Large Zeeman Splitting in Out-of-Plane Magnetic Field in a Double-Layer Quantum Point Contact
D. Terasawa, S. Norimoto, T. Arakawa, M. Ferrier, A. Fukuda, K. Kobayashi, Y. Hirayama
aa r X i v : . [ c ond - m a t . m e s - h a ll ] J a n Journal of the Physical Society of Japan
Large Zeeman Splitting in Out-of-Plane Magnetic Field in aDouble-Layer Quantum Point Contact
Daiju Terasawa , Shota Norimoto , Tomonori Arakawa , , Meydi Ferrier , , Akira Fukuda ,Kensuke Kobayashi , , and Yoshiro Hirayama Department of Physics, Hyogo College of Medicine, Nishinomiya 663-8501, Japan Graduate School of Science, Department of Physics, Osaka University, Toyonaka560-0043, Japan Center for Spintronics Research Network, Osaka University, Toyonaka, Osaka 560-8531,Japan Laboratoire de Physique des Solides, CNRS, Universit´e Paris-Sud, Universit´e Paris Saclay,91405 Orsay Cedex, France Institute for Physics of Intelligence and Department of Physics, The University of Tokyo,Tokyo 113-0033, Japan Graduate School of Science and CSIS, Tohoku University, Sendai 980-8578, Japan
In this study, we observe that the conductance of a quantum point contact on a GaAs / AlGaAsdouble quantum well depends significantly on the magnetic field perpendicular to the two-dimensional electron gas. In the presence of the magnetic field, the subband edge splitting dueto the Zeeman energy reaches 0.09 meV at 0.16 T, thereby suggesting an enhanced g -factor.The estimated g -factor enhancement is 17.5 times that of the bare value. It is consideredthat a low electron density and high mobility makes it possible to reach a strong many-bodyinteraction regime in which this type of strong enhancement in g -factor can be observed.
1. Introduction
Tunnel-coupled double-layer two-dimensional electron gas (2DEG) systems exhibit sev-eral interesting phenomena due to two internal degrees of freedom, spin and pseudospin(layer index). A well-known phenomenon in single-layer systems, such as quantum Hall ef-fect (QHE), reveals further richness in double-layer systems .
For example, a predictionof Kosterlitz-Thouless transition in association with the dissociation of pseudospin vortices(“meron”s) is discussed .
Interestingly, these topologically protected pseudospin quasi-particles are considered to be non-Abelian. Furthermore, recent theoretical studies on topo- /
17. Phys. Soc. Jpn. logical quantum computing explored double-layer QHE systems that can host numerousnon-Abelian quasiparticles .
However, the role of spins in double-layer QHE systemsis unclear , because controlling spin and pseudospin degrees of freedom individually isdi ffi cult. This di ffi culty in controlling the spin and pseudospin degrees of freedom hampersprecise identification of QHE ground states and topological quasiparticles. Therefore, thedevelopment of a selective spin filtering technique is required for double-layer systems.For this purpose, using a quantum point contact (QPC) is a feasible technique . Aprevious double-layer QPC study suggests that the system has an excessive interactionregime, in which a strong potential gradient produces an enhanced spin-orbit interactionand e ff ective screening, as a result of high mobility electrons with a very low density. Thisstrong interaction regime possibly leads to an enhanced Land´e’s g -factor because the lowelectron density and strong confinement increases the electron-electron interaction, andthereby increases the exchange interaction. Such a situation is preferable for manipulatingspins using the Zeeman e ff ect. In the aforementioned study , we used in-plane magneticfields and observed a g -factor of twice that of the bare GaAs value. Enhancements in g -factor for in-plane fields are also observed in previous studies . However, g -factor is notisotropic .
32, 38–40)
Thus far, only a few conductance measurements in the presence of out-of-plane magnetic fields are conducted to date ,
36, 41–48) and there is a paucity of reports ondouble-layer QPCs. In the experiments above, enhancements in g -factor were reported thatdeserved theoretical attention .
