Quantum pumping and nuclear polarization in the integer quantum Hall regime

Europhys. Lett.,69 (1), pp. 109–115 (2005) DOI:10.1209/epl/i2004-10315-2

Quantum pumping and nuclear polarization

in the integer quantum Hall regime

M. Blaauboer

Kavli Institute of NanoScience, Delft University of Technology Lorentzweg 1, 2628 CJ Delft, The Netherlands

received 19 July 2004; accepted in final form 3 November 2004 published online 8 December 2004

PACS.72.20.-i – Conductivity phenomena in semiconductors and insulators. PACS.73.43.-f – Quantum Hall effects.

PACS.72.25.Rb – Spin relaxation and scattering.

Abstract. – We study pumping of charge in a 2DEG in the quantum Hall regime at filling

factor ν = 2 (2 spin-split levels of the lowest Landau level). For pumpingfrequencies that match the Zeeman energy splitting, quantum pumping together with hyperfine interaction between electrons and nuclei induces transitions between the spin-split levels. These lead to a step-like change in the pumped current and to polarization of the nuclei. We present quantitative predictions for both. Our model provides the first quantitative tool to both control and measure the amount of local nuclear polarization in a 2DEG.

Charge pumping involves the generation of a d.c. current in the absence of an applied bias voltage. The current is generated through application of two, or more, time-varying signals with a phase difference φ between them. In recent years, attention has focussed on pumping of charge in phase-coherent mesoscopic systems [1–4] such as, e.g., quantum dots, small conducting islands embedded in a semiconductor material [5]. The pumped current in these systems depends on quantum-mechanical interference, hence the name “quantum pumping”, and exhibits some fundamental properties, such as sinusoidal dependence on the phase difference φ and linear dependence on the pumping frequency ω for ω
τ_d−1, with τ_d the dwell time of the electrons in the system [1, 3].

In this work, we investigate pumping of charge in a two-dimensional electron gas (2DEG) in the integer quantum Hall regime. Transport in this system occurs via edge states corre-sponding to quantized energy levels, the Landau levels (LLs) [6–8]. Each LL is macroscopically degenerate, holding many electrons, and the number of occupied LLs is characterized by the filling factor ν ≡ neh/(eB), the number of electrons per flux quantum in the system (ne denotes the electron density and B the applied magnetic field). Our motivation to study quantum pumping in such a system comes from the idea that this way of transport might lead to interesting effects in the regime of spin-split LLs [9]. Since pumping involves exchange of energy quanta ¯hω, as described below, it can cause transitions between these LLs in combina-tion with spin-dependent interaccombina-tions in the system. At the same time, the pumped current can be used to monitor such transitions. This opens new possibilities to manipulate and measure spin-flip processes and nuclear polarization in a 2DEG.

c

B 1 3 C 2 B 1 3 C 2 1 3 B C 2 (a) (b) (c) x y

Fig. 1 – Schematic picture of a Hall bar at bulk filling factor ν = 2 (spin-split levels of the lowest Landau level). The split-gates A-B and A-C form two QPCs with conductance GAB, GAC≤ e2/h and are used for pumpingby applyingtwo time-varyingvoltages with a phase difference φ to B and C. The current leads 1–3 are all held at the same chemical potential µ. (a)-(c) show the edge channels originating from, respectively, terminal 1, 2 and 3. Solid (dashed) lines correspond to the spin-up (spin-down) edge channel. In the shaded region, spin-flip scattering can take place.

Using a Floquet scattering approach, we calculate the pumped current in the Hall geometry of fig. 1. Pumping is induced by applying a.c. voltages to the split-gates A-B and A-C on top of the bar, which results in current flowing into contacts 1, 2, and 3. The gates form 2 quantum point contacts (QPCs) that transmit at most one edge channel. When the pumping frequency matches the Zeeman energy splitting, pumping together with hyperfine interaction between electrons and nuclei in the system causes scattering from the ↑-spin to the ↓-spin LL. This manifests itself as a sudden change of the pumped current and leads to polarization of the nuclei. For small pumping amplitudes, we calculate the change in current and the corresponding nuclear polarization analytically. The relationship between these two provides a tool to measure the amount of local nuclear polarization in the system. We conclude by discussing the validity of our model and comparing it to previous work on hyperfine interaction in quantum Hall systems.

