University of Pisa
Department of Physics
Master’s thesis
Spin-orbit interaction in
suspended InAs nanowires
Andrea Iorio
External supervisors
Dr. E. Strambini
Dr. F. Giazotto
Internal supervisor
Prof. S. Roddaro
Academic year 2017/2018
c
Andrea Iorio - 2018
andreaiorio@me.com
Department of Physics University of Pisa
Largo Bruno Pontecorvo 3, 56127 Pisa, Italy NEST Laboratory
National Enterprise for nanoScience and nanoTechnology Scuola Normale Superiore
Piazza San Silvestro 12, 56127 Pisa, Italy
front page
A C K N O W L E D G E M E N T S
Dedicated to my beloved family I would like to first thank my thesis advisors: Dr. F. Giazotto, Dr. E. Strambini and Prof. S. Roddaro. Thanks to Franz, for the opportunity of carrying out my master’s thesis in its amazing group. Your enthusiasm and passion for science is contagious for anyone is working with you. Thanks to Elia for all your help, guidance, advice and patience. Your door was, literally, always open for me. Thank you for the countless sessions of questions and for your day-to-day support and encouragement during the whole year. Thanks to Rod for all the fruitful discussions and for the availability to clarify my doubts despite your busy schedules. Your photography skills captured the magical moment in which I submitted my first paper!
I want to extend my gratitude to Mirko Rocci, Valentina Zannier and Lucia Sorba for having fabricated the devices measured in this thesis: without you this work would not have been possible. My sincere gratitude goes also to Matteo Carrega and Lennart Bours for all the help and the valuable suggestions during the last year.
Finally, I would like to acknowledge the Department of Physics at the University of Pisa and the NEST Laboratory for making this dissertation possible.
C O N T E N T S
1 introduction and motivations 1
1.1 Spin-orbit qubits 2
1.2 Solid-state Majorana fermions 3
1.3 Thesis outline and objectives 4
2 theory and state of the art 7
2.1 Spin-orbit interaction 7
2.1.1 Rashba & Dresselhaus interaction 7
2.1.2 Spin relaxation mechanisms 11
2.2 Electron interference effects 14
2.2.1 Weak localization 14
2.2.2 Universal conductance fluctuations 17
2.2.3 Weak anti-localization 18
2.3 Quantitative models of weak anti-localization 19
2.3.1 Magnetic dephasing time 22
3 results and discussion 25
3.1 Measurement setup 25
3.2 Electrical characterization 26
3.3 UCF and averaging techniques 30
3.4 Weak anti-localization analysis 35
3.5 Temperature dependence 37
3.6 Magnetic field orientation dependence 39
3.6.1 Transverse plane 40
3.6.2 Longitudinal plane 43
3.7 Tuning of the Rashba effect 45
4 device fabrication and experimental setup 49
4.1 Device fabrication 49
4.1.1 Nanowire growth 49
4.1.2 Electron beam lithography 50
4.1.3 Device fabrication process 51
4.2 Dilution refrigerator unit 52
5 conclusion and outlook 57
a appendix 59
a.1 Magnetic dephasing time: different geometries and field orientation 59
a.2 WAL in the x−y plane 60
a.3 Manuscript submitted for publication 61
1
I N T R O D U C T I O N A N D M O T I V A T I O N S
Semiconductors form the foundation of today’s electronic devices. Over the past decades the development of the semiconductor technology has been aimed to in-crease the performance and complexity of electronic circuits [1]. The tremendous
number of transistor in state-of-the-art integrated circuits comes along with a cor-responding shrinkage of device dimensions which nowadays features the size of a dozen of nanometers. At this scale, quantum effects are becoming pronounced in how electronic devices and signals behave and unusual and unexpected changes can compromise their functionality [2].
Stepping into the world of quantum physics, however, can also offer a broad range of novel opportunities. If the manipulation of electric charge in semiconductors forms the basis of all contemporary electronic devices, spin-dependent phenomena in semi-conductors have now opened the door to technological possibilities that harness the spin of the electron [3,4]. In addition to providing spin-dependent analogies that
ex-tend existing electronic devices into the realm of semiconductor spintronics, the spin degree of freedom also offers prospects for fundamentally new functionality within the quantum domain [3,5]. Since the suggestion of Feynman [6], physicists have been
working on the idea to build entirely novel types of information processors, namely quantum computers, in which the computation is encoded in quantum bits (qubits). The discovery of quantum algorithms that run up to exponentially faster than their classi-cal counterparts suggested that a huge advantage may come by performing operations exploiting the quantum mechanical properties of nature [7,8].
Several systems have been proposed for implementing such a quantum computer, from the most “natural” quantum systems, like single atoms or ions [9, 10], to more
challenging but practical platforms like solid-state structures [5, 11, 12]. In this
con-text, systems that have recently demonstrated promise for quantum computing are semiconductor nanowires (NWs) with strong spin-orbit interaction (SOI) [13, 14]. On
the one hand, qubits based on the coherent manipulation of electron’s spin (spin qubits) in nanowires yield the fastest electrical spin manipulation times and offer an easy in-tegration in today nanofabrication process [15,16]. On the other hand, semiconductor
NWs with strong SOI, when proximized by a superconductor, are predicted to host
RF burst 1 2 3 4 5 Gate finger InAs nanowire dot 1 dot 2
Figure 1: Schematic illustration of a double quantum dot formed in an InAs nanowire. Gate 1,
3, and 5 are employed to define the barriers of the dots. Gate 2 is used to apply the rf-burst for the EDSR transitions.
Majorana fermion quasiparticles which represent the building blocks of topological qubits [17–19].
Driven by these perspectives, the investigation of such systems is one of the most active and exciting area of research combining both profound fundamental physics and a potential for applications.
1.1
spin-orbit qubits
A schematic illustration of a typical nanowire spin qubit is shown in Fig. 1. A
sin-gle InAs nanowire, a material with a strong SOI, is placed over an array of fine gate electrodes. Applying negative voltages to the gates allows to selectively deplete re-gions of the nanowire, thereby establishing a double quantum dot. One of the dots defines the qubit, while the second is used as a detector of the spin state. Detection relies on the fact that in the two-electron regime the current through the double dot is highly sensitive to the relative spin orientation of the two electrons due to the Pauli exclusion principle in a phenomenon called spin-blockade [20]. For the manipulation
be-tween spin-up and spin-down states, a standard technique is electron spin resonance (ESR) [21]. A high frequency magnetic field is generated by a current in a coil
fabri-cated near the quantum dot and the duration of the microwave pulse controls the spin rotation angle. However, this approach is not easily scalable as coils tend to be large and magnetic fields affect multiple nearby qubits. Materials with strong SOI come in
1.2 solid-state majorana fermions 3 E B Beff k 1 2 3 1 2 3 x y t a) b)
Figure 2: a) Example of a braiding operation. The order in which a series of exchanges occurs
determines the particle final state. b) Schematic illustration of a “Majorana nanowire” with the directions of the electric field E due to the presence of the substrate, the pinning of the spin-orbit field Beffand the external Zeeman field B.
our help since spin resonance can also be induced by local electric fields on the gate electrodes (electrical dipole spin resonance, EDSR) [22,23]. As the electron is moved back
and forth by the gate, its spin rotates under the influence of the effective spin-orbit field (see Section 2.1). Spin rotation frequencies beyond 100 MHz can be achieved
enabling fast qubit manipulations for nanowire-based spin-orbit qubits [24].