31, 49–53)
Thus, we believe that further elaborated studies fordouble-layer GaAs / AlGaAs QPC systems in the presence of an out-of-plane magnetic fieldwill provide valuable information on spin manipulation and spin filtering.In this study, we fabricated a QPC in a double-layer 2DEG of GaAs / AlGaAs doublequantum well (DQW) sample and examined a small out-of-plane magnetic field e ff ect on it.To explore the possibility of manipulating electron spins, we determined the remaining basicspin-splitting properties of a double-layer QPC system by investigating the Zeeman gap basedon the g -factor in the out-of-plane direction. Owing to the magnetic confinement, subbandedges (SBEs) are parabolically bended towards higher energy .
54, 55)
We observe a strong per-pendicular field dependence of SBEs. Furthermore, the Zeeman gap splitting of SBEs clearlyappears at 0.1 T, and becomes 0.09 meV at 0.16 T. From the magnetic field dependence ofthe Zeeman splitting, we derive an enhanced g -factor of 7.7 (17.5 times the bare value). Wediscuss the possible contribution of electron-electron interaction to the enhancement of the g -factor. The results are promising for spintronics and quantum computation.The remainder of this paper is organized as follows. In Section 2, the sample structure and /
17. Phys. Soc. Jpn.
Si- d (cid:3) dopingSi- d (cid:3) doping Si (cid:3) doped GaAs Si- d (cid:3) doping Al Ga AsAl Ga AsGaAsGaAsAlAsAlAs(2nm)/GaAs(2nm) x
40 Super LatticeAl Ga AsAlAs(2nm)/GaAs(2nm) x
40 Super Lattice
AlAs(2nm)/GaAs(2nm) x
45 Super Lattice
180 nm300 nm1.2 m m20 nm DQW
20 nm20 nm
GaAs BufferGaAs(100) SubstrateAl Ga AsAl Ga As [Si] = 5 x 10 cm -3 cm -2 cm -2 cm -2 Fig. 1. (Color online) Schematic illustration of the DQW sample. the experimental methods are described. In Section 3, the experimental results and discussionare presented. Finally, brief concluding remarks are presented in Section 4.
2. Experimental Details
The sample used in the study is identical to the sample that was used in another liter-ature .
Figure 1 shows a schematic illustration of the sample layer sequence. The DQW /
17. Phys. Soc. Jpn. heterostructure is grown by molecular beam epitaxy on the GaAs (100) surface in NTT BasicResearch Laboratories. Epitaxial layers with DQW (two 20-nm-wide GaAs quantum wellsseparated by a 3-nm-wide AlAs barrier layer) is located 605 nm below the surface, and isdoped from both sides via 1 × cm − Si δ − dopings that are 200 nm away from each sideof DQW. The electron density in the symmetric state corresponds to 0 . × cm − and thatin the anti-symmetric state corresponds to 0 . × cm − , with an energy gap of 0.29 meVbetween them . The low temperature electron mobility is approximately 2 . × cm / (Vs).A standard Hall bar is fabricated with AuGe / Ni ohmic electrodes in contact with both layers.A pair of split gates with a width of 500 nm and a length of 100 nm is fabricated at the centerof the Hall bar via electron beam lithography technique. The sample is mounted upside-downon the cold finger of the mixing chamber of a dilution refrigerator with a base temperaturecorresponding to 20 mK.Figure 2 shows a scanning electron microscopy image of the split gates and schematic im-age of the measurement. Two-terminal di ff erential conductance G = dI sd / dV sd (where I sd and V sd denote the source-drain current and voltage, respectively) and transconductance dG / dV g ( V g denotes the gate voltage applied to the split gates) are simultaneously measured usingtwo lock-in amplifiers. First, G is measured via the first lock-in amplifier with a frequency of387 Hz and amplitude of V acsd = µ V r.m.s., and a small AC gate modulation V acg = V acg ) is input to the second lock-in amplifier. This method enables precise direct measurementof transconductance. However, conductance slightly becomes noisy. Therefore, conductancechanges that are not supported by concurrent transconductance changes can possibly be ex-perimental noise. A DC gate voltage V dcg is also applied to the sample, and thus the totalvoltage applied to the split gate V g is V g = V dcg + V acg . Additionally, we apply a DC voltageto the source to cancel the voltage of the Seebeck e ff ect and to induce a nonequilibrium bias.Hence, the total voltage applied to the source V sd is V sd = V acsd + V dcsd , where V dcsd denotes thetotal DC voltage applied to the sample. In the graphs and image plots, we ignore the ACcomponent of V g and V sd for practical reasons. The x , y , and z -directions are as follows: x -direction is perpendicular to the current and in-plane to 2DEG (see also Fig. 1); y -directionis parallel along the current and in-plane to 2DEG; and z -direction is perpendicular to 2DEG.A z -directional magnetic field B z is applied using a superconductor (vector) magnet, with themaximum field of B z = /
17. Phys. Soc. Jpn. V sd V g x y
20 mKQPC
100 nm500 nm B z Lock-in amp. sig.in ref.in sig.out sig.in ref.in sig.out
AC ref. DC+AC DC+AC AC ref.
