Coulomb drag : An experimenter's handle to the electron- electron interaction problem

Transcription

1 Coulomb drag : An experimenter's handle to the electron- electron interaction problem Kantimay Das Gupta Dept. of Physics, IIT Bombay Semiconductor Physics Group Cavendish Laboratory Cambridge United Kingdom Low Dimensional Quantum Systems. HRI, Oct 11 (Wed 1.)

2 Q. What is the interaction energy between two electrons? vacuum slightly polarisable lattice but with no other free electrons slightly polarisable lattice with more free electrons 1 e 4 r 1 e 4 r r Explicit expression for V(r) is not possible in general

3 V(r) is not very important, we need its Fourier transform V(q) in most cases. Why? Any quantum mechanical problem would require matrix elements like 1 ~ τ(q) = k f V (r ) k i DOS dv V (r )e i(k i k f ).r DOS between free electron states q=kf - ki = V (q) DOS The simplest textbook case of one static charge and all other moving ones goes quite far ext V= r r EF CB place for more electrons less electrons =e. ev D E F =qtf V

4 How is the bare potential of a charge modified due to the presence of other electrons? Fourier transforming the equation leads to the solution: ( V (q)= In general ) 1 ρext (q) 1 ϵ ϵ r q 1+qTF /q V ext q V q = q Assumption is that the source of the extra potential is static What if the source of the potential also varies in time? V ext q, V q, = q, Q: Can the denominator be zero? Density waves, Plasmon modes...

5 How is the electron-impurity scattering different from electronelectron scattering as far as screening is concerned? Impurites don't move about. So we use : q, Electrons move about, So we must use: q, This is also why electronimpurity scatt rate will not tell a lot about ε(q,ω) How does the experimenter try to measure this (in clean metals)? T = ep T Aee T residual resistance due to impurities (Mathiesen & Vogt 1964) electron-phonon Bloch 193 Gruniessen 1933 electron-electron Landau & Pomeranchuk (1936) McDonald et al (1981)

6 Resistivity of ultra-pure Silver. What is the power law for ρ(t)? l=v F 5 m In very clean samples e-e and electron-impurity interactions might become comparable. BUT Power law with N = to 4 have been reported. Comparison of data from different groups. no agreement!! Besides we can't change density of electrons in a metal. They cannot be gated. Khoshnevisan et al, Jl of Phys F: Metal Phys. 9, L1 (1979)

7 A new idea: Instead of trying to measure the momentum lost by the particle try to measure the momentum gained by them. M.B Pogrebinsky, Sov Phys Semicond. 11,37 (1977) P. Price, Physica 117B, 75 (1983) V Consider parallel layers. Moving electrons in one layer transfers momentum to the other layer. Try to measure that. There was no way to do that in Two films of metal won't work! Quantum well and modulation doping had not yet come.

8 How to analyse the amount of momentum transferred and what can be inferred about the interactions from that? F v.. v r t m f 1 r, v, t f 1 r, v, t = t e E 1 f1. k f 1= ℏ t Small deviation from Interlayer equilibrium due to scattering current flow in layer, ' I ' ' k 1 + k k 1 + k f1 = t d k collision 1 V ' d k1 W 1, 1', ' 1 1 ' ' f 1 f 1 f 1' 1 f ' 1 1 ' ' Jauho & Smith PRB 47,44 ( 1993) Zheng & MacDonald PRB 48, 83 (1993) Yurtsever et al. Solid State Comm. 15,575 (3) Hwang & Das Sarma PRB 78, 7543 (8)

9 The deviation functions: momentum conservation gives ' 1 1= = ee v x = kt This is a shift in the Fermi circle m v x m v ' x =m 1 v1 ' x m1 v1x We want to isolate the electric field in layer 1, so multiply both sides of the equation by k1x and integrate over all k-space. e E1 d k 1 f1 e E1 f 1 n1e E1 LHS = k = D d = 1x ℏ k 1x ℏ ℏ d k 1 f 1 RHS = k = 1x t ℏ e E d k d k '1 d k1 W 1, 1 ', ' k 1x k 1x k 1 ' x m kt f 1 f 1 f 1 ' 1 f ' 1 1 ' ' Symmetries of this expression allow simplification

