Download Development of electrostatically controlled quantum Hall

Development of electrostatically controlled quantum Hall
ferromagnet, a new platform to realize high order non-Abelian
excitations
Aleksandr Kazakov,1 George Simion,1 Yuli Lyanda-Geller,1, 2 V. Kolkovsky,3
Z. Adamus,3 G. Karczewski,3 Tomasz Wojtowicz,3 and Leonid P. Rokhinson1, 2, 4
1
Department of Physics and Astronomy,
Purdue University, West Lafayette, IN 47907 USA
2
Birck Nanotechnology Center, Purdue University, West Lafayette, IN 47907 USA
3
Institute of Physics, Polish Academy of Sciences,
Al. Lotnikow 32/46, 02-668 Warsaw, Poland
4
Department of Electrical and Computer Engineering,
Purdue University, West Lafayette, IN 47907 USA
(Dated: Submitted to Nature Materials on August 28, 2015)
1
Search for non-Abelian excitations is motivated by both scientific curiosity and a practical desire to alleviate decoherence problems of conventional
qubits[1, 2]. While current efforts are primarily focused on the discovery of
Majorana fermions, it is understood that braiding of Majoranas is not sufficient
to perform universal quantum operations[3]. Proposals to realize higher order
non-Abelian excitations with denser rotation group require properties not yet
demonstrated in real systems. Here we report development of new heterostructure materials where quantum Hall ferromagnetic (QHFm) transitions can be
induced locally by electrostatic gating. Helical domain walls (h-DW), formed at
phase boundaries in the integer quantum Hall effect (QHE) regime, are similar to helical channels at the edges of two-dimensional topological isolators [4]
and, coupled to an s-wave superconductor, should support Majorana fermions.
In the fractional QHE regime h-DW are predicted to support parafermions[5]
and, possibly, Fibonacci fermions[6], higher order non-Abelian excitations required for the realization of a fully protected topological quantum computer[3].
Beyond the study of topological excitation, new heterostructures can serve as
model systems to study arbitrary configurations of Neél’s domain walls formed
from excitations with fermionic or fractional anyonic statistics.
In a quantum Hall effect (QHE) regime kinetic energy of electrons in a two-dimensional
electron gas (2DEG) is quantized into Landau levels (LL), which are further split due to
the presence of spin and electron-electron interactions[7]. Polarization of a 2DEG and, more
importantly, of the top filled energy level, depends on the number of occupied energy levels
ν = n/nφ (the filling factor is a ratio of electron n and magnetic flux nφ = eB/h densities),
and changes as the system undergoes phase transitions between QHE states with different
filling factors. If a 2D gas is separated into regions with different ν’s by, e.g., electrostatic
gating, chiral current-caring states are formed at the boundary. These states are distinctly
different from domain walls separating magnetic domains in ordinary ferromagnets.
The actual order of spin-split energy levels is determined by an intricate balance between
Zeeman, cyclotron and exchange energies. By shifting the balance it is possible to induce
magnetic phase transitions between different QHE states with the same filling factor. One
of the first studied examples was a transition between unpolarized and polarized ν = 2/3
states[8]. More quantum Hall ferromagnetic (QHFm) transitions have been studied by tilting
2
𝐸𝑠↓ < 0
𝐸𝑠↑ < 0
1.0
𝑄𝐻𝐸
0.5
𝐸𝑠↑↓ [K]
h-DW
𝐸𝑠↑↓
𝐸𝐹
0.0
𝑥
-0.5
𝐵2∗ 𝐵1∗
-1.0
0
3
6
9
12
magnetic field (T)
(b)
60
𝜈=2
40
20
0
𝐸𝑛𝐶𝐹 + 𝐸𝑠↑↓ (meV)
|𝑛, 𝑠
|3, ↑
|3, ↓
|2, ↑
|2, ↓
|1, ↑
|1, ↓
|0, ↑
|0, ↓
80
1
𝑛 + 2 ℏ𝜔𝑐 + 𝐸𝑠↑↓ (meV)
(a)
|4, ↓
|3, ↓
|2, ↓
0.2
|1, ↓
|4, ↑
|3, ↑
|2, ↑
|1, ↑
0.0
1 5
,
3 3
-0.2
-20
0
(c)
|𝑛, 𝑠
2 4 2 8
𝜈= , , ,
3 3 5 5
0.4
5
7
10
magnetic field (T)
(d)
8
9
10
magnetic field (T)
