1. Electron Transport of Metal Gated Devices in GaAs/AlGaAs Heterostructure K. M. Liu( 劉凱銘 ), W. R. Chen( 陳偉仁 ), Y. M. Lin ( 林玉敏 ), and S. Y. Hsu ( 許世英 ) Low Temperature Laboratory, Department of Electrophysics, National Chiao Tung University, Hsinchu, Taiwan, R.O.C.

2. Outline Introduction –GaAs/AlGaAs Heterostructure –Transport in mesoscopic system –Quantum Point Contact –Gate Defined Quantum Dot Fabrication Double quantum point contacts in series Electron Pumping –Adiabatic Quantum Pumping Summary

3. Introduction - GaAs/AlGaAs Heterostructure There is a very thin layer called 2-Dimensional Electron Gas(2DEG) at the interface of GaAs and AlGaAs. Which is a conducting layer. GaAs/AlGaAs 0.3K(5-70) carrier density n s 1.88x10 11 cm -2 mobility μ0.8x10 6 cm 2 /Vs Fermi wavelength λ f 57.8 nm mean free path l e 5.9  m 2DEG E1E1 energy EcEc EfEf 0.2eV Structure of GaAs/AlGaAs grown by MBE 10 nm, GaAs Cap 15 nm, δ- doping layer, Si, 2.6x10 18 cm -2 60 nm, spacerAlGaAs, x=0.37 1500 nm, buffer layer GaAs 0.3mm GaAs substrate 8 nm, spacer AlGaAs The mean free path is much larger than the limit length scale of modern technology. Thus we can obtain a system where the transport of electron is coherent and ballistic through lithographic fabrications.

4. Transport in Mesoscopic System – A Theoretical Description For an Ideal 1D system with one conducting channel  Current passing through a conductor can be expressed as v(k): electron velocity f(k): distribution function T(E): Transmission In the limit of low temperature, small (μ 1 - μ 2 ), and assume T(E) independent with E μ2μ2 reservoir μ1μ1 T R T e-e- conductor Transport is conductance! 2. Multi-Channel Conductor -Landauer Formula 1. Conductor with one channel

5. Quantum Point Contact Applying a negative voltage on the metal split-gates fabricated above a 2DEG, depleting the electron gas, a quasi-1D quantum wire is formed. And the electron state in the conductor is quantized. e-e- SourceDrain VgVg EfEf E(k x ) kxkx x y Each plateau corresponds to an additional mode as integer multiples of half the Fermi wavelength.

6. Gate Defined Quantum Dot e-e- source drain The modeled circuit: VgVg V sd R l C l R r C r ΣC g - + - + dot The energy is quantized as soon as the quasi-0D dot is formed. And the transport is blocked as shown: The energy potential of QD can be tuned by varying V g, and electron tunneling occurs when there is a state aligning with the Fermi level at source or drain. Coulomb Blockade 0.5μm e 2 /C eq charging energy N N+1 VgVg μsμs μDμD

7. PRL. 80, 4522(1998) Coulomb Staircase For the I-V curve of the QD, the value of current corresponds to the number of states in the energy window V sd and is quantized. Z. Phys. 85, 367(1991) Weak Localization Coulomb Oscillation e-e- 6-102a-I2 Number of electrons~1500

8. Fabrications Part I. Photolithography Hot plate 90°C sample PR Coating & Prebake UV Light Exposure sample PR mask Develop sample MesaEtch the wafer with solution H 2 SO 4 :H 2 O 2 :H 2 O=1:8:160 Ohmic Contact Deposit Ni/Au/Ge/Ni=100Å/2000Å/1000Å/700Å Annealing:450 o for 13min GateDeposit Au/Ti =1200Å/100Å mesa contact pads metal gates

9. Part II. E-beam lithography sample Develop (MIBK:IPA=1:3) sample Metal Deposition (Ti/Au) metal sample metal Lift off in the Acetone sample PMMA Coating & Prebake sample PMMA Electron Beam Exposure electron beam

10. -J. Phys. C 21, L887 (1988) g1g1 g2g2 (a)V g2 =-1V (b)V g2 =0V The second channel must impose a more severe constriction on the transverse momentum (Collimation)  additional geometry resistance As both QPCs are confined, the plateau index start from the smallest number among them  the resistance through two QPCs is determined by the narrowest of the two constrictions. Double QPCs in Series Channel length:0.3μm 1 μm Remove the anomalous resistance