Consider the Hall bar of fig. 1. In the absence of the gates A, B, and C, the dynamics of the electrons is described by the Schr¨odinger equation i¯h_∂t∂ψ(x, y, t) = Hψ(x, y, t) with

H = 1 2m∗
i¯h∇ + e A2 + V (y) + 1 2g ∗_µ BBSz+1₂  i A_iI_i· S . (1) Here m∗ denotes the effective mass, A the vector potential, V (y) the transverse confining potential, g∗ the effective g-factor, and µB the Bohr magneton. The last term in eq. (1) is the hyperfine interaction between the electron spin S and the nuclear spin I_i, with A_i the corresponding hyperfine interaction strength. If V (y) is constant on the scale of the magnetic length l_m≡ ( ¯h

m∗√_ω2 0+ω2c

)1/2_{, with ω}

c ≡ |e|B/m∗ the cyclotron frequency, it can be described to a good approximation by a harmonic potential V (y) = 1₂m∗ω2

0y2 [7, 10]. Taking the Landau gauge A_x=−By, the Schr¨odinger equation can be solved exactly in the mean-field approximation of the hyperfine interaction, with the result [11]

ψ_λ,σ(x, y, t) = C_λχ_λ(y) ei
kx−Eλ,σ_¯h t_{, (2)} χ_λ(y) = e 1 2l2_m  y+ ω2c ω2_{0 +ω}2_cyk 2 ∂ ∂y λ e−l21_m  y+ ω2c ω2_{0 +ωc}2yk 2 , (3) E_λ,σ=  λ+1 2  ¯ hω2 0+ω2c+ 1 2m ∗ ω02ω2c ω2 0+ωc2 y2 k+1₂g∗µBBσ+ 1 2AIzσ, λ = 0, 1, 2, . . . . (4)

E_0,1 E_0,1 E_0,1 E_0,1

h

ω

h

ω

I

< >

g *

µ

+

A

(a)

(b)

Fig. 2 – Energy level diagrams for (a) ¯hω  |g∗µBB + AIz| and (b) ¯hω ∼ |g∗µBB + AIz|. See the

text for the discussion.

Here C_λdenotes a normalization constant, y_k≡ ¯hk/eB, σ = 1 (−1) for electrons with spin-up (down), and A≡ _iA_i. I_z is the average nuclear polarization and E_λ,σrepresent the energy levels (LLs) measured from the lowest band imposed by the confinement in the z-direction. The electrons travel near the edges of the 2DEG along one-dimensional (1D) channels that are formed at the intersection of the Fermi energy E_F with the LLs (4) [8].

In order to study quantum pumping in this system, we model the QPCs in fig. 1 as oscillating δ-function potentials, VAB(x, t) = [ ¯VAB + 2 δVABcos(ωt + φAB)] δ(x + L/2) and V_AC(x, t) = [ ¯V_AC+ 2 δV_ACcos(ωt + φ_AC)] δ(x− L/2).