FIG. 1. Formation of helical domain walls in a quantum Hall regime. (a) Schematic of edge
states in a device where spin of the top level is changed under the gate (yellow). At the boundary
counter-propagating edges form a helical domain wall (h-DW). Coupled to a superconducting
contact (green), these h-DW should support non-Abelian excitations (magenta dots), which will
be localized at the edges in the presence of sp in-orbit interactions. (b) Magnetic field dependence
of spin subbands (Eq. 1) in a CdTe QW with xef f = 1.5% (solid lines) and 1.2% (dashed lines) of
Mn paramagnetic impurities T = 100 mK. (c) Fan diagram of Landau levels for xef f = 1.5%). (d)
Fan diagram for composite fermions Λ levels for xef f = 0.15%
magnetic field[9, 10], introducing hydrostatic pressure[11], varying density[12], or inducing
level crossing in dilute magnetic semiconductors [13–18]. If a system is developed where a
QHFm transition can be locally controlled, helical domain walls (h-DW), which consist of
two counter-propagating chiral channels with opposite polarization, will be formed at the
phase boundary, see schematic in Fig. 1. In the presence of spin-orbit interactions these
h-DW resemble Neel’s walls and allow investigation of electron transport in an arbitrary
network of domain walls (unlike ordinary ferromagnets, the bulk of QHFm domains is not
conducting). Even more excitingly, formation of h-DW from QHE states with integral and
fractionalized excitations opens a new class of systems where Majorana and higher order non3
Abelian excitations can be realized. Recent demonstrations of induced superconductivity
from superconducting contacts with high critical field in HgTe [19] and GaAs [20] makes
demonstration of non-Abelian excitations in this system plausible. Unlike implementations
bases on hybrid wire/superconductor structures [21, 22] QHE protection of h-DWs insure
that exactly one helical 1D channel is present at the gate boundary. A network of these
channels can be easily reconfigured in a multi-gate structure providing a path to the braiding
of non-Abelian excitations.
In the following we report development of new heterostructures where QHFm transitions
can be controlled by electrostatic gating. High mobility two-dimensional electron gas is
formed in CdTe quantum well (QW) heterostructures with engineered placement of paramagnetic impurities (Mn, spin S = 5/2). QWs with uniform Mn-doping have unusual
spin splitting with spin levels crossing at high magnetic fields[23], and QHFm transition
in both integer and fractional QHE regimes have been observed in tilted magnetic fields
experiments [15, 16]. In the presence of magnetic field B spin-dependent energy in dilute
magnetic semiconductors is [24]:
g ∗ µB SB
1 ∗
↑↓
,
Es = ± g µB B + xef f Esd SBs
2
kB (T + TAF )
(1)
where the first term is the Zeeman splitting and the second term is due to an s-d exchange
between electrons in the QW and d-shell electrons localized on Mn. Here g ∗ ≈ −1.7 in
CdTe, exchange Esd ≈ 220 meV [25, 26], xef f is an effective Mn concentration, and TAF is
due to the Mn-Mn antiferromagnetic interaction. At low fields spin splitting is dominated
by a large positive exchange term, while at high fields and low temperatures the Brillouin
function Bs (B, T ) ≈ 1 and B-dependence is dominated by the negative Zeeman term. In
Fig. 1 we plot spin splitting of energy levels (1) and spectrum of Landau levels (LL) for
electrons (n + 1/2)~ωc + Es↑↓ and composite fermions (CF) EnCF + Es↑↓ , where energy gaps
√
CF
between CF levels En+1
− EnCF ≈ αC Ec /(2n + 1) ∝ B [27]. Here ~ is the reduced Plank’s
constant, ωc is the cyclotron frequency, Ec = e2 /` is the charging energy, ` is the magnetic
length, constant αC ≈ 0.01 − 0.03 depends on the confining potential [28], and n = 0, 1, 2, ...