1.2
solid-state majorana fermions
Majorana fermions were originally derived in the context of particle physics as real solutions to the Dirac equation [25]. Despite they have been researched in elementary
particle physics [26], it has been also shown that Majoranas may arise also in certain
condensed matter systems [17, 27]. What has spurned the increased interest in their
research is not just their remarkable particle/antiparticle property (they are their own antiparticles), but also their properties under exchange. Namely, their exchange statis-tics is non–Abelian (i.e. non–commuting), which means that exchange of particles can lead to a final state that is a unitary transformation of the initial one [13, 28]. In
other words, the effect of exchanging, also known as braiding, is analogous to perform quantum state operations because the final state after several exchanges depends on the order in which exchanges have been performed (Fig. 2a). This allows to encode
and process quantum information in a topologically protected way since the braiding operation is insensitive to local perturbations.
Among the several recipes for how to generate Majorana fermions, one is relatively straightforward because it contains just four common ingredients: a nanowire
sys-tem, strong spin-orbit interaction, superconductivity and magnetic fields [17, 19]. It
is predicted that a pair of Majorana fermions can be produced one at each end of a proximized nanowire (i.e. in which superconductivity is induced in the semiconductor by proximity effect [29]), when a Zeeman field is applied orthogonal to the spin-orbit
field (Fig.2b) [17]. First signatures of Majoranas have been found by Mourik et al. [19],
in which the relevance of the SOI was confirmed by observing that zero-bias peaks (a peak in the tunneling current, at zero applied voltage, signature of the presence of the two Majorana fermions), vanish when the magnetic field is aligned with the spin-orbit field.
1.3
thesis outline and objectives
Despite the vast interest and the variety of new exotic phenomena based on SOI, ex-perimental evidences regarding its origin in semiconductor NWs and the vectorial de-pendence of the spin-orbit coupling are limited. Indeed, in one dimensional systems, the electron properties are strongly affected by the surface states and the confinement potentials that depend on the nanowire geometry and position with respect to the underlying substrate [23,30,31]. For instance, confinement can strongly enhance the
electron gyromagnetic ratio [32] and the contact between the nanowire and substrate is
believed to induce an asymmetry in the confinement potential. In turn, this is the main source of the spin-orbit coupling acting via the Rashba effect [33] and causes the
pin-ning of the spin-orbit field in the plane of the substrate, as demonstrated for nanowire quantum dots and for Majorana nanowires [19,23,34].
In this thesis work, we take a different perspective and investigate the SOI in freely suspended NWs which are not expected to display any intrinsic electrostatic asymme-try. In this way, we are able to reestablish the natural geometrical degeneracies since the suspended wires offer an ideal platform for studying the intrinsic SOI. The vecto-rial dependence of the SOI is investigated by tracking the weak anti-localization (WAL) peak [35,36] in the magnetoconductance while rotating the magnetic field orientation
respect to the NW symmetry axes.
The remainder of this thesis is organized as follows:
• Chapter 2 is concerned with the theoretical background necessarily to under-stand the measurements that were conducted. First, SOI in solid-state structures
1.3 thesis outline and objectives 5
and the spin relaxation mechanisms relevant for our discussion are presented. Then, quantum interference phenomena and the theory of WAL are discussed.
• Chapter 3 is focused on the experimental findings of the devices that were in-vestigated. The typical measurement setup is presented. Preliminary electrical characterization of the NWs has been done prior to conduct the magnetocon-ductance measurements. The vectorial features of the spin-orbit coupling are discussed in presence and not of external electric fields.
• Chapter 4 contains the description of the device fabrication. It gives complete steps in how the suspended wires were fabricated for this project and a brief description of the dilution refrigerator unit.
• Chapter 5 summarizes the most important findings of this work and gives out-looks in relation to new devices to be fabricated and measured.
2
T H E O R Y A N D S T A T E O F T H E A R T
2.1
spin-orbit interaction
The aim of this section is to review the basic physics of spin-orbit interaction in solid-state systems. We will especially focus on the aspects of SOI that are relevant for the measurements that are presented in this thesis. Therefore, we will discuss the origin of SOI in semiconductors and put emphasis on the spin relaxation mechanisms that can be induced by SO-coupling.
2.1.1 Rashba & Dresselhaus interaction
The spin-orbit interaction has been first discovered in the context of atomic physics as a relativistic correction coupling electron spin and momentum [37], but appears
in similar fashion also in solid-state systems [33, 38]. It originates when an electron
moving in an electric field E= −∇V experiences, in its rest-frame, an extra magnetic field B= −(v×E)/c2 whose magnitude and direction depends on the velocity v and travel direction of the electron itself (c is the vacuum speed of light). The coupling of the electron spin to the magnetic field through the Zeeman interaction gives rise to the spin-orbit hamiltonian [39]
HSO =− ¯h
2
4mec2
σ· (p× ∇V), (1)
where meis the free electron mass, ¯h is the reduced Planck’s constant, p is the electron
momentum and σ the Pauli spin matrices. In traditional atomic physics, the electric field produced by the charged nucleus gives rise to the SOI responsible of the atomic fine-structure splitting. In this context, for a single electron in the electric field of a charge Ze, Eq.1can be written as [39]:
HSO ∝
Z
r3(r×p)·σ, (2)
where r is the electron position vector. Since the strength of the field from the nucleus depend on its charge Ze, the effect of SOI is larger for elements with high atomic number. In these cases, the electric field close to the nucleus is strong enough so that its effects are significant even at non-relativistic momenta.
Similarly in solids, when there is a potential gradient on the average, effective SO interactions arise [33, 38]. In a single particle picture, it is possible to describe the
electron motion in presence of SOI by including the contribution of Eq. 1in the
one-electron hamiltonian [40,41] H = p2 2me +V(r)− ¯h 2 4mec2σ· (p× ∇ V), (3)
in which V(r) includes the periodic crystal potential and, eventually, an external ap-plied potential (e.g. gate voltage).
When the electric field causing the SOI originates from a structural inversion asym-metry (SIA), i.e. is due to an asymmetric confinement potential, we talk about Rashba interaction [33]. A typical example is given by a two-dimensional electron gas (2DEG)
formed at the interface of two-dimensional semiconductor heterostructures in which the asymmetry of the triangular confinement potential in the z-direction gives rise to an electric field E = E ˆz (Fig. 3a) [40]. In analogy to Eq.1, the corresponding
hamilto-nian (Rashba hamiltohamilto-nian), can be written as
HR =αRˆz· (σ×k), (4)
where the Rashba parameter αRquantify the strength of the spin-orbit coupling. More
stringent calculations show that the Rashba parameter is considerably larger than naïvely expected for a 2DEG propagating in an electric field E since αR depends not
only on the electric field E due to the macroscopic potential, but also on the local po-tential gradients due to the electron orbitals of the atoms forming the crystal [40, 42].
Thus, given Eq. 2, the Rashba interaction is particularly strong for narrow band-gap
crystals containing atoms with a large atomic number like In, As, Sb or N. A precise derivation for αR can be obtained by using the envelope function approach (see for
instance Fabian et al. [41]).