We then use Floquet theory [12] to calculate the pumped current flowing along the edge channels. This approach has recently been employed to study 1D mesoscopic quantum pumps [13, 14] and describes pumping in terms of gain or loss of energy quanta ¯hω at the two oscillating barriers. Following Moskalets and B¨uttiker [14], the pumped current into lead α, with α = 1, 2, 3, is given by I_α= e h _∞ 0 dE β  En>0

|tαβ(En, E)|2(f (E)− f(En)) , (5)

where f (E) = (1+exp[(E−µ)/(kBT )])−1is the Fermi distribution, with µ the chemical poten-tial in the leads, and t_αβ(E_n, E) denotes the Floquet scattering amplitude for a particle origi-nating with energy E in lead β to gain n energy quanta ¯hω and be scattered with energy E_n ≡ E +n¯hω into lead α. For two oscillating barriers, these scattering amplitudes are calculated by matching wave functions (see eq. (2)) ψ(x, t) = ∞_n=−∞(a_neiknx_+bne−iknx_)e−i(E0,1+n¯hω)t/¯h_at each of the two barriers [14]. For details of this calculation we refer to ref. [15]. Here we restrict ourselves to the situation of weak pumping, defined by q_AB(AC)≡ δV_AB(AC)m∗/¯h2
k0, with k_n =  2m∗ ¯ h2 ω 2 0+ω2c ω2 0 [EF+n¯hω− 1 2¯h  ω2

0+ ω2c−12g∗µBB −12AIz]1/2. In this case, it is sufficient to take only the two lowest sidebands n = 0,±1 into account [15] and the pumped current can be obtained analytically. Before analyzing this current, let us take a closer look at the energy levels in the region between the QPCs. These are drawn schematically in fig. 2 for pump-ing frequencies much less than (fig. 2(a)) and approximately equal to (fig. 2(b)) the effective Zeeman splitting. In each picture the left side represents the energy levels corresponding to the spin-up and spin-down edge channels before scattering at a QPC (describing the situa-tion in, e.g., the upper left corner of fig. 1(a)), while the right half represents the situasitua-tion after scattering at a QPC. In the latter, electrons in the spin-up channel either remain at energy E0,1 or are scattered into sidebands with energies E0,1 ± ¯hω. The ↓-spin electrons,

↑-spin, edge channel), all remain at E0,−1. This lifting of degeneracy of the↑-spin level has an interesting consequence: for appropriate level splitting ¯hω, it can combine with hyperfine interaction to enable scattering from the ↑-spin to the ↓-spin band. If ¯hω is less than the effective Zeeman splitting|g∗µBB + AIz|, as in fig. 2(a), this is not possible, since the ↓-spin

level is too high in energy. If, however, ¯hω is equal to |g∗µBB + AI_z| (fig. 2(b)), electrons in the (E_0,1+ ¯hω)-level can flip-flop scatter into the ↓-spin level. As a result, part of the spin-up electrons that would normally, i.e. in the absence of spin-flip scattering, flow into lead 2, now end up as spin-down electrons in lead 3 [16]. One would expect this to lead to a visible change in I₃, the current pumped into contact 3.

We now calculate I₁, I₂ and I₃ from eq. (5). Assuming equal barriers (q_AB = q_AC ≡ q and ¯V_AB= ¯V_AC≡ ¯h2p/m∗), low temperatures (kBT
¯hω) and no spin-flip scattering in the region between the QPCs, we obtain, in the adiabatic limit,

Ino sf 1 = 0, (6a) Ino sf 2 = − 2eω π k3 0pq2(k20+ p2− 2q2) sin(4-k0L + φ) [(k2 0+ p2)2+ 4q2(k20− p2) + 4q4]2 , (6b) Ino sf 3 = 2eω π k3 0pq2(k02+ p2− 2q2) sin(4-k0L + φ) [(k2 0+ p2)2+ 4q2(k02− p2) + 4q4]2 . (6c) Here - ≡ ¯hω/(4[EF−1₂¯hω2

0+ ω2c −12g∗µBB −12AIz]) [17] and φ ≡ φAC− φAB. In the opposite limit of full spin-flip scattering, i.e. when each electron in the ↑-spin channel in the region between the two QPCs is scattered into the↓-spin channel, the result is