GdG/dV g DC+AC DC+AC
IV converter + - W Fig. 2. (Color online) Scanning electron microscopy image of split gates (QPC) and schematic image of themeasurement setup. Sample is placed upside down on the cold finger of the mixing chamber, and its drain isgrounded. Magnetic field B z is applied perpendicular to the sample.
3. Results and Discussion
Figure 3 (a) shows G in the unit of G = e / h ( e is elementary charge, and h is Planck’sconstant) as a function of V g for di ff erent B z values ranging from 0 to 0.8 T (0.05 T step). Fun-damental conductance properties are similar to those of the single-layer GaAs 2DEGs .
56, 57)
The energy level of electrons becomes quantized due to nanometer-scale lateral confinementat QPC, and thus, the conductance is described by the Landauer-B¨uttiker model .
58, 59)
A clearconductance plateau at G ≃ . G is observed at B z = G ≃ . G gradually develops as B z increases. As subsequently demonstrated, the 1 . G plateau pertains to Zeeman splitting, and thus we focus on its subsequent change. When weincrease B z , the G plateau shifts for larger V g values and the plateau region is extended.Further, conductance exhibits many small plateau-like features for G < G .These features are also evident in the measurement of dG / dV g . Figure 3 (b) shows the dG / dV g profile as a function of V g for B z = dG / dV g plot,the plateau in G and the crossing of SBEs correspond to minima and maxima, respectively.With respect to B z = /
17. Phys. Soc. Jpn.
00 .51 .01 .5 G ( e / h ) -2 .8 -2 .7 -2 .6 -2 .5 -2 .4 -2 .3 V g (V ) B z =
0 T B z ( T ) -2 .8 -2 .7 -2 .6 -2 .5 -2 .4 -2 .3 V g (V )1 2 3 4 5 d G / d V g ( a . u . ) d G / d V g ( a . u . ) V g (V ) G < G G > G SBE2SBE1 SBE3 SBE2SBE1 SBE3 * (cid:66) Fig. 3. (Color online) (a) G in the unit of 2 e / h ( = G ) as a function of V g for several B z values from 0 (boldline) to 0.8 T by 0.05 T step. The dashed line indicates G for B z = . . G plateau. (b) dG / dV g as a function of V g for B z = ff set for clarity. ∗ mark indicates the minimum of 1 . G plateau. The dotted lines are visual guidelines. (c) Image plot of dG / dV g as a function of B z and V g for V sd = G . The dash-dotted line divides G ≥ G and G < G regions. The dottedlines indicate SBE peaks of interest in Fig. 5. The dashed line indicates the corresponding B z value for the nextfigure, Fig. 4. 6 /
17. Phys. Soc. Jpn. soon resolved into three peaks (see ref. ), and a broad peak for the second integer SBE(SBE2) and the third integer SBE (SBE3). These peaks were resolved into two Gaussianpeaks as shown in Fig. 5 (a). Specifically, a small minimum appears in the dG / dV g profile for G = . G (indicated by ∗ ) at B z = . dG / dV g as a function of V g and B z . SBEs show a rapidincrease relative to V g as B z increases. Apparently, SBE2 and SBE3 split into the two mainpeak lines indicated by dotted lines. As shown later in Fig. 5 (a), the broad peak consists oftwo smaller peaks at B z =
0, which indicates the existence of other spin-splitting contributionsto this system. The dG / dV g bifurcation that corresponds to the 1 . G plateau in Fig. 3 (a) isclearly observed near a small field of B z ≥ . require B z = ff erent, namely, a GaAs heavy hole system that is considered toyield a larger Zeeman splitting. The parabolic dependence of the SBEs is typically describedas an additional e ff ective confinement due to the cyclotron motion , that is, m ∗ ω x , where ω = ω + ω c ( ω c = eB z / m ∗ ), at the center of the QPC region. In this system, a strong potentialgradient along the z direction is expected, and