The hamiltonian in Eq. 1 can also be expressed by defining an effective magnetic
field Beff(k) = 2α¯hRk׈e, with ˆe the electric field direction:
2.1 spin-orbit interaction 9 Beff v E a) InGaAs InP x z y b)
B
eff ky kx EFigure 3: a) Schematic illustration of an asymmetric quantum well responsible of the Rashba
interaction. The electron spin precesses around the magnetic field Beff,
perpendicu-lar to the confining electric field E and to the electron travel direction. b) Direction of the effective magnetic field Beffover the Fermi surface due to the Rashba term with
electric field along the z-axis.
showing that the Rashba interaction leads to a momentum-dependent spin-orbit field around which the spin precesses with angular frequency ΩR = 2αR¯hkF. The direction
of the Rashba field is always pointing perpendicular to k and to the confining electric field (Fig. 3b). The length over which the electron precesses a full cycle, the spin
precession length lR, is given by lR = π¯h
2
mαR.
The importance of the Rashba effect lies in the fact that the asymmetry in the con-finement potential can be varied by electrostatic means (e.g. by external gates) allow-ing the tunallow-ing of the spin-orbit strength and, then, of the electron spin precession length [43, 44]. This mechanism leads to a variety of experimental and theoretical
in-vestigations which exploits the manipulation of the spin orientation in spintronic devices by electrical means [4,45,46]. The most famous example is the Datta-Das proposal for
the realization of a spin-transistor (Fig.4a) [47].
rashba soi in semiconductor nanowires While much initial studies of SOI effects were conducted on planar 2DEG systems, recently there has been an increasing interest in SOI in semiconductor nanowires synthesized by bottom-up approach, as the InAs wires fabricated for this work. In this case, the origin of the Rashba effect, in absence of external fields, is more subtle. One contribution is related to the forma-tion of an accumulaforma-tion layer due to the Fermi level pinning at the nanowire surface (Fig. 4b) [42, 48]. Here, the electrons are confined in a triangular-like quantum well
Source Gate Drain E F EC EV b) a)
Figure 4: a) The electron spin precession due to the Rashba SOI is exploited in a spin
field-effect transistor (Datta-Das proposal). Here spin-polarized electrons are injected through ferromagnetic electrodes and coherently precesses around the Rashba field. The precession length can be varied by applying a gate voltage making the spin orientation of the electrons arriving at the drain to fit or not its magnetization. b) In-terface band bending at vacuum-crystal inIn-terface for an InAs nanowire. The pinning of the Fermi energy into the conduction band gives rise to a surface accumulation layer responsible of the electron confining potential.
second contribution arises for wires placed over substrates. Due to the breaking of the inversion symmetry in the electrostatic environment, a net electric field perpendic-ular to the underlying surface occurs [34]. This has been demonstrated for nanowire
quantum dots and for Majorana nanowires [19, 23], which show the pinning of the
spin-orbit field in the direction perpendicular to the motion of the electrons (the NW axis) and to the electrostatic asymmetry.
Besides the Rashba coupling, a further important contribution to the SOI for crystals characterized by a bulk inversion asymmetry (BIA) is the Dresselhaus interaction [38].
This contribution is due to the internal electric field in the crystal which is not canceled out for unit cells that lack a center of inversion, such as zinc-blende and wurzite (Fig.5a). Again, prominent examples of such a category are III-V semiconductors like
GaAs, GaP, InAs, InSb. Differently from the Rashba hamiltonian, which is linear in momentum, the Dresselhaus one is cubic and is described by [38]
HD =γD[σxkx(k2y−k2z) +σyky(k2z−k2x) +σzkz(k2x−k2y)], (6)
with γDthe Dresselhaus spin-orbit coefficient (for a detailed derivation of Eq.6see Wu
2.1 spin-orbit interaction 11
Beff(k) to a given wave vector, as is illustrated in Fig. 5b, which in this case is not
necessarily perpendicular to k. a) Si b) Ga As
B
eff ky kxFigure 5: a) Diamond (left) and zincblende (right) structures consist of two interpenetrating
face centered cubic lattices. The first is typical of Si or Ge, while the second of InAs or GaAs. In the diamond structure an inversion center is indicated in red. In the zincblende, since the two sublattices are occupied by different types of atoms, no inversion center can be found. b) The effective magnetic field Befffor the Dresselhaus
contribution in a two-dimensional system.
The total spin-orbit field in a sample is given by the sum of both Rashba and Dres-selhaus contributions leading, in general, to a complex spin dynamics. However, for transport along certain crystallographic directions, such as [111] or [001], the Dressel-haus term is absent [50,51]. Since the nanowires investigated in this thesis are grown
along the [111] direction, we will assume the Rashba contribution to be the dominant term of SOI.
2.1.2 Spin relaxation mechanisms
The basic idea behind spintronics is to manipulate the electron spin thanks to the co-herent spin precession induced by SOI. In this sense, the spin precession length lR is a
very import quantity. Impurities and dislocations present in any real system, however, can lead to a randomization of the spin orientation making spintronics ideas ineffec-tive. Different mechanisms can induce spin relaxation, but the ones relevant for our materials are the Elliott-Yafet (EY) [52, 53] and the Dyakonov–Perel (DP) [54],
illus-trated in Fig. 6. It is convenient to introduce a characteristic time scale that quantify
anal-b) Dyakanov-Perel Elliott-Yafet
a)
Figure 6: Illustration of the spin relaxation mechanisms. a) In the Elliott-Yafet mechanism,
scattering can induce spin-flips while preserving the spin orientation while travel-ing between two impurities. b) In the Dyakonov–Perel, the electron spin precesses around the internal spin-orbit field in between the scattering events. In both cases, after several random scattering events, the spin orientation is randomized.
ogously, the spin relaxation length lSO = √DτSO, where D is the diffusion constant.
In the EY mechanism, the spin relaxation originates from a finite spin-flip probability when the electron scatters on an impurity (Fig. 6a). The spin-flip is possible due
to the fact that, in the presence of SOI, the exact crystal Bloch states are not spin eigenstates but a superposition of spin states [40]. Thus, there is finite probability of
spin-flip that may occur when a collision event changes the electron wavevector since the spin orientations associated with different k-states are not mutually parallel. As a consequence, the EY mechanism results in a spin relaxation time proportional the momentum scattering time τEY
SO ∼τe(more frequent scattering will randomize the spin
earlier) [46], and in which the spin orientation is preserved while traveling between
the scattering events.
On the contrary, in the DP relaxation mechanism the electron spin is coherently changing its orientation while precessing around the internal spin-orbit field in its path from one scattering center to the next (Fig. 6b). However, when the scattering
occurs, the electron momentum is randomly changed and similarly the axis around which the spin precesses. Since the scattering sequence is a stochastic process, the electron spin effectively precesses around a sequence of random axes that leads it to lose the information on the spin orientation.