10 The symmetry allows replacing The current and voltages in layer are easily related 1 1 k 1x (k 1x k 1 ' x ) (k 1x k 1 ' x ) q 4 E= I m n e τ n1 e E 1 ℏ τ e E d k d k '1 d k 1 = W (1, 1 ', ' )k 1x (k 1x k 1 ' x ) σ ℏ m kt ( π) ( π) ( π) f 1 f (1 f 1' )(1 f ' )δ(ϵ1 +ϵ ϵ 1 ' ϵ ' ) Notice that individual layer scattering times are going to disappear from the ratio between E1 and I. This is immensely important - because we have now related a transport measurement to electron-electron scattering. The effect of disorder has somehow disappeared - at least within the relaxation time approximation. Usually the disorder scattering is 1-1 times stronger than e-e even in very clean samples.

11 Expression for Coulomb drag including dynamic screening: DRAG = ℏ k B Te q np 3 dq d V q, I m e q, I m h q, sinh ℏ /k B T V bare det q, det ε (q,ω) can be zero or very small and these collective modes of the - component plasma can contribute very significantly to Coulomb drag me mh 3 k B T DRAG = npe 16 ℏ k Fe d k Fh d qtfe d qtfh d holds for high densities and large interlayer separations k F d 1 T /T F 1

12 What difficulties has been swept under the carpet? Why did everyone not start doing this? A. The conditions under which the effect can be appreciable are not trivial. Also there are possible sources of errors.

13 What difficulties has been swept under the carpet? 1. Independent contacts to two layers spaced by about 1 nm ohmic ohmic EF depletion gate AlGaAs AlGaAs GaAs depletion gate AlGaAs CB Eisenstein et al. APL (199) Gramila et al PRL, 66, 116 (1991) Linfield et al Semicond. Sci & Tech. 8, 415 (1993) NPR Hill et al PRL, 78, 4 (1997) x DEG GaAs Requires two-sided lithography OR Focussed Ion Beam patterning Q. What would be realistic gate voltages needed? How can we see both sides of a GaAs wafer?

14 Top side of chip with scribes on glass (a) 5 µm. Aligning the gates on the top and bottom side is absolutely necessary, with better than 5 micron accuracy Acetone wash (b) Thinned to ~6µm Back side of chip Devices GaAs chip Crystalbond-59 Cover slip ~5µm wide scribes on glass 1µm Alignment marks for backside lithography Croxall et al JAP 14, (8)

15 3. Even a small leakage between the two layers would produce a spurious signal, masking the real one. Shift the bias point and check if the signal changes... Less than 1pA over 1x1 microns is generally necessary Often less than 1 in 5 devices would meet this requirement

16 Let's summarise the three key requirements 1. Independent contacts. Very low leakage barrier (1.5V, 1nm, 1x1 micron) 3. Gating from both top and bottom 1 cm

17 The relation between Coulomb drag and Onsager reciprocity relation for four-terminal linear response. I- I+ V V+ V- OR V- V+ V I+ I- H.B.G. Cassimir Rev. Mod. Phy. (1945)

18 The first measurement of drag effect in an electron-electron bilayer ρdrag <.1 ρlayer 11 - N=1.5 x 1 cm µ=3.5x16 cmv-1s-1 Gramilla, Eisenstein et al PRL 66, 116 (1991) But that is no longer a problem.

19 The many significances of ε(q,ω): relation to density-density response and local field corrections Total potential = External potential + potential due to induced charges ϵ(q, ω)=1 v q χ (q, ω) δ n=χ V tot e Density-density response or charge susceptibility Fourier transform of the Coulomb potential. 1/q in D, 1/q in 3D The Thomas-Fermi form predicts a constant χ(q) for all q. This cannot be correct. Implication is that the system responds equally well to all frequencies. In fact it leads to some problems...