Isf 1 = 0, (7a) Isf 2 = − eω π k04q2 
k2 0+ p2  +
k2₀+ p2− 2q2cos(4-k0L + φ) + +_kp 0
k2 0+ p2− 2q2  sin(4-k0L + φ) /
k2 0+ p2 2 + 4q2
k2₀− p2+ 4q4 2, (7b) Isf 3 = eω π k40q2 
k2 0+ p2  +
k₀2+ p2− 2q2cos(4-k0L + φ) + +_kp 0
k2 0+ p2− 2q2  sin(4-k0L + φ) /
k2 0+ p2 2 + 4q2
k2₀− p2+ 4q4 2. (7c) Note that in both cases the total current I1+I2+I3= 0. Equations (6) and (7) are proportional to the pumping frequency ω and amplitude q2_{, which is characteristic of adiabatically pumped} currents [1, 2]. The currents in general do not vanish if the phase difference φ = 0, though. This is due to the 3-terminal geometry considered here: for the corresponding 2-terminal Hall bar, one obtains I1,2 ∼ q2sin φ sin(2-k0L) k0∼L1 q2ω sin φ. In the presence of a third (symmetry-breaking) terminal, current can be pumped even if the two barriers oscillate in phase and with the same amplitude.

Equations (6) and (7) simplify in case of nearly open QPCs, corresponding to low barriers with p
k0and 2q < p [18]. The pumped currents then become, up to O(_kp

0), Ino sf 1 = 0, (8a) Ino sf 2 = −2 eω π q2 k2 0 p k₀ sin(4-k0L + φ), (8b) Ino sf 3 = 2 eω π q2 k2 0 p k₀ sin(4-k0L + φ), (8c)

1 1.5 2 ω (1011 s-1) 0 0.5 1 <I 3 > (nA) ω_down ω_up

Fig. 3 – The averaged pumped current I3 (eqs. (8c) and (9c)) for a linear sweep of the pumping frequency ω, see the text for explanation. Parameters used are B = 4 T , φ = π/2, q2_/k2

0 = 0.1,

g∗_{=−0.44, m}∗_{= 0.61 10}−31_{kg, A = 90 µeV [19], t}

sweep= 10−5s, tdwell= 10−5s, v = 1016s−2 and

Nn= 107. and Isf 1 = 0, (9a) Isf 2 = − eω π q2 k2 0  1 + cos(4-k0L + φ) + p k₀sin(4-k0L + φ)  , (9b) Isf 3 = eω π q2 k2 0  1 + cos(4-k₀L + φ) + p k₀sin(4-k0L + φ)  . (9c)

Note the dramatic change inI3 (I3 averaged over k0L), from 0 for no spin-flip scattering to the finite value eω_π q_k22

0 at the onset of full spin-flip scattering. This change inI3 can be used

to detect the amount of nuclear polarization in this region [20]. In order to illustrate this, fig. 3 shows the behavior ofI3 if the pumping frequency is linearly swept up during a time tsweepas ω(t) = ω(0) + vt (with v the sweep rate), then kept at the value ω(t_sweep) during a time tdwell, and subsequently swept down again as ω(t) = ω(t_sweep)−v(t−tsweep−tdwell) for t_sweep+t_dwell≤ t ≤ 2tsweep+ t_dwell. Starting from ω(0) = 1· 1011_s−1_{, the pumped current into lead 3 is first}

zero, since for low frequencies no spin-flip scattering can occur (fig. 2(a)). Then, assuming the nuclei to be initially unpolarized, at frequency ω_up=|g∗|µBB/¯h electrons begin to scatter from the↑-spin band to the ↓-spin band (fig. 2(b)), leading to a jump in I3to the value eω_π q_k22