We can make some simple quantitative estimations on the DP mechanism by consid-ering a two-dimensional system with Rashba SOI. Between two scattconsid-ering events the
2.1 spin-orbit interaction 13
spin will precess by an angle δθ = ΩRτe. After a time t, the number of scattering
events will be t/τeand the total precession angle
δθ(t) =ΩRτe
r t
τe, (7)
since, for uncorrelated events, the total squared precession angle will be given by sum of the squared precession angles. Defining the spin relaxation time τSODP as the time after which the total precession angle becomes of the order of unity, we get
1 τSODP = Ω2Rτe. (8) W lR lR W a) b)
Figure 7: Effects of dimensional confinement. a) When the channel width W lRthe motion
is effectively two-dimensional. It can be shown that, in this limit, the spin relaxation length corresponds to the spin precession length lSO = lR [42]. b) In the opposite
case W lR, spin relaxation is suppressed by the boundary scattering. In this case
lRhas a strong width dependence.
From Eq.8we can see that for the DP mechanism, contrary to the EY, τDP
SO is inversely
proportional to the elastic scattering time τe, which is quite counterintuitive. This is
due to the inability of the electron spin to follow the internal spin-orbit field when it is varying its direction too rapidly. This characteristic implies that introducing additional scattering (for example from the walls of a nanowire, see Fig.7) results in an increase
of τSO. Indeed, suppression of spin relaxation due to the lateral confinement has been
both theoretically and experimentally investigated [55–57]. Since lSO is the quantity
that can be experimentally determined in our wires, the conversion between lSO and
lR is a delicate task and needs to take into consideration these non-trivial confinement
2.2
electron interference effects
This section is devoted to briefly review a series of phenomena which appear when the wave nature of the electrons becomes important. An important prerequisite for observing these effects is that the electron phase coherence is preserved. In solid-state materials, preserving electron coherence and observing interference effects is a ma-jor challenge: this is due to the fact that inelastic scattering processes break phase coherence, e.g. the electron-phonon scattering which is abundant at room tempera-ture. Thus, transport phenomena due to electron interference can be observed only by lowering the temperature to a few Kelvin or lower. Moreover, if the electron spin orien-tation is altered while the electron moves, the spin contribution has to be included in the interference process. A well-known example is the weak anti-localization (WAL) effect which will be used in this thesis as a versatile tool to get information on the properties of the SOI in our nanowire devices.
2.2.1 Weak localization
Weak localization is a quantum interference effect where the electrons, after series of elastic scattering events, experience an increased probability for returning to the origin point [35, 36, 58]. Let us consider a diffusive conductor where the electron
partial waves propagate through all the possible paths from a starting point A to a second point B while preserving their phase coherence (Fig.8a). We can describe each
path by a complex amplitude Ai = Cieiϕi, where ϕi =
R
γik·dl is the phase that the
electron of momentum k acquires on its way from A to B along path γi [59].
A
B
a) b)
Figure 8: a) A set of electron wave paths for diffusive propagation from point A to B. Each
path contributes to the total transmission through the sample, but have in general uncorrelated phase. b) The closed loop leading to the weak localization effect. The time-reversed paths are indicated by the green/red arrows.
2.2 electron interference effects 15
The total probability PABfor an electron to move from A to B is obtained by squaring
the modulus of the sum over all the partial wave amplitude contributions PAB =|
∑
i Ai|2=∑
i |Ci|2+2∑
i<j CiCjcos(ϕi−ϕj). (9)The first term of Eq.9corresponds to the classical diffusion probability while the last
one describes the quantum mechanical interference of the partial waves. Usually the contribution of the interference term is not important because when we take many different paths of different lengths, each pair of paths will have a phase difference ∆ϕij = ϕi −ϕj that will be randomly distributed and, when we average over many
pairs, we get that the second term averages to zero. However, there are some special paths: if we consider indeed the closed loops (Fig.8b), we are able to group them in
pairs of identical paths in which an electron can propagate only in two opposite di-rections (time-reversed paths) with the corresponding complex amplitudes A± =C±eϕ±.
When the electron system is invariant under time-reversal, these paths are identical in terms of scattering sequence and length and their transmission coefficient Ci and
phase ϕi are the same, i.e. C+ = C− and ϕ+ = ϕ−. Thus, the electron probability to
return to the starting point of the loop is
Pret=|A++A−|2 =|C+|2+|C−|2+2C+C−=4|C+|2, (10)
which is a factor of two greater than the classical phase incoherent transport regime in which the probability would simply be 2|C+|2. It is important to note that constructive
interference occurs for all possible closed loops in the conductor (as long as their lengths are smaller than the phase coherence length) and their contribution is thus not averaged out. As a result, a larger probability to return to the origin implies that the current through the sample is reduced and the total resistance is increased compared to the classical case, hence the name weak localization.
1.0
0.5
0.0
0.5
1.0
B [T]
0.25
0.20
0.15
0.10
0.05
0.00
G
[e
2/h
]
Diameter W
150 nm
100 nm
50 nm
Figure 9: Typical magnetoconductance curves showing the weak localization correction to the
conductance obtained with the one-dimensional model discussed in Section2.3. As
the wire diameter W gets smaller, higher magnetic fields are needed to suppress the interference effect as in according to the phase shift acquired in Eq.12.
Since weak localization is dependent on the time-reversibility of the electron paths, the application of a magnetic field can suppress the constructive interference effect. Indeed, if the sample is penetrated by a magnetic field B, the phase accumulated by the path γi is modified due to the additional contribution of the vector potential A
δϕi = e ¯h Z γi A·dl. (11)
Then, the phase difference ∆ϕ between two time-reversed paths along a closed loop will be ∆ϕ= Z γ k+·dl− Z γ k−·dl= 2e ¯h Z (∇ ×A)·dS= 2eBS ¯h =4π Φ Φ0 = 2S l2 m , (12) where Φ = BS denotes the flux enclosed by the loop of area S, Φ0 = h/e is the
mag-netic flux quantum and lm =
√
¯h/eB is the magnetic length. In a diffusive conductor usually many loops of different sizes are found and then, if the sample is exposed to a magnetic field, a phase shift according to Eq.12occurs for each closed loop. However,
since each loop encircles a different area, the phase shift is different for each loop and the constructive interference, which occurs for all loops at B = 0, is lost on average.
2.2 electron interference effects 17
30
15
0
15
30
V
G[V]
4.5
5.0
5.5
6.0
6.5
G
[e
2/h
]
e
2/h
Figure 10: Example UCF pattern obtained during this thesis. The NW conductance G is shown
as a function of the applied gate voltage VGfrom a nearby electrode, with black/red
curves corresponding to the back/forth sweeps in VG. The fluctuation are
repro-ducible since they are not instrumental noise but are related to the nanowire impu-rity distribution. The conductance is seen to fluctuate with a typical amplitude of the order of∼e2/h.
As a result, the conductance gradually increases with increasing magnetic field: this is schematically illustrated in Fig. 9, where the conductivity has a minimum at zero
field, while it increases monotonously when a magnetic field is applied.
It is worth to note that here it is assumed that the magnetic field acts only by the phase contribution given by the vector potential in Eq. 11 while effects on spin, due to for
instance a Zeeman field, are neglected.