this gradient in the potential causes electronsto populate in one layer (back layer) of the DQW . Hence, the low electron density of thissample ( ∼ . × cm − per layer) accelerates the depopulation from the higher energyin the presence of a magnetic confinement. Consequently, this sample embodies one of thelowest density regime in the QPC region in which a strong electron-electron interaction isexpected.However, in the G < G region, as B z increases, the peak lines that belong to SBE1 showcomplicated bifurcations. These lines have features that are not easily associated with thespin-resolved SBE lines. As discussed later, these features are probably attributable to eitherFabry-P´erot resonances
44, 60) or transmission resonances , or to spin-dependent transmis-sions due to many-body interactions. Although these observations are interesting and maybroaden our understanding of a previous study , it is di ffi cult to discuss the Zeeman split-ting of SBE1 based on these lines. Therefore, we focus on the SBE2 and SBE3 peaks.We use the non-equilibrium bias e ff ect to convert the peak separation in V g into Zeemanenergy. Figure 4 shows dG / dV g as a function of V sd and V g for B z = .
16 T. We connect themaxima (SBE) in dG / dV g and draw solid lines for integer series and dash-dotted lines forsplit SBEs. As shown in Fig. 3 (b), we observe three split SBE lines for G ≤ G . Importantly,a small diamond that corresponds to the 1 . G plateau appears due to the increase in Zeeman /
17. Phys. Soc. Jpn. -2.8-2.7 -2.6-2.5 -0.5 0.0 0.5 V sd (mV) B z = 0.16 T Zeeman V g ( V ) dG/dV g (a.u.) Fig. 4. (Color online) Image plot of dG / dV g as a function of V sd and V g at B z = .
16 T. The numerals on theimage plot represent approximate conductance values in G . The white solid lines indicate primary SBE lines;further splitting is indicated by dash-dotted lines, which were drawn based on dG / dV g maxima. energy (indicated by the white arrow). From this diamond pattern, the estimated magnitude ofZeeman splitting, along with the contributions of other spin splitting factors, corresponds to0.09 meV. The bare Zeeman energy ǫ Z = | g | µ B B with | g | = .
44, is calculated as ∼ .
004 meVfor B z = .
16 T, where g denotes the Land´e g -factor, and µ B denotes the Bohr magneton.Subsequently, we extract the Zeeman splitting peak positions and estimate the enhance-ment in g -factor. Figure 5 (a) shows an example of two-peak Gaussian curve fit for two con-volved peaks. Then the extracted peak positions (approximately, the broken lines in Fig. 3)are converted into energies using the relationship between the energy gap and the diamondwidth at B z = .
16 T. In addition, it is necessary to consider the “lever-arm” correction tocompare diamonds in di ff erent V g value regions (Appendix A). The result is shown in Figure /
17. Phys. Soc. Jpn. ∆ E Z for SBE2 and SBE3 as a function of B z . As shown inthe figure, ∆ E Z for both SBEs remain finite at B z = B z region( B z . .
17 T), if we express e ff ective Zeeman energy E z in the following form: E z = q E + ( g ∗ µ B B z ) , (1)where E represents the spin-splitting contributions of the e ff ective magnetic fields otherthan the z -directional applied field; this includes the contribution of the spin-orbit interaction,strong confinement, and electron-electron interaction, which are presumably oriented in the x -direction . g ∗ denotes the g -factor in the z -direction. Then, the fit using the above equation(the solid lines in Fig. 5 (b)) yields E = . ± .
002 meV and g ∗ µ B = . ± .
02 meV / T forSBE2, and E = . ± .