11. Phys. Rev. B 39, 10445 (1989) QPC constriction With zero magnetic field: T r =T l =0  a.If T d =0  Ohmic addition of conductance. b.For direct transmission(T d >1)  (h/2e2)G series = T d =N s -R s  G≈ N (2e 2 /h)  G series is identical to single QPC L TlTl TrTr TdTd Nonadditivity of point contact resistances in series From Landauer-Büttiker formula describing current in a lead of a four-terminal conductor set I r =I l =0 and I s =-I d =I 

12. Preliminary Summary Transport through single QPC demonstrates quantized conductance in units of 2e 2 /h. If the transport is ballistic, the total conductance across double QPCs is determined by the smallest one. The values are also integer multiples of 2e 2 /h. It is theoretically predicted that: When one of the QPC is in the tunneling regime (N<1), the transport should behave ohmic addition. Source e-e- V2V2 L V1V1 Drain qpc 2 qpc 1

13. L=0.8μm The traces have fewer plateaus with narrower qpc 2. It has only 1 plateaus with qpc 2 set in N=2.  It’s ballistic when L=0.8μm. Destruction of coherence in double quantum point contacts (QPCs) in series

14. L=1μm The number of plateaus doesn’t relate to the channel width of qpc 2. The transport doesn’t behave ballistically.  Ohmic addition.

15. For larger L, subtracting the contribution from qpc 2, the single QPC’s conductance quantization is restored. These two QPCs are almost independent with each other. L=2.9μm

16. When the separation L is much larger than the mean free path, identical traces were obtained.  These two QPCs are completely independent with each other. L=20μm

18. As the transmission mode is set to zero (N<1) and L small, we can regard qpc 2 as a barrier, and there’s no coherence between QPCs.  Plateaus completely vanish.

19. Summary Transport through double QPCs in series : The transport behaviors are determined by two factors: (a) separation between two QPCs, L. (b) number of transmission modes N. As L is larger than a specific length, order of e, the transport behaves completely as that of two independent QPCs. As L is less than e and N is less than one, the quantized conductance vanishes.  Coherence between QPCs can be destroyed.

20. Electron Pumping- Photon-assisted Tunneling Etched narrow wire with width ~0.8 μ m. Use gate 1 and 2 to define a quantum dot containing about 100 electrons. Couple a microwave signal to gate via a capacitor near the sample. The electron can tunnel through the dot by absorption or emission with a photon→a shoulder along with a Coulomb Oscillation peak Curves in different Excitation amplitude Curves in different Excitation frequency The position of shoulder is independent with excitation amplitude but scaling with frequency. PRL. 73, 3443(1994)

21. Turnstile Pumping I :used gates μrμr μrμr N+1 N μrμr μlμl μrμr N μlμl μlμl μrμr Configure the barriers to oscillate with a phase difference π. Exactly one electron passes in one cycle. While Increasing the bias V increases number of electron states between μ l and μ r I=nef quantized current The pumped current scales with frequency and quantized with respect to V sd. PRL. 67, 1626(1991)

22. Adiabatic Quantum Pumping System: Electron reservoirs are held at same voltage.(zero bias) Each QPC have N channels at the Fermi level E F. The scattering matrix of the system has dimension 2N×2N and is a function of X 1 and X 2 X 1 and X 2 are two parameters modifying the wavefunction of the open dots. Which may be magnetic field or gate voltage. Small harmonic variation: The charge δ Q(m) entering or leaving the cavity through contact m(m=1,2) in an infinitesimal time: For two parameters X 1 and X 2 Integrate over one period and use Green’s Theorem or PRB, 58, 10135(1998) emissivity

23. a.For a phase coherent quantum system, the out-of-phase variation will give rise to a dc current. b.The current scales as the area enclosed by X 1 and X 2 in phase space or say the current varies as sinφ. Science 283, 1905(1999) Experiments: I sd =0

24. V Pumped current in different dot size Open dot Closed dot qpc3 qpc4 The pumped current reduces with increasing barrier height between dot and reservoirs.

25. Pumped current with different excitation amplitude The pumped current enhances with increasing excitation amplitude. Non-sinusoidal form when V pp becomes too large.

26. Pumped current with different frequency The pumped current is roughly linear with frequency.

27. Summary A mesoscopic system is easily achieved through GaAs/AlGaAs heterostructures due to it’s long mean free path. The transport of electrons in such systems is characterized by transmission or conductance. Quantum phenomenon: Quantized Conductance in QPC, periodic Coulomb Oscillations, Weak Localization. Double QPCs in series is also studied, where the behavior is characterized by distance between QPCs. Adiabatic Pumping can generate a DC voltage or current without external bias.