2.2.2 Universal conductance fluctuations
It is clear from the above discussion that when a sample is exposed to a magnetic field each closed loop will contribute with a phase-dependent correction to the conductiv-ity. However, the summation over these loops may or may not ensemble average to zero depending on the number of such loops contained within the sample. Universal conductance fluctuations (UCFs) occur indeed if the sample only contains a small finite number of scattering centers, like in case of a single wire [42,60,61]. Since the
impu-rity distribution is reasonably fixed for a given configuration, these random-looking but reproducible fluctuations are unique for each individual sample. The same fluc-tuation pattern in the conductance appears also by a change in the Fermi wavelength (e.g. obtained by means of gating), as this cause a change in the phase accumulation of the electron waves during the travel through the scattering loops (see Fig. 10).
Per-haps the most remarkable feature of these oscillation is that the resulting conductance fluctuation are in the order of e2/h regardless of the sample size nor the strength and configuration of the elastic scatterers.
2.2.3 Weak anti-localization
So far spin effects were neglected, which means that the spin orientation was assumed to be preserved when the electron propagated along a closed loop. If the spin ori-entation is altered, however, the interference, that is always constructive at zero mag-netic field, can be suppressed leading to an observable destructive interference even at B=0 [35,36]. One of the possible mechanisms that alters the spin dynamics is the
spin-orbit coupling discussed in Section2.1, which leads to a precession of the electron
spin during its propagation around the internal spin-orbit field.
G
WAL
WL
B
a) b)
Figure 11: a) Electron travel along a closed loop in presence of SOI. The spin precession now
needs to be accounted while calculating the electron return probability in Eq. 12.
b) When spin-orbit coupling is present, an enhanced conductivity is found (weak anti-localization). An increasing magnetic field will gradually suppress the effect of the SOI similarly to what described for WL.
To get an idea on how SOI affects the quantum interference of the time-reversed paths, we can assume a partial wave starting with the initial spin state |si(Fig. 11a).
After the clockwise propagation along a loop, the final spin state will become
|s0i = R|si, (13)
where R∈SU(2)is a rotation matrix given by the product of all the rotations RN· · ·R2R1
2.3 quantitative models of weak anti-localization 19
state along the loop traveled counter-clockwise will undergo the spin rotations in the opposite order and each rotation angle will be inverted
|s00i =R−1|si, (14) with R−1 = R1−1R2−1· · ·R−N1. Since we want to estimate the localization effects, we are interested in the amplitude of the interference contribution in Eq. 10 which is
multiplied by
hs00|s0i = hs|(R−1)†R|si = hs|R2|si, (15) since (R−1)† = R for a unitary operator. If SOI is weak, the spin does not rotate
much and R2 =1 so that the quantum interference is not altered. When SOI is strong, instead, the expectation value of R2 has to be computed on a general spinor state
|si = (a, b)t (with |a|2+|b|2 = 1 and with t denoting the transpose). By averaging
on all the passible trajectories, i.e. averaging over all the possible angles, one finally arrives to [62]:
hs|R2|si = −1
2, (16)
which means that, owing to the negative sign, the quantum return probability Pret of
Eq.10is reduced with respect to the classical one due to the destructive interference in
the presence of SOI. The increased conductance is called weak anti-localization (Fig.11b).
2.3
quantitative models of weak anti-localization
the important length scales The above discussions give us a general idea about the effects of quantum interference on transport phenomena and how these are modified by magnetic fields and spin-orbit coupling. Now, we have a number of lengths scales describing the magnitude of the effect of different parameters on the return probability for an electron to its origin point:
• the magnetic length lm determines the maximum length of paths contributing
to any form of localization in a given magnetic field since time-reversibility is broken for paths longer than lm due to the magnetically induced phase shift;
• the phase coherence length lϕ sets the upper limit on the lengths of paths
• the spin relaxation length lSO gives the distance over which the anti-localization
correction is lost due to spin relaxation.
The relative sizes of these lengths determine whether and to what extent positive and/or negative magnetoconductance is observed.
Below we will briefly sketch how the correction to the conductance can be quantita-tively obtained. Taking the above mentioned parameters into account, it can be written as the time integral of the return probability C(t)[58,61,63]:
∆G(B) =−2e 2 h DW L Z ∞ 0 dtC (t)e−t/τϕhP B(t)ihPSO(t)i. (17)
where D is the diffusion constant, L and W the length and the width of the sample. Here, the first damping term e−t/τϕ takes into account the loss of phase coherence
due to phase breaking mechanisms. hPB(t)i accounts for the phase picked up by
the magnetic field while averaged over all the paths that close after time t. It can be shown [60] to be given by:
hPB(t)i = hei∆ϕ(t)i = exp 2e ¯h Z γ(t)A·dl ≡e−t/τB, (18)
with τB being the magnetic relaxation time. Generally, τB depends on the type of
conductor, i.e. its dimensionality or the kind of boundary scattering and thus has to be determined for each situation separately (see Section2.3.1).
Similarly, one has to include an average over the phase shifts owing to the spin-orbit induced effective magnetic fieldhPSO(t)i. Chakravarty and Schmid [36] showed that:
hPSO(t)i =
1 2
3e−3τSO4t −1, (19)
which contains two interference contributions corresponding to the triplet and singlet state of the the total spin of the two counter-propagating partial waves. It can be shown, indeed, that only the triplet contribution is affected by spin-orbit coupling and then the first exponential decay accounts for the spin relaxation for the three triplet states. The singlet state is not and it is reflected in the constant 1 with negative sign accounting for its destructive interference effect.
The dimensionality of the sample becomes important when determining the form of the return probability C(t). One speaks of 2D or 1D weak localization depending on whether the return probability C(t)is determined by the 2D or 1D diffusion equation.
2.3 quantitative models of weak anti-localization 21
0.50
0.25
0.00
0.25
0.50
B [T]
0.15
0.10
0.05
0.00
G
[e
2/h
]
l = 300 nm
L = 2 m
W = 90 nm
Spin relaxation length l
SO100 nm
200 nm
400 nm
Figure 12: Magnetoconductance curves obtained with Eq.21 for different strength of SOI. A
crossover from WAL to WL is obtained when lSO ∼ lϕ since, when spin relaxation
is less frequent than phase relaxation, the spin orientation is preserved during the electron travel along time-reversed paths.
By using the return probability in one dimension, i.e. C(t) = W−1(4πDt)−1/2 [58],
one arrives to ∆G(B) =−2e 2 h DW L Z ∞ 0 dtW −1(4πDt)−1/2e−t/τϕe−t/τB1 2(3e − 4t 3τSO −1). (20)
The full expression for the magnetoconductance is finally given by integrating the above equation [63]: ∆G(B) =−2e 2 hL 3 2 1 l2 ϕ + 4 3l2 SO + 1 DτB !−1/2 − 12 l12 ϕ + 1 DτB !−1/2 . (21) Eq. 21 is commonly known as the dirty or diffusive limit expression in which le W
and where the electron is scattered many times while traveling across the wire (Fig.12).
In the opposite limit le W, the condition that the electron has to be scattered at
W lφ
le
(DτB)1/2
Figure 13: Typical closed electron trajectory contributing to 1D weak localization in the
diffu-sive regime le W.