for electrons and n = 1, 2, 3... for CFs. The field of spin subbands crossing B ∗ for the same
LL (|n, ↑i and |n, ↓i) or neighboring LLs (|n, ↑i and |n + 1, ↓i) depends on the strength of
the s-d exchange interaction xef f Esd . Thus, engineering heterostructures with gate-tunable
s-d exchange will allow local control of spin polarization in both integer and fractional QHE
4
Barrier
QW
FG, [Mn1]
Back Gate
0.8
Front Gate
Barrier
0.4
0.0
300
400
Barrier
Mn2
0.4
0.0
FG [Mn2]
BG [Mn2]
1.0
0.4
Quantum Well
Barrier
Ec [eV]
200
distance (nm)
1.2
0.8
100
0.3
0.2
 (z)|
0
BG [Mn1]
1.1
(VG)(0)
Ec [eV]
1.2
0.9
(b)
0.9
0.1
1.0
1.1
n(Vg) / n(0)
Mn1
0.0
(a)
0.3
0.8
Mn2
0.2
0.4
0.0
0.1
Mn1
100
 (z)|
Ec [eV]
Sample A (xeff=1.7%)
0.0
130
300
Samples B,C,D (xeff < 0.5%)
(c)
0
100
110
120
130
distance (nm)
distance (nm)
FIG. 2. Modeling of a non-uniform dilute magnetic semiconductor CdMnTe quantum
well heterostructure. (a) Band diagram of a 30nm CdTe QW heterostructure device is modeled
using nextnano3 package [29] (top panel). Electron wavefunction is calculated for different voltages
on the top (middle panel) and back (bottom panel) gates. In (b) an integral overlap χ(Vg ) between
Mn-doped regions [M n1 ] and [M n2 ], normalized to the value at zero gate voltage χ(0), is plotted
as a function of the 2D gas density change for front (FG) and back (BG) gates. (d) Mn doping
distribution (red regions) in different wafers.
regimes.
The second term in (1) is a mean-field approximation to the exchange Hamiltonian
hR
i
P
~
~
~ where interaction of an electron at a poJsd R~ i δ(~r − Ri )Si · ~σ ∝ [M n] |ϕ(z)|dz hSi,
~ i is approximated as an overlap of
sition ~r with a large number of Mn ions at positions R
the electron wavefunction ϕ(z) with a uniform Mn background within z ∈ [M n] and an
~ = hSz i = SBs (B, T ). For quantum wells with homogeneous
average magnetization hSi
R
Mn distribution throughout the whole QW region an integral χ = [QW ] |ϕ(z)|dz has weak
dependence on the shape of ϕ(z) and level crossing field B ∗ is almost independent of a gate
voltage [16, 17].
We now consider non-uniform distribution of Mn inside a QW, e.g. Mn is confined
5
to regions [M n1 ] or [M n2 ] within the QW, see Fig. 2. In these regions ϕ(z) has strong
dependence on the out-of-plane electric field and χ becomes gate dependent, χ = χ(Vg ).
Application of positive (negative) voltage to the front gate shifts electron wavefunction
closer to (away from) the surface, dχ/dVf g > 0 for [M n1 ] and dχ/dVf g < 0 for [M n2 ]. Gate
voltage also changes electron density dn/dVf g > 0, thus dχ/dn > 0 for [M n1 ] and dχ/dn < 0
for [M n2 ] for the front gate. Application of a back gate voltage results in a density change
dn/dVbg > 0 but electrical field shifts wavefunction in the opposite direction, thus dχ/dn < 0
for [M n1 ] and dχ/dn > 0 for [M n2 ] for the back gate. For the formation of well defined
h-DWs we want to control B ∗ with a minimal change of n in order to remain at the same
filling factor ν, or maximize |dχ/dn|.
Quantum Hall effect in sample A is shown in Fig. 3a. A small peak at B ∗ = 7 Tesla
in the middle of the ν = 2 state is a phase transition between |1 ↓i and |0 ↑i states which
changes polarization of the top filled energy level. In the color plots magnetoresistance is
plotted as a function of voltage on the front and back gates (Fig. 3b,c). For both gates
density increases with gate voltage and the peak between ν and ν + 1 shifts to higher B.
The QHFm transition B ∗ shifts in opposite directions as a function of Vf g and Vbg , consistent
with the modelling in Fig. 2. Note that for B = 7 Tesla polarization of the top level can be
tuned between |1 ↓i and |0 ↑i states by electrostatic gating, thus realizing the theoretical
concept of Fig. 1.
For small Mn concentration, relevant for the QHFm transition in the fractional QHE
regime[15], gate-dependent xef f (Vg ) can be extracted from the modulation of Shubnikov-de
Haas oscillations, where nodes are defined by the condition (n + 1/2)~ωc = |Es↑ − Es↓ | [30].
