " v 2 # kt 2 ln $ p ' & ) + 3 % n ( 4 kt * ln m p * m n * intinsic semiconducto p=n moe complicated dependence fo doped systems. Special case: degeneate semiconducto with vey high doping N D "10 19 cm -3 quasi-metallic conductivity many defects in the lattice (scatteing) 9.2 Chage tanspot in semiconductos Chage tanspot is descibed by Boltzmann equation as in the case of metals. Thus, the cuent is limited though the dift velocity, which is detemined by the influence of scatteing pocesses. The man scatteing pocesses ae: phonons (Si, Ge acoustic phonons µ~t -3/2 ionized defects (dopant) GaAs optical phonons µ~t 1/2 neutal defects (lattice) GaN Scatteing fom ionized defect can also be seen as 3-step pocess electon moves though cystal electon ecombines with dono (ecombination time) electon is emitted fom dono (emission time) fo shallow impuity levels: ecombination time ~10-7 sec emission time ~10-11 sec in vey clean (undoped!) semiconductos ecombination time dominates and the chage caies may have extemely long scatteing lengths of up to micometes! ballistic tanspot! : Gundlagen und Anwendung spinabhängige Tanspotphänomene 4

5 Winte 05/06 Speciality in semiconductos: Because of low chage caie concentation, dift and diffusion tems in the tanspot can have simila size diffusion in metals usually neglected, but must be consideed in semiconductos. Consequence: J N = enµ n E + ed n " n n=n ( v ) dift diffusion J P = epµ p E " ed p # p p=p( v ) mobility µ and D ae linked by the Einstein elations ed N =µ N kt ed P =µ P kt Total cuent density J = e( nµ n + pµ p ) E + ed n " n # edp" p! Full teatment of the semiconducto situation must also include chage geneation and ecombination n=n(,t ) "n "t = 1 e # div J n + g eh + eh "p "t = 1 e # div J p + g eh + eh with g eh and eh geneation and ecombination ates of electon-hole pais. 9.3 Spin tanspot in SC Obsevations Semiconductos ae non-magnetic, theefoe, spin-polaized electons do not exist in conventional semiconductos in the gound state! Exception: feomagnetic semiconductos Let us concentate on Si o GaAs: low caie density inefficient electostatic shielding low spin density weak shielding of magnetic fields We have basically isolated spins (ensemble) moving though the cystal. : Gundlagen und Anwendung spinabhängige Tanspotphänomene 5

6 Winte 05/06 Intinsic semiconductos do not contain spin-dependent scatteing centes: Si (Z=14) light element weak s.o. effects Ge (Z=23) and GaAs simila GaN may be even bette due to lowe SOC, but has much moe stuctual defects. GaAs eveals spin diffusion lengths of " s # 100µm (4.2K) " s # 200ps $10ns Viewgaph 7 ZnSe, GaN show simila values, pesisting even up to 300 K. Obsevation: Spin dephasing time τ s becomes lagest at the MIT (upon doping). 9.4 Spin tanspot in semiconductos Fo the moment, we assume to have an ensemble of spin-polaized electons in the semiconducto no matte, how it has been ceated! Time evolution of the spin density S in the solid is descibed by Bloch equation (which has a fom simila to that of the Boltzmann equation, but deals with the vecto quantity spin polaization), which descibes pecession of the spin aound the magnetic field axis " s "t = s # B S µ B g/h $ $ & ' J s % s Viewgaph 8 pecession damping spin cuent contibution As in the case of the chage cuent in semiconductos, the spin cuent composed of two contibutions and takes the fom of a 2. ank tenso J S = v " s # D s $ " s (dyadic pod.) with dift velocity v = j /( q n) and spin diffusion const. D s. J S is In ode to get some insight into the poblem, we conside a simple one-dimensional : Gundlagen und Anwendung spinabhängige Tanspotphänomene 6