20 A theory of charge susceptibility is also a theory of pair correlation Singwi-Tosi-Land-Sjolander (1968) 1 d k k. q G q = [ S k q 1] n q χ χ= 1 v q (1 G (q))χ ℏ S q = I m Need to fix this K.S. Singwi & M. Tosi: Correlations in electron liquids Generalisation to bilayers : Liu, Swierkowski. Neilson, Szymanski PRB 53, 793 (1996) Zheng & Macdonald PRB 49, 55 (1994)

21 XDEG (1nm barrier) Similar data exists for holehole bilayers Vig n Sin ale gw i STLS Verifying the local field correction in a bilayer ic m na y D A P R RPA c i t Sta V Data from M. Kellog et al Solid State Comm 13, 515 (3) Calculated curves: Yurtsever. Moldaveanu, Tanatar Solid State Comm 15, 575 (3) Measure drag at low densities large rs

22 Does a fermi liquid at large r_s continue to be a fermi liquid? This question can be asked using Coulomb drag as a probe. Can local field corrections explain the huge enhancement in hole-drag? We can ask these questions without worrying about disorder. R. Pillarisetty at al. PRL 89, 1685 ()

23 Why interaction effects can appear in a bilayer more easily than they do in a single layer? l E ee e N = 4 EF ℏ N = meff If n=1 111 cm- then E ee 1 rs = = EF ab N l ~ 3 nm rs 1.8 in GaAs Q. Effective mass in Si is higher, but not ideal for these, why?

24 Why interaction effects can appear in a bilayer more easily than they do in a single layer... Total potential = External potential + contribution due to polarization charges in same layer + contribution due to polarization charges in other layer tot ext qd V e =V e +v q δ n e +v q e δ n h ext qd V tot =V +v δ n +v e δ ne h h q h q e h δ ne =χ V δ n h=χ V tot e tot h e v q= ϵ ϵr q [ 1 v q χ qd v q e e qd v q e χh χ 1 v q χ h h ][ ] [ ] V V tot e tot h = V V ext e ext h Q. What if the matrix has det =? A. Spontaneous density modulations possible, with well behaved single layers

25 What are the other possible phases in a bilayer? d ~ ab l l d l = = 1 n kfd ~ 1 assuming equal densities in both layers... Such low densities and high mobilities are possible only in GaAs/AlGaAs 1- nm separation needed. Bound states are not the only possibility...density waves may occur...

26 Coulomb drag measurements in an electron-hole bilayer The strength of the interlayer to intralayer interaction... 1 n d / l 1.8 l= Croxall et al. PRL 11, 4681 (8)

27 Coulomb drag measurements in an electron-hole bilayer nm barrier EHBL Data from Seamons et al PRL 1, 684 (9)

28 No phase space is expected for scattering at T= Between two Fermi liquids. How does the electron-hole bilayer seem to have a finite scattering rate at T=?

29 What kind of questions can be addressed using the Coulomb drag measurements? Electron-electron scattering rates are not masked by electron-impurity scattering rates, we can use this in many situations. Like.. Fermi-liq / non-fermi liq at low densities? Emergence of density wave modes/bound states/bilayer Quantum Hall states in a bilayer. Correctness/verification of local field corrections. The idea can be extended to experiments on narrow channels, where the experimental evidence of Luttinger liquid phases is sketchy... Acknowledgements. The work was done in the semiconductor Physics group of Prof David Ritchie and Michael Pepper. The MBE growth was done by Christine Nicoll, Harvey Beere & Ian Farrer. The devices were fabricated and measured by KDG with Andy Croxall, James Keogh, Mamta Thangaraj & Joanna Waldie at various times. The work was funded by EPSRC, UK.