Positions of nodes are found to be gate dependent, as shown in Fig. 3d for the sample with
xef f ≈ 0.34%. From the T -dependence of nodes positions we extract TAF ≈ 80 mK with no
measurable gate dependence. Gate control of s-d exchange xef f (Vg )/xef f (0) = χ(Vg )/χ(0)
is summarized in Fig. 4a for several wafers. Slopes dxef f /dn are in a good agreement with
band simulations, the agreement is further improved when increase of Ec in the Mn-doped
region is taken into account.
Finally, we want to note that no peak is observed at ν = 1 (crossing of |0 ↑i and |0 ↓i
levels), which is most likely due to a large exchange-induced anticrossing which we estimate
to be ∼ 12 K. Exchange interactions are suppressed for the crossing of levels with different LL
indexes. Nevertheless, we measured a small anticrossing of ≈ 1 K at ν = 2, see Fig. 4b, which
6
a)
c)
0.75
5.0
0.50
2.5
0.25
Vfg (V)
7.5
0.3
5
6
7
8
1.0
b) 200
0.0
=3
d) 3.2
0
=2
|>
-100
-200
5
6
0.0
6
8
B (T)
VBG
T = 30 mK
200 V
2.4
Rxx (k)
100
Vbg (V)
4

>

|>
=2
9
B (T)
2.0
0.1
0.00
4
(k)
0.2
|1 |0
0.0
Rxx

>

Rxy (h/e2)
1.00
Rxx (k)
10.0
1.6
0.8
7
8
0.0
0.1
9
B (T)
-200 V
0.2
0.3
0.4
0.5
B (T)
FIG. 3. Gate control of QHFm transition. (a) Longitudinal (Rxx ) and Hall (Rxy ) magnetoresistances in sample A measured at T = 400 mK. An additional maximum at ν = 2 is a QHFm
transition between |1, ↑i and |0, ↓i spin-polarized levels. In (b) and (c) Rxx is plotted as a function
of a voltage on front Vf g or back Vbg gates and magnetic field B. Position of the QHFm transition
is highlighted by a white dotted line. For B = 7 T (green line) polarization of the top LL can be
switched between ↑ and ↓ by the gate. Both plots have the same color scale. Measurements are
performed at T = 300 mK. (d) Shift of SdH oscillations modulation in a sample with xef f = 0.34%
with a back gate voltage measured at the base temperature T ≈ 30 mK.
we attribute to the spin-orbit (SO) coupling between neighboring LLs. Energy spectrum
in the presence of SO interactions is calculated by adding Dresselhaus γD κ · σ and Rashba
γR E · (σ × k) spin-orbit terms to the single-particle Hamiltonian of a 2D gas in magnetic
field in the presence of s-d coupling (1), square well confinement potential in z direction, and
electric field potential eφ(z) ≈ eEz z, see Supplementary Material for details. Here γD and
γR are the Dresselhaus and Rashba constants, and κ is defined as ({k̂x , k̂y2 − k̂z2 }, {k̂y , k̂z2 −
k̂x2 }, {k̂z , k̂x2 − k̂y2 }). Energy spectrum near |0 ↑i and |1 ↓i levels crossing is plotted in the
7
Rxx (k)
800mK
Rxx (k)
xeff (VG) / xeff (0)
1.1
1.0
0.34%
0.2%
0.085%
1.71%
1.71% (FG)
0.9
(a)
0.9
200mK
2
1
0
6.5
7.0
7.5
8.0
B (T)
19.0
E (meV)
Mn xeff
4
0.1
18.5
18.0
17.5
7.1
7.2
7.3
7.4
7.5
B (T)
1.0
(b) 200
1.1
n(VG) / n(0)
300
400
600 800
T (mK)
FIG. 4. Gate control efficiency and effect of spin-orbit interaction. (a) Gate dependence
of the measured effective Mn concentration for different wafers. Efficiency of s-d exchange control
depend on the |dxef f /dn| slope: for QHFm transition in the integer QHE regime (B ∗ (Vg )−B ∗ (0)) ∝
(xef f (Vg ) − xef f (0)). (b) Arrhenius plot of the Rxx T-dependence at the QHFm transition, the
activation energy is 0.096 meV. Top inset: temperature dependence of Rxx near ν = 2. Bottom
plot: anticrossing of |0, ↑i and |1, ↓i levels calculated using spin-orbit Hamiltonian, see text.
insert in Fig. 4b. The value of the anticrossing gap is found to depend only on the Rashba
spin-orbit coupling
∆SO
√
2 2|γR hEz i|
=
.