002 meV and g ∗ µ B = . ± .
03 meV / T for SBE3. The magnitudeof the enhanced g -factor g ∗ extracted from the fit for SBE2 is g ∗ ≈ .
7, which is approximately17.5 times the bare value, and g ∗ ≈ . ≈ . . . g ∗ . . and 3 . . g ∗ . . ) and GaAs two-dimensional hole systems (3 . g ∗ . . ). As previously reported , the g ∗ value decreases as the subband index increases,because the 1D confinement becomes stronger for lower subbands. In addition, the di ff erencein g ∗ between SBE2 and SBE3 is probably attributed to the density di ff erence . With respectto the higher B z region ( B z > .
17 T), the data deviate from the fit, which indicates a furtherenhancement in the g ∗ value. As we can approximate dE Z / dB z ≃ g ∗ µ B for the higher B z region, we obtained a considerably higher value of g ∗ µ B = . ± .
08 meV / T from the slope(the dashed line in Fig. 5 (b)). Such large values were obtained using the density functionaltheory .
For this region, the parabolic confinement due to B z contributes to the apparentenhancement in the splitting .
31, 50)
We consider that the electron-electron many-body interaction is the underlying cause ofthe enhanced g -factor. In an earlier experiment, Thomas et al . observed 0.5 plateau thatsuggests a spin polarized state in zero magnetic field, in which the lower density enhancesthe many-body interaction. Considering that the low electron density and high mobility, thesample used in this experiment o ff ers a unique opportunity to realize a state with signif-icant many-body interaction e ff ects. Nuttinck et al . also indicated a 0.5 plateau at zeromagnetic field along with a 0.7 shoulder. Considering that the present sample has the qual-ity that is comparable to the sample used by Nuttinck et al ., a similar regime in terms ofimpurity e ff ect is realized in the experiment. Subsequently, the large Zeeman gap and 0.5plateau should be attributed to a large electron-electron exchange interaction, as proposed in /
17. Phys. Soc. Jpn. d G / d V g ( a . u . ) -2.80 -2.70 -2.60 V g (V)SBE2 SBE3 B z = 0 TSBE1(a) (b) 0.300.250.200.150.100.050.00 E Z ( m e V ) B z (T) SBE2 SBE3 Fig. 5. (Color online) (a) dG / dV g as a function of V g at B z = ∆ E z as afunction of B z for SBE2 and SBE3. Ref.
33, 62, 63) and theoretically indicated in Ref.
31, 49–51)
Along with the exchange interaction,the Zeeman gap can have the contribution of the spin-orbit interaction, as speculated in ourprevious work .
However, as Ref. shows that a large g -factor enhancement is account-able from the exchange contribution, we infer that the spin-orbit interaction contributes lessto the z -directional g -factor enhancement than the contribution by exchange interaction. Itis immediately observed that the enhanced value is larger than that of a single-layer system( g ∗ . . ). Regarding this result, the DQW constriction may constructively a ff ect the g -factor enhancement; however, no theoretical studies to support this idea exist to date. Ad-ditionally, we observed Zeeman splitting approximately corresponding to 0.09 meV in thepresence of an in-plane and perpendicular-to-current magnetic field B x of 2.0 T . In thiscase, the value is approximately twice the bare Zeeman splitting, and thus the di ff erence be-tween x and z -directions is evident, as observed in two-dimensional hole systems .
46, 64)
Asdiscussed in Ref. , this remarkable di ff erence in the g -factor is attributed to the di ff erencein the electric confinement between the x - and z -directions.Further, the strong many-body interaction may a ff ect the conductance. As shown in Fig. 3(a) and (c), we observe the conductance plateaus for G < G and dG / dV g peak lines for B z ≥ . //
Asdiscussed in Ref. , this remarkable di ff erence in the g -factor is attributed to the di ff erencein the electric confinement between the x - and z -directions.Further, the strong many-body interaction may a ff ect the conductance. As shown in Fig. 3(a) and (c), we observe the conductance plateaus for G < G and dG / dV g peak lines for B z ≥ . //
17. Phys. Soc. Jpn. results suggest the deficient involvement of the pseudospin degree of freedom .
Theoret-ically, the possibility of forming a quasi-bound state of electrons in the strong interactionregime was discussed .