C(t) = C(t)(1−e−t/τe). In this case, integration gives the so-called pure or ballistic
limit expression [63]: ∆G(B) =−2e 2 hL 3 2 1 l2 ϕ + 4 3l2 SO + 1 DτB !−1/2 − 1 2 1 l2 ϕ + 1 DτB !−1/2 −3 2 1 l2 ϕ + 1 l2 e + 4 3l2SO + 1 DτB !−1/2 + 1 2 1 l2 ϕ + 1 l2 e + 1 DτB !−1/2 . (22)
In terms of the phase coherence length, the 1D criterion stands also for lϕ W:
indeed, if for instance lϕ W, the electrons lose their coherence before “feeling” the
sidewalls of the wire effectively acting like in 2D. Similarly, at lm ∼W a crossover from
1D to 2D weak localization occurs. The reason for this crossover is that the lateral confinement becomes irrelevant when lm < W since trajectories of duration τB then
will have a typical extension(DτB)1/2 <W. This crossover from 1D to 2D restricts the
available magnetic field range that can be used to study the magnetoconductance in one dimension (weak-field limit).
2.3.1 Magnetic dephasing time
Since the effectiveness of a magnetic field in suppressing the localization effects is con-tained in the functional dependence of τB on B, the magnetic dephasing time is a key
parameter in the MC models. Physically, τB represents the time scale over which a
flux of the order of ¯h/e is enclosed, corresponding to a phase difference of 1 between a closed path and its time-reverse.
For instance, we can qualitatively obtain the expression for τB in two and one
2.3 quantitative models of weak anti-localization 23 W W
B
W lφ a) b)B
Figure 14: Parallel (a) and perpendicular (b) field configuration showing the different areas
picking up a magnetic flux.
to the plane of motion. In the former case, the typical area S enclosed by a backscat-tered trajectory on a time scale τB is of the order S ∼ DτB. The corresponding phase
shift, in view of Eq.12, is∆ϕ∼ DτB/l2
m. The criteria ∆ϕ∼1 thus yields:
τB ∼
l2m
D. (23)
On the contrary, in one dimensional systems, the backscattered trajectories on a time scale τB have a typical enclosed area S∼W(DτB)1/2, due to the lateral compression of
the trajectories (see Fig.13). Consequently, the condition S ∼l2
m for a unit phase shift
implies:
τB ∼
l4m
DW2, (24)
which shows a different functional dependence of τB on B with respect to the
two-dimensional case. The specific prefactor in Eq.24has to be computed both by
numer-ically averaging over random electron paths (see for instance van Weperen et al. [64])
or by analytical expressions (see Appendix A.1), and needs to account the nanowire geometry, magnetic field orientation and type of boundary scattering. In the diffusive regime, for a wire of hexagonal cross section and magnetic field perpendicular to the NW axis, it is τB =C 4l
4 m
DW2 with C=1.8.
angular dependence of τB In contrast to two dimensional systems, where only
a magnetic field perpendicular to the carrier plane suppresses the localization effects, in semiconductor nanowires both the parallel and perpendicular field orientations (with respect to the nanowire axis) are relevant. For diffusive motion one expects
τB⊥ < τBk [65] (see Appendix A.1), which means that a perpendicular field is more
effective than a parallel field in suppressing the localization effect. This difference is related to the 1D size confinement (Fig.14): in the parallel field configuration, electron
B
B
a) b)
Figure 15: a) Illustration of the flux-cancellation effect for a closed trajectory in the ballistic
regime le W. The trajectory is composed of two loops of equal area but opposite
orientation, so it encloses zero flux. b) For tubular transport inside the NW, trajec-tories are allowed to wind multiple times around the magnetic field leading to an increased flux pickup.
diffusion paths enclosing a magnetic flux are well confined within the cross sectional area of the NW, while, for the perpendicular field case, they are mostly confined by the nanowire diameter W in the radial direction and lϕ in the axial one. So the magnetic
fluxes enclosed by electron diffusion paths, for a fixed magnetic field strength, are, on average, greater in the perpendicular field configuration than in the parallel one .
flux cancellation/pick-up phenomena In the above discussion the nature of electron transport and boundary scattering did not play a role since, there, the channel walls only serve to impose a geometrical restriction on the lateral diffusion. However, more complex mechanisms can alter this basic picture. One example is the flux cancel-lation effect [58,60,66], characteristic of the ballistic regime le W, where electrons
move ballistically from one wall to the other with boundaries causing trajectories to fold back upon themselves (Fig. 15a). This effect leads to a reduction of the
effec-tiveness of a magnetic field in suppressing the localization effects due to the flux-cancellation of areas travelled in opposite orientation and leads to an increased value of τB with respect to the diffusive case. On the opposite direction, a reduction of τB is
expected if the transport is dominated by electron states at the tubular surface of the nanowire. Here, indeed, winding trajectories can increase the effective area encircled by time-reversed paths for a magnetic field along the wire axis (Fig.15b) leading to a
3
R E S U L T S A N D D I S C U S S I O N
This chapter is focused on the low temperature magnetotransport properties of the suspended InAs nanowires fabricated for this project (see Chapter4for more details on
the fabrication). Here, we will discuss the main results and achievements of this thesis, targeting the vectorial nature of the spin-orbit coupling in III-V nanowire devices.
In Section3.1we first introduce the measurement setup, the main nanowires
charac-teristics and how the transport measurements are performed. Then, some preliminary characterizations are reported in Section 3.2 by means of transconductance
measure-ments extracting the relevant transport parameters for our wires. In Section 3.3 the
magnetotransport measurements are introduced by discussing typical UCF patterns of our sample obtained by different gating configurations and magnetic field orienta-tions. Since UCFs masks the WAL/WL correction, averaging techniques are employed in order to recover the localization effects.
We start by investigating the temperature dependence of WAL in Section3.5,
show-ing the peculiar WAL/WL crossover as the temperature raises. Some of the main results of this thesis work will be reported in Section 3.6 in which, by studying the
angular maps of WAL, we show that the average SOI within the suspended nanowire is isotropic. In Section 3.7, we remove this isotropy by applying an external electric
field and thus introducing an additional vectorial Rashba spin-orbit component whose strength can be controlled by gating.
3.1
measurement setup
The experimental results are obtained using either two and four-probe transport mea-surements. In the first case, an excitation VSD is applied between the source-drain
elec-trodes and the resulting ISD current is measured. In the four-terminal configuration,
instead, the current ISD is imposed between the source-drain and the corresponding
voltage drop ∆V is measured by two other inner probes. This allows to obtain the
nanowire resistance R = ∆V/ISD avoiding the voltage drop over the contact resis-tances and setup filtering stages (see Section4.2).
The chosen input current of 10 nA is generated from the 10 mV voltage drop sourced by a NF LI5640 lock-in on a 1 MΩ resistor. The lock-in frequency is chosen as an
odd number, typically 129.49 Hz, for a best filtering of 50 Hz or multiples source of noise from the local power-line. The measured voltage signal is amplified by a room-temperatureDL 1201preamplifier and read by theHP 34401A digital multimeter. Ad-ditional DC channels are used to control the electrostatic gates by means of low noise voltage sourcesYokogawa GS200.
A schematic illustration of a suspended InAs nanowire with a sketch of the four-terminal measurement setup is shown in Fig.16. The measurements were controlled
by a PC equipped with LabVIEW software and National Instruments data acquisition card enabling digital-to-analog and analog-to-digital signal conversion. The sample is mounted into a Leiden Cryogenics cryostat with a base temperature of 10 mK and equipped with a 3-axis superconducting vector magnet (for more details on the cryo-setup see Section4.2).