`
(2)
2
For an average electric field of hEz i = 35000 V/cm, B = 7 T and γR = 6.9 eÅ the gap
∆SO = 70 µeV, in a good agreement with the experimentally measured activation gap of 96
µeV. We note that an ability to open a topologically trivial (spin-orbit) gap is required for
the localization of non-Abelian excitations [5].
Methods
In order to demonstrate electrostatic control of QHFm transition several Cd1−x Mnx Te/
Cd0.8 Mg0.2 Te QW heterostructures were grown by molecular beam epitaxy (MBE), see [15,
16] for details. Mn was introduced into the QD region either as a digital δ-doping (for
xef f > 1%, wafer A) or as a continuous doping (for xef f < 0.5%, wafers B-D), as shown
schematically in Fig. 2c. Samples were patterned into Hall bar geometry and semitransparent
10 nm Ti front gate was thermally evaporated. Copper foil glued to the back of samples
8
served as a back gate. Devices were illuminated with a red LED at 4 K, low temperature
electron density and mobility were in the range of 3.2 − 3.5 · 1011 cm−2 and 2 − 3 · 105
cm2 /V·s in different samples. Electron transport was measured in a dilution refrigerator
using standard ac technique with 10 nA excitation.
Acknowledgements
Authors acknowledge support by the U.S. Department of Energy, Office of Basic Energy Sciences, Division of Materials Sciences and Engineering under Awards DE-SC0008630 (A.K
and L.P.R.), by the Department of Defence Office of Naval research Award N000141410339
(A.K, T.W., G.S. and Y. L-G.), by the National Science Centre (Poland) grant DEC2012/06/A/ST3/00247 (V.K., Z.A., G.K., and T.W.), and by the Foundation for Polish
Science (T.W.).
Authors contribution
L.P.R. and T.W. conceived and designed the experiments, V.K, Z.A., G.K. and T.W. have
grown heterostructures, A.K. and T.W. performed measurements and data analysis, G.S.
and Y.L-G. provide theoretical analysis, A.K. and L.P.R written the manuscript with input
from all authors.
[1] A. Y. Kitaev, Ann. Phys. 303, 2 (2003).
[2] M. H. Freedman, Proceedings of the National Academy of Sciences 95, 98 (1998).
[3] C. Nayak, S. H. Simon, A. Stern, M. Freedman, and S. D. Sarma, Rev. Mod. Phys. 80, 1083
(2008).
[4] M. Z. Hasan and C. L. Kane, Rev. Mod. Phys. 82, 3045 (2010).
[5] D. J. Clarke, J. Alicea, and K. Shtengel, Nat. Commun. 4, 1348 (2012).
[6] R. S. K. Mong, D. J. Clarke, J. Alicea, N. H. Lindner, P. Fendley, C. Nayak, Y. Oreg, A. Stern,
E. Berg, K. Shtengel, and M. P. A. Fisher, Phys. Rev. X 4, 011036 (2014).
[7] R. E. Prange and S. M. Girvin, eds., The quantum Hall effect (Springer-Verlag, New York,
1990).
[8] J. P. Eisenstein, H. L. Stormer, L. N. Pfeiffer, and K. W. West, Phys. Rev. B 41, 7910 (1990).
[9] E. De Poortere, E. Tutuc, J. Papadakis, and M. Shayegan, Science 290, 1546 (2000).
9
[10] G. Gusev, A. Quivy, T. Lamas, J. Leite, O. Estibals, and J. Portal, Physical Review B 67,
155313 (2003).
[11] J. Eom, H. Cho, W. Kang, A. Gossard, and W. Wegscheider, Physica B: Condensed Matter
298, 106 (2001).
[12] J. Smet, R. Deutschmann, W. Wegscheider, G. Abstreiter, and K. von Klitzing, Physical
Review Letters 86, 24122415 (2001).
[13] J. J. Jaroszyński, T. Andrearczyk, J. Wróbel, G. Karczewski, T. Wojtowicz, D. Popović, and
T. Dietl, in Noise as a Tool for Studying Materials, Vol. 5112, edited by M. B. Weissman,
N. E. Israeloff, A. S. Kogan, M. B. Weissman, N. E. Israeloff, and A. S. Kogan (SPIE, 2003)
pp. 165–173.