0

(as-suming maximal spin-flip scattering, see the discussion below). Sweeping the frequency further up continues to increase the amount of nuclear polarization through spin-flip scattering [21]. Upon reversing the sweep direction, the current decreases while the nuclear polarization still increases, until at frequency ω_downthe highest energy level of the↑-spin falls below that of the ↓-spin, reducing spin-flip scattering and the current to zero. Because the nuclei are now polar-ized with average polarizationI_z(t) > 0, ωdown is smaller than ω_up and a hysteresis loop is formed, see fig. 3. For Ga and As nuclei with I = 3/2,I_z(t) = 3 × the number of flipped spins per unit of time× (t−tup)/ total number of nuclear spins, where tupcorresponds to the time at which ωupis reached, so tup= (|g∗|µBB−¯hω(0))/(¯hv). In case of nearly open QPCs this yields

Iz(t) = 3_πNωav n q2 k2 0
t − tup (10)

∆ω = M ¯ h(M − ¯hv)[ 2|g∗|µBB − 2¯hω(0) − ¯hv(2tsweep+ tdwell)] , (11) with M ≡ 3 ωav πNn q2 k2

0A and provided M(tdown− tup)≤ |g

∗_{|µBB. ∆ω is thus a direct measure}

of N_n. This opens the possibility to both control (through tsweep, tdwelland v) and accurately estimate (via eq. (10)) the amount of nuclear polarization by changing the pumping frequency ω and measuring I₃. The sharpness of the steps in fig. 3 is due to the mean-field approximation of the nuclear polarization Iz and is reduced if one goes beyond that approximation.

Finally, let us discuss spin-flip times due to hyperfine interaction and previous work on nuclear spin effects in 2DEGs. We have recently calculated the spin-flip time tsf due to hyperfine interaction in a quantum channel in the presence of a magnetic field [22]. Using this result to calculate the spin-flip time t↓↑ from the ↑-channel to the ↓-channel we obtain for typical GaAs parameters t_↓↑ < 10 ns, while the reverse scattering time t_↑↓  t_↓↑ due to full occupancy of the ↑-spin edge channel. The travel time of an electron in the region between the two QPCs is ttravel ∼ L/vch ∼ 5 · 10−8s for L = 50 µm and vch = _m¯hk∗ ω

2 0

ω2 c+ω20

with ω₀≈ 0.08 ωc [10]. Since t_↓↑< t_travel
t_↑↓, the probability that an electron with spin-↑ will scatter to the↓-spin channel is very large and we can to good approximation assume full spin-flip scattering. We then find that I₁, I₂, I₃∼ 0.1 − 1 nA and for timescales t_dwell∼ 10−5s shifts of∼ 30% of the Zeeman energy gap, which is well within reach of observation.

Nuclear spin effects at the edges of a 2DEG have been studied before in the past decade [23– 25]. In ref. [24], Dixon et al. examine a similar 3-terminal geometry at ν = 2. They measure I–V curves, which show hysteretic behavior and provide strong evidence for dynamic nuclear polarization through hyperfine interaction at the edges. The new element in our work here is the use of quantum pumping instead of applying a bias voltage to generate a current. This allows, as far as we know for the first time, to quantitatively predict the current flow between spin-split edge states and the amount of local nuclear polarization. Our simple model is based on a number of assumptions, for example neglect of exchange interactions and correlation effects between the edge channels. A rigorous theory should take these into account. However, based on experimental results in similar geometries [24], we believe that the non-interacting particle picture presented here provides a valid first-order description of the basic physics of quantum pumping in spin-split edge states.

In conclusion, quantum pumping is a useful tool to study and probe local dynamic nuclear polarization at the edge of a 2DEG in the quantum Hall regime. Our model provides the first quantitative technique to accurately manipulate and detect local nuclear spin polarization. This is a topic of great recent interest, not in the least in view of possibilities to develop spin qubits in solid-state systems [26].

∗ ∗ ∗

Stimulating discussions with M. B¨uttiker_{and M. Heiblum are gratefully acknowledged.} This work has been supported by the Stichting voor Fundamenteel Onderzoek der Materie (FOM), and by the EU’s Human Potential Research Network under contract No. HPRN-CT-2002-00309 (“QUACS”).

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