VSG1 ISD VSG2 V VBG z x y
Figure 16: a) Schematic illustration of a suspended InAs nanowire with a sketch of the
four-terminal measurement setup. The nanowire is parallel to the x-axis and the sub-strate is in the x−y plane.
3.2
electrical characterization
We first investigate the electrical transport properties of our suspended InAs nanowires. Typical source-drain I−V sweeps obtained by two-wire measurements for all the
con-3.2 electrical characterization 27
a)
b)
Figure 17: a) Source-drain current ISDas a function of the applied excitation between source
and drain electrodes VSD. The good ohmic behavior indicates the absence of
Schot-tky barriers and the presence of good interfaces between metal and semiconductor. The slopes obtained for the different contact combinations are due to the differ-ent contact resistances. b) Typical magnetoresistance curves obtained for differdiffer-ent source-drain excitations VSDfrom 0.01–3 mV at the temperature of 20 mK.
tact combinations of device H1 are shown in Fig. 17a. The linearity between
source-drain voltage VSD and current ISD indicates the presence of good ohmic contacts and
a sufficiently transparent interface between metal and semiconductor for our devices. In order to choose a proper source-drain bias to the nanowire, some typical magne-toresistance curves are shown in Fig.17b for different VSD for the device B9. Here the
wire resistance R is measured while sweeping the magnetic field B along the z-axis: as the source-drain voltage increases, the fluctuation in the resistance flattens due to the higher electronic temperature with respect to the thermal bath (VSD >kBT/e) and, to a
lesser extent, to the effect of Joule heating into the kilo-ohm nanowire resistance. Since a VSD =0.1 mV is large enough to minimize measurement noise but not too much to
induce heating effects, this bias value will be used in all the following discussion (the equivalent ISD =10 nA will be used in the four-point measurements).
For the nanowire devices investigated in this thesis, characteristic parameters such as carrier mobility µ, carrier density n, mean free path le, Fermi velocity vF and
dif-fusion constant D can all be estimated from the evolution of the NW conductance as a function of the gate voltage (transconductance), controlling the charge density in the NW channel via field-effect [69,70]. By employing semiclassical models of transport,
and the assumptions beyond these models, the above parameters can be extracted once the capacitive coupling of the nanowire to the back gate electrode CBG is known.
a) b)
Figure 18: a) Transconductance of device H1 measured in the 4-terminal configuration
show-ing the conductance G as a function of the back gate potential VBG. The red line is
the fit with Eq.25. b) Transconductance of device B9 with only 3 terminals
avail-able. The data are now fitted with Eq.26considering a resistor Rs in series to the
nanowire. The fitted value for Rs is consistent with the 2.1 kΩ resistance of the
filtered line of the dilution unit.
Some typical transconductances are shown in Fig. 18, where the conductance G is
studied as a function of the applied back gate voltage VBG. In panel a) the
transcon-ductance is obtained in the four-terminal setup for device H1 while, in panel b), the one obtained in the three-terminal configuration for device B9 (one of the contacts is missing, see Section4) at T = 20 mK. In both cases, the transport channels begin to
deplete for negative applied voltage in agreement with n-type carrier transport of the wires. In order to electrically characterize the devices, the conductance G is fitted with the expression
G(VBG) =
µCBG
L2 (VBG−Vth), (25)
where CBG is the back gate capacitance, L the separation between the contacts, µ the
nanowire mobility and Vth the threshold voltage. Since Eq. 25 only expresses the
conductivity for the nanowire channel, in the three-terminal configuration it needs to be further modified by adding a resistor Rsconnected in series
G(VBG) = Rs+ L2(VBG−Vth) µCBG −1 . (26)
3.2 electrical characterization 29
The value of CBG has been estimated by the analytical expression for the capacitance
of a cylinder of diameter W and length L, with center at a height h above an infinite, charged plane, which yields [69]
CBG =
2πe0erL
cosh−1(2h/W), (27)
where e0 is the permittivity of free space and er is the dielectric constant of the gate
oxide (er = 3.7 for SiO2). However, since our nanowires are suspended, a slightly
lower value of er ' 3 has been used in Eq. 27, estimated by taking the weighted
average dielectric constant between the 300 nm thick oxide and∼100 nm air. By using L=2 µm and W '90 nm (as evaluated by the SEM images in Section4), the capacitive
coupling between back gate and nanowire is estimated to be CBG '120 aF.
The resulting fits, obtained by fixing the values of L and CBG and by leaving free
the other parameters, are shown by the red lines in Fig. 18 and lead to an estimate
of µ ' 800–1200 cm2
/Vs compatible with similar nanowires fabricated at the NEST laboratory [71, 72]. From the mobility value, it is possible to retrieve information on
the carrier density n = (eρµ)−1, where ρ = Rπ(W/2)2
L is the wire resistivity, which
yields n'2–4×1018cm−3for VBG =0 V. Similarly it is possible to evaluate the Fermi
velocity vF = m¯h∗(3π2n)1/3 ' 2×106m/s, where the effective mass m∗ = 0.023me for
InAs has been used. Moreover, the electron mean free path le = vFτe is estimated to
be 20–30 nm, with τe= m
∗µ
e , and the diffusion constant D = vfle
3 to be'200 cm 2
/s. It is worth to note that the capacitance CBGused in the fit is just a rough estimate and
may have been overestimated both due to the suspension and to the screening effects of the back gate electric fields by the electrons of the heavily doped nanowires [73,74]. A
lower value of CBG would result in an even higher mean free path making legenerally
comparable with the nanowire diameter W. Therefore, the transport parameters do not allow us to conclusively put our devices in neither one of the regimes le W nor
W le in relation to the one-dimensional theory for weak anti-localization discussed
in Section2.3.
Our devices are also fabricated with pairs of local side gates placed on the opposite sides of the nanowire. To fully characterize them, we have also estimated the coupling of both the lateral gates to the NW. In Fig.19we show the conductance G obtained by
sweeping the two side gates voltages VSG1,2for the device B9 at 10 K (in the following
we will refer always to this device). On the left, two example conductance traces are measured by using just one side gate at time; on the right, the full map obtained with both gates is presented. The two traces in Fig. 19a clearly indicate a different
capac-a) b)
Figure 19: a) The nanowire conductance G as a function of the side gates voltages VSG1(black
trace) and VSG2 (red trace). b) The value of G is shown in a colorplot for different
VSG1,2. The black and red cut of a) are indicated. The blue trace indicates a curve
of constant transconductance.
itive coupling of the two SGs with respect to the wire due to a small asymmetry of the lateral electrodes as it can be seen by the SEM images in Section 4. In Fig. 19b
a constant transconductance curve is indicated by the dashed blue line: keeping the nanowire conductance, i.e. the electrostatic potential on the wire, fixed allows to in-duce a tunable electric field inside the NW without changing its overall charge density. As we will discuss in Section 3.7, this guarantees the tuning of the Rashba effect by
acting only on the external electric field without involving other possible mechanisms linked to a change of the wire carrier density [75,76].
3.3
ucf and averaging techniques
As described in Section2.2.2, universal conductance fluctuations occur in samples with
a small finite number of scattering centers, as in the case of a single nanowire. Here the phase differences between different electronic paths result in aperiodic fluctuations of the conductance of the order of e2/h when an external magnetic field is applied. The same interference effect can also occur when the electron Fermi wavevector is changed through gate potentials.