[14] F. Teran, M. Potemski, D. Maude, T. Andrearczyk, J. Jaroszyński, T. Wojtowicz, and G. Karczewski, Physical Review B 82, 245120 (2010).
[15] C. Betthausen, P. Giudici, A. Iankilevitch, C. Preis, V. Kolkovsky, M. Wiater, G. Karczewski,
B. A. Piot, J. Kunc, M. Potemski, T. Wojtowicz, and D. Weiss, Physical Review B 90, 115302
(2014).
[16] J. Jaroszyński, T. Andrearczyk, G. Karczewski, J. Wróbel, T. Wojtowicz, E. Papis,
E. Kamińska, A. Piotrowska, D. Popović, and T. Dietl, Physical Review Letters 89, 266802
(2002).
[17] J. Jaroszyński, T. Andrearczyk, E. Stringer, G. Karczewski, T. Wojtowicz, J. Wróbel,
D. Popović, and T. Dietl, in AIP Conference Proceedings, Vol. 772 (AIP, 2005) pp. 1271–1272.
[18] F. Teran, M. Potemski, D. Maude, Z. Wilamowski, A. Hassan, D. Plantier, J. Jaroszynski,
T. Wojtowicz, and G. Karczewski, Physica E: Low-dimensional Systems and Nanostructures
17, 335 (2003).
[19] J. B. Oostinga, L. Maier, P. Schüffelgen, D. Knott, C. Ames, C. Brüne, G. Tkachov, H. Buhmann, and L. W. Molenkamp, Phys. Rev. X 3, 021007 (2013).
[20] Z. Wan, A. Kazakov, M. J. Manfra, L. N. Pfeiffer, K. W. West, and L. P. Rokhinson, Nature
Communications 6, 7426 (2015).
[21] R. M. Lutchyn, J. D. Sau, and S. D. Sarma, Phys. Rev. Lett. 105, 077001 (2010).
[22] J. Alicea, Y. Oreg, G. Refael, F. von Oppen, and M. P. A. Fisher, Nat Phys 7, 412 (2011).
[23] T. Wojtowicz, M. Kutrowski, G. Karczewski, J. Kossut, F. J. Teran, and M. Potemski, Phys.
Rev. B 59, R10437 (1999).
10
[24] J. Furdyna, Journal of Applied Physics 64, R29 (1988).
[25] J. Gaj, W. Grieshaber, C. Bodin-Deshayes, J. Cibert, G. Feuillet, Y. Merle d’Aubigné, and
A. Wasiela, Physical Review B 50, 5512 (1994).
[26] A. Kiselev, E. Ivchenko, A. Sirenko, T. Ruf, M. Cardona, D. Yakovlev, W. Ossau, A. Waag,
and G. Landwehr, Journal of Crystal Growth 184-185, 831 (1998).
[27] K. Park and J. K. Jain, Phys. Rev. Lett. 80, 4237 (1998).
[28] Y. Liu, S. Hasdemir, A. Wójs, J. K. Jain, L. N. Pfeiffer, K. W. West, K. W. Baldwin, and
M. Shayegan, Physical Review B 90,, 085301 (2014), arXiv:1406.2387.
[29] “nextnano3 nanodevice simulator,” http://www.nextnano.de/nextnano3/.
[30] F. Teran, M. Potemski, D. Maude, T. Andrearczyk, J. Jaroszynski, and G. Karczewski,
Physical Review Letters 88, 186803 (2002).
11
Supplementary Materials
Development of electrostatically controlled quantum Hall ferromagnet,
a new platform to realize high order non-Abelian excitations
A. Determination of TAF
Antiferromagnetic coupling TAF in Eq. 1 of the main text can be obtained from the temperature dependence of nodes positions in the modulation of Shubnikov-de Haas oscillations:
gµB B
1
∗
~ωc = g µB B + xef f SEs−d Bs
n+
2
kB [T + TAF ]
Temperature dependence of magnetoresistance for a wafer with xef f = 0.34% is shown in
Fig. S1a. From the fits to the temperature dependence of nodes positions we obtain TAF = 84
mK for this sample, we found that TAF ≈ 80 mK in all samples with xef f < 0.5% (in samples
with larger Mn concentrations there is no easy way to determine TAF ).
b)
#081214
33 mK
offset - 0,25 k
Rxx [k]
3
2
1
625 mK
0
0,0
0,2
0,4
0,6
# 081214
node 1
node 2
node 3
node 4
node 5
node 6
node 7
node 8
0,6
node position [T]
a)
0,4
0,2
xeff = 0.34%
0,0
0,0
0,8
B [T]
TAF = 84 mK
0,2
0,4
0,6
0,8
1,0
T [K]
FIG. S1. a) Magnetoresistance in a sample with xef f = 0.34% as a function of temperature. b)
Temperature dependence of nodes extracted from the beating of Shubnikov-de Haas oscillations.