7 Winte 05/06 geomety: the semiconducto stats at x=0 and and a spin-polaized cuent (chaacteized by the cuent density j and the polaization P) flows into the x>0 diection. The bounday condition fixes J s (x = 0) = 1 q j " P Because of the vecto chaacte of the spin polaization, J s takes a complicated fom " v x S x v x S y v x S z % " $ ' $ j = v S y S x v y S y v y S z ( D S $ $ # v z S x v z S y v z S ' $ z & # )S x )x )S x )y )S x )z )S y )x )S y )y )S y )z )S z )x )S z )y )S z )z % ' ' ' & $ #S v x S x " D x & S L #x j = & M O M S & #S & v z S x " D x S L % #x v x S z " D S #S z #x v z S z " D S #S z #x ' ) ) ) ) ( Viewgaph 9 In the one-dimensional poblem, we can solve these diffeential equation fo the steady state and in the absence of a magnetic field, i.e., "s "t = 0, B = 0 " s = # s $ % $ J s The solutions have to be consideed fo diffeent situations and caie types. Majoity caies: ecombination can be neglected and n " n(x) S P s (x) = n = 2" d P " d + " 2 d + 4" exp $ # x ' & " d + 4" 2 s # " d s 2" s ( )) % ( " d = v # $ s dift length " s = D s # s diffusion length nondegeneate semiconducto: " d /" s = qe" s /(k B T) >> 1 (dift dominated) P s (x) = P $ exp &" x % # d ' ) ( Viewgaph 10 : Gundlagen und Anwendung spinabhängige Tanspotphänomene 7

8 Winte 05/06 opposite limit " d /" s <<1 (diffusion dominated) P s (x) = " % d P # exp ' $ x " s & " s the pefacto leads to an inteface-induced eduction of the polaization in the semiconducto minoity caies diffusion is moe impotant, as long as n min. <<n maj. ( * ) P s = " % % s P # exp ' $x' 1 $ 1 " & &" s " (( ** )) λ s λ minoity caie spin diffusion length minoity caie chage diffusion length The caie density vaies as a function of x accoding to n(x) = " J D exp $ # x ' & ) % ( thus λ includes effects fom both ecombination + elaxation. " : Gundlagen und Anwendung spinabhängige Tanspotphänomene 8

9 Chapte 9 Spin Tanspot in Semiconductos

10 Field effect tansisto elatively simple device equies intefacial engineeing voltage epels holes fom the mateial and changes conductivity chaacte below the gate electode (p-type intinsic n-type) two-dimensional electon gas (2-DEG) Winte 05/06 2

11 Datta & Das poposal FET stuctue feomagnetic electodes chage contol spin contol (otation due to Rashba effect) not poposed as a device!! spin injection spin tansfe spin detection S. Datta and B. Das, Appl. Phys. Lett. 50, 665 (1990). Winte 05/06 3

12 Conductivity in intinsic semiconductos intinsic semiconductos themal activation of chage caies fom EV to EC conductivity depends on band gap of semiconducto Winte 05/06 4

13 Doping semiconductos Loch shallow impuity levels easily to ionize contol of chage density up to ND~10 20 cm -3 (degeneate) Winte 05/06 5

14 Conductivity: tempeatue dependence 3 conductivity egimes feeze-out extinsic intinsic Winte 05/06 6

15 Spin dephasing times spin dephasing times up to 100 ns! T2 highest close to the metal-insulato tansition simila behavio fo GaAs, GaN, ZnSe Winte 05/06 7

16 Bloch equations time evolution of spin-polaization: S t = S B µ B g/h S τ S J S pecession damping spin cuent spin cuent: J = v S D S S S equation simila to Boltzmann equation fo f complication: spin polaization is a vecto Winte 05/06 8

17 Spin cuent 3x3 tenso to descibe spin cuent density: j S = S v S D x L x x S x M O M S v S D x L z x S x v x S z D S S z x v z S z D S S z x Winte 05/06 9

18 Spin cuent majoity caies: P S (x) = S n = λ d + 2λ d P λ 2 d + 4λ exp x 2 s 2λ s 2 ( λ 2 d + 4λ 2 s λ ) d minoity caies: P (x) = λ s S λ P exp x 1 1 λ λ s Winte 05/06 10

19 Spin cuent majoity caies: P S (x) = S n = λ d + 2λ d P λ 2 d + 4λ exp x 2 s 2λ s 2 ( λ 2 d + 4λ 2 s λ ) d P (x) = λ d S λ s P exp x λ s equation simila to Boltzmann equation fo f complication: spin polaization is a vecto Winte 05/06 11