In Fig. 20 some of such UCF measurements are shown. Each trace is obtained by
sweeping the magnetic field oriented along the z-axis for different values of VBG. It
3.3 ucf and averaging techniques 31
a) b)
Figure 20: a) Sample magnetoresistance curves showing the NW resistance R vs Bzfor
differ-ent applied back gate voltages. b) All the UCF traces are shown in a colorplot for
|VBG| ≤1 V. The cuts of the curves in a) are indicated.
verifying the Onsager symmetry relations [77]. These fluctuation patterns are
deter-mined by the specific impurity configuration that are made accessible by changing the localization of the charge inside the wire through the back gate potential. Different impurity configurations can also be investigated by changing the magnetic field orien-tation. For instance, in Fig.21, the UCF are shown as a function of the azimuth angle θ between the magnetic field and the x-axis and for different values of VBG.
The study of UCF allows to retrieve important information on the mesoscopic prop-erties of the samples (in a larger magnetic field range compared to the one here in-vestigated), but they are detrimental to our experiment since they mask the WL/WAL correction. Averaging techniques are then needed in order to recover the localization effects, e.g. by measuring different nanowires connected in parallel [78] or by
averag-ing the conductance G of a saverag-ingle wire on the different UCF patterns obtained by gat-ing [79,80]. Indeed, the above measurements have been performed as a very first try
to obtain the averaged magnetoconductance hG(B)iV by performing two-parameter sweeps for G(B, V)and subsequent numerical averaging the MC. However, care must be taken to ensure the suppression of the fluctuations, meaning that the average range must be large enough compared to the characteristic scale of the UCF (see Fig.22).
Indeed, the voltage window of only 2 V initially employed (|VBG| ≤1 V) was too
nar-row, resulting in an averaged hG(B)iV still strongly affected by fluctuations. Further-more, performing two-parameter sweeps for G(B, V)presents different critical issues: on the one hand, the voltage range must be large enough to suppress the UCF while the voltage steps must be sufficiently small to have enough data to average on. Thus
0
90
180
[d
eg
]
V
BG= -1.00 V
V
BG= -0.75 V
V
BG= -0.50 V
0
90
180
[d
eg
]
V
BG= -0.25 V
V
BG= 0.00 V
V
BG= 0.25 V
0.4
0.0
0.4
B [T]
0
90
180
[d
eg
]
V
BG= 0.50 V
0.4
0.0
0.4
B [T]
V
BG= 0.75 V
0.4
0.0
0.4
B [T]
V
BG= 1.00 V
6.5
7.1
G [e
2/h]
Figure 21: The evolution of the UCF is shown in each subplot as a function of the magnetic
field orientation along the x−z plane for different applied gate voltage VBG.
the magnetic field has to be swept several times over a wide range (±0.2 T) making the measurements highly time-consuming. On the other hand, sweeping the gate voltage takes less time but results in a significantly higher noise level.
In order to resolve the above issues, instead of performing two-parameter sweeps for G(B, V), we decided to directly measure the gate-averaged magnetoconductance
hG(B)iV by using an AC modulation to the back gate with a low frequency voltage of f = 10.320 Hz and peak-to-peak amplitude Vavgpp = 6 V. A time constant on the
3.3 ucf and averaging techniques 33
10
5
0
5
10
V
BG[V]
4.5
5.0
5.5
6.0
6.5
G
[e
2/h
]
V
avgppFigure 22: Conductance G as a function of the back gate voltage VBGat T =20 mK in which
the typical UCF scale can be inspected. In red the voltage window Vavgpp =6 V used
for averaging with the AC back gate modulation is indicated.
1.0
0.5
0.0
0.5
1.0
B
z[T]
6.4
6.5
6.6
6.7
R
[k
]
V
avgpp= 0 V
V
avgpp= 4 V
V
pp avg= 6 V
V
avgpp= 8 V
Figure 23: Magnetoresistance curves for different averaging voltages Vavgpp are shown. In order
to clear out the effects of UCF, a voltage window of Vavgpp =6 V has been used since
it guarantees the convergence of the averaging procedure.
lock-in of 1 s is set in order to allow ∼ 10 periods from the back gate modulation to be recorded for averaging. The amplitude value of Vavgpp = 6 V has been chosen in
order to guarantee the convergence of the averaging procedure (see Fig.23). After this
process, the localization effects become clear and the dip in the magnetoresistance at low magnetic fields can be distinctly related to the presence of weak anti-localization in our nanowire.
30
15
0
15
30
V
SG1= 0.4V
SG2[V]
4.5
5.0
5.5
6.0
6.5
G
[e
2/h
]
Without UCF averaging (back)
Without UCF averaging (forth)
With UCF averaging
Figure 24: The conductance G as a function of the two SG voltages asymmetrically swept. In
black/gray the back/forth curves obtained without UCF averaging. In red, the trace measured with the AC back gate modulation turned on and in which the conductance fluctuations are completely averaged.
The UCF suppression is even more evident by looking at Fig. 24 in which the
con-ductance is shown while sweeping the two side gate potentials with and without the averaging back gate. In the former case, UCF are superimposed to the conductance while, in the latter, they are completely averaged out. The constant transconductance curve is obtained by sweeping the two SGs as explained in Section 3.2. Here, the
nature of UCF can also be noticed by the back & forth sweeps of the gate voltage (blue/gray traces in Fig.24): the fluctuations are reproducible since they are not due
to instrumental noise but are related to the specific impurity configuration of the sam-ple.
3.4 weak anti-localization analysis 35
3.4
weak anti-localization analysis
Having eliminated the effects of UCF, a typical WAL curve is shown in Fig. 25 at
the temperature of 50 mK and with magnetic field along the z-axis. A clear negative magnetoconductance is observed at low fields interpreted as an effect of weak anti-localization, thus showing that a significant spin-scattering mechanism takes place in our nanowire. Here the correction to the conductance ∆G(B) = G(B)−G(0) is
0.2
0.1
0.0
0.1
0.2
B
z[T]
0.12
0.06
0.00
G
[e
2/h
]
Exp
Fit prefactor A
l 300 nm
lSO
170 nm
A = 3
W = 90 nm
Fit renormalized
Bl 1200 nm
lSO
700 nm
A = 1
W = 20 nm
Figure 25: Example of a WAL trace taken with the magnetic field along the z-axis. The
cor-rection ∆G(B) to the conductance is fitted with the 1D model for WAL: the fit including a prefactor A in Eq.28is indicated by the red line, while the fit obtained
by renormalizing τBby the light blue line.
studied and compared to the 1D theory of WAL for disordered systems described in Section2.3, which yields:
∆G(B) =−A2e 2 hL 3 2 1 l2 ϕ + 4 3l2SO + 1 l2 B !−1/2 −1 2 1 l2 ϕ + 1 l2 B !−1/2 , (28)
where B is the external magnetic field, lϕ the phase coherence length, lSO the spin
re-laxation length, lB =
√
DτB the magnetic dephasing length and L=2 µm the distance
between the contacts. A prefactor A has also been included in Eq.28in order to obtain a satisfactory fit to the data (we postpone the discussion on this prefactor later in the text).