Solid lines are two-parameter fittings (xef f and TAF ) to the nodes positions.
sup-1
B. Calculation of spin-orbit–induced anticrossing of LLs in the presence of s-d
exchange
A general single-particle Hamiltonian can be written as:
H0 =
2
−i~∇ + ec A + 21 gµB B · σ − eφ(r) + Vb (z)
P
− Jσ · i Si δ (Ri − r) + γD κ · σ + γR E · (σ × k) ,
1
2m∗
(S1)
where σ is a vector containing Pauli matrices, φ is an electric potential, and Vb is a
confinement potential in z direction, the fifth term is s-d exchange with Mn impurities
and the last two terms are Dresselhaus and Rashba spin-orbit coupling.
κ is defined
as ({k̂x , k̂y2 − k̂z2 }, {k̂y , k̂z2 − k̂x2 }, {k̂z , k̂x2 − k̂y2 }), where {A, B} = (AB + BA)/2 and k =
−i~∇+eA/c. Magnetic field B = (0, 0, B) corresponds to a vector potential A = (0, Bx, 0).
For exchange interaction we use a mean-field model described in the main text −Jρi hSi ·
σ = −Jρi BS (gµB S|B|/kB T )σ ·b , where ρi is the density of ions, B is the Brillouin function,
kB is Boltzmann’s constant, and b is a unit vector in the direction of magnetic field. We
consider high-field limit BS (x) = 1. We also assume uniform Mn doping in the range
zmin < z < zmax . Electric potential is φ(x, z) ≈ −Ez z, we consider Ez > 0.
The Hamiltonian describing motion in z direction is
Hz = −
~2 ∂ 2
+ Vb (z) − e|Ez |z − Jθ(z − zmin )θ(zmax − z)σz .
2m ∂z 2
(S2)
Its eigenvalues for the lowest subband are:
"
31 #
~2
Λs = e|Ez | −w −
ai1
(S3)
2me|Ez |
0
sJ
2
2
0
2
2
+ 0
2 Ai [Ξ (zmin )] − Ai [Ξ(zmax )] − Ai [Ξ (zmin )] Ξ (zmin ) + Ai [Ξ (zmax )] Ξ (zmax )
Ai (ai1 )
where Ai is the Airy function, ai1 is its first zero, Ξ(z) = (w + z)(2me|Ez |/~2 )1/3 + ai1 , and
s = 1 for spin up states and s = −1 for spin down.
In the presence of perpendicular magnetic field an effective Hamiltonian is

√ γR |Ez | 
†
†
3
2 2 †
D
~ωC a† a† + 21 + 21 (gµB B + δΛ)
i √γ2`
â
ââ
−
a
−
2`
k
â
+ 2 ` â
3
z
,
H0 = 
√
γR |Ez | †
γD
1
1
†
† 3
2 2
† †
√
−i 2`3 ââ â − (â ) − 2` kz â + 2 ` â
~ωC a a + 2 − 2 (gµB B + δΛ)
(S4)
√
√
where lowering and rising operators are defined as a† = (k̂y − ik̂x )/ 2, a = (k̂y + ik̂x )/ 2,
p
` = eB/~ is the magnetic length, and ~ωC = ~eB/m is the cyclotron energy.
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We treat spin-orbit couplings as perturbations and found that only Rashba term has a
non-zero matrix element between |0 ↑i and |1 ↓i energy levels. In the vicinity of crossing
the energy spectrum is
1
E± = ~ωC ±
2
r
(~ωC − gµB B − δΛ)2 +
and the anticrossing gap is
∆SO
8γR2 Ez2
,
`2
(S5)
√
2 2|γR hEz i|
.
=
`
(S6)
2
For an average electric field of hEz i = 35000 V/cm, B = 7 T and γR = 6.9 eÅ , ∆SO ≈ 70
µeV. A similar value is obtained from a numerical solution of Eq. S4.
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