If this is the case, the transmission coe ffi cient depends on thespin configuration of the quasi-bound state whether the quasi-bound state is singlet or triplet.Further investigation is required to clarify the relationship between the conductance and spinconfigurations.
4. Concluding Remarks
In conclusion, the results of the study indicate a rapid and strong SBE dependence on B z in a double-layer QPC. The results reveal that the Zeeman gap opening begins to appear at B z = .
10 T. It is important to note that the estimated enhancement in the g -factor is 17.5times the bare value. We attribute the g -factor enhancement to a strong electron-electroninteraction due to low electron density and high mobility. We believe that the results areprofitable to manipulate spins in double-layer systems. Acknowledgment
The authors express their gratitude to K. Muraki and T. Saku of the NTT basic re-search laboratories and A. Sawada for providing a high-mobility sample. This study wassupported by the JSPS KAKENHI (JP15K17680, JP15H05854, JP18H01815, JP19H05826,JP19H00656).
Appendix A: “Lever Arm” Corrections
As shown in a certain reference , the V g dependence between subband edges (the dif-ference between dG / dV g maxima) can be converted into energy gaps by using the corre-sponding V sd di ff erences. However, the conversion coe ffi cient A (termed as the “lever arm”)varies depending on V g , and thus it needs corrections as a function of V g . We deduced thislever arm correction in A from the slopes of subband edge lines dV n g / dV sd . Specifically, V n g denotes the n -th integer SBE line. Figure A·1 shows deduced dV n g / dV sd as a function of V g at 0 T. The slopes are derived from − . < V sd < − . dV g / dV sd = α + α V g , and subsequently the lever arm coe ffi cient as follows: A ( V g ) = A α + α V g , (A·1)where α = .
56 and α = .
54 V − . Thus, the Zeeman gap ∆ E Z is modified as follows: ∆ E Z = A · ∆ P , (A·2)where ∆ P denotes observed V g gaps (between two dG / dV g peaks in V g ). /
17. Phys. Soc. Jpn. d V g / d V s d ( x ) -2.8 -2.7 -2.6 -2.5 -2.4 -2.3 V g (V) Fig. A·1. (Color online) Slope correction of the subband edge lines dV g / dV sd as a function of V g . Appendix B: Conductance at B z = .
30 T
For further information on conductance change at B z = . dG / dV g and G at B z = . V sd and V g where equi-conductance contour lines are incorporated. The black-and-white scale in(a) starts from -10, and thus dark black regions indicate decreases in the conductance. Asshown in the figure, G increases by approximately 0 . G at each dG / dV g maxima (SBE) linefor the G < G region. //
For further information on conductance change at B z = . dG / dV g and G at B z = . V sd and V g where equi-conductance contour lines are incorporated. The black-and-white scale in(a) starts from -10, and thus dark black regions indicate decreases in the conductance. Asshown in the figure, G increases by approximately 0 . G at each dG / dV g maxima (SBE) linefor the G < G region. //
17. Phys. Soc. Jpn. -2.8-2.7-2.6-2.5-2.4 V g ( V ) -0.5 0.0 0.5 V sd (mV) . . . . dG/dV g (a.u.) -2.8-2.7-2.6-2.5-2.4 -0.5 0.0 0.5 V sd (mV) . . . . G (2 e /h ) . (a) (b) Fig. B·1. (Color online) (a) and (b) Image plot of dG / dV g and G as a function of V sd and V g with incorporatedequi-conductance lines. 13 //
17. Phys. Soc. Jpn. -2.8-2.7-2.6-2.5-2.4 V g ( V ) -0.5 0.0 0.5 V sd (mV) . . . . dG/dV g (a.u.) -2.8-2.7-2.6-2.5-2.4 -0.5 0.0 0.5 V sd (mV) . . . . G (2 e /h ) . (a) (b) Fig. B·1. (Color online) (a) and (b) Image plot of dG / dV g and G as a function of V sd and V g with incorporatedequi-conductance lines. 13 //
17. Phys. Soc. Jpn.
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