arXiv:cond-mat/0408137v2 [cond-mat.mes-hall] 8 Dec 2004
Proposal for a spintronic femto-Tesla
magnetic field sensor
S. Bandyopadhyayaand M. Cahayb
aDepartment of Electrical and Computer Engineering, Virginia Commonwealth
University, Richmond, VA 23284, USA
bDepartment of Electrical and Computer Engineering and Computer Science,
University of Cincinnati, OH 45221, USA
We propose a spintronic magnetic field sensor, fashioned out of quantum wires,
which may be capable of detecting very weak magnetic fields with a sensitivity of
∼ 1 fT/√Hz at a temperature of 4.2 K, and ∼ 80 fT/√Hz at room temperature.
Such sensors have commercial applications in magnetometry, quantum computing,
solid state nuclear magnetic resonance, magneto-encephalography, and military ap-
plications in weapon detection.
Key words: spintronics, magnetic sensors, spin orbit interaction
PACS: 85.75.Hh, 72.25.Dc, 71.70.Ej
Preprint submitted to Elsevier Science2 February 2008
There is considerable interest in devising magnetic field sensors capable of
detecting weak dc and ac magnetic fields. In this paper, we describe a novel
concept for realizing such a device utilizing spin orbit coupling effects in a
Consider a semiconductor quantum wire with weak (or non-existent) Dressel-
haus spin orbit interaction , but a strong Rashba spin orbit interaction 
caused by an external transverse electric field. The Dresselhaus interaction ac-
crues from bulk inversion asymmetry and is therefore virtually non-existent in
centro-symmetric crystals, whereas the Rashba interaction arises from struc-
tural inversion asymmetry and hence can be made large by applying a high
symmetry breaking transverse electric field. We will assume that the wire is
along the x-direction and the external electric field inducing the Rashba effect
is along the y-direction (see Fig. 1).
This device is now brought into contact with the external magnetic field to be
detected, and oriented such that the field is directed along the wire axis (i.e.
x-axis). Assuming that the field has a magnetic flux density B, the effective
mass Hamiltonian for the wire, in the Landau gauge A = (0, −Bz, 0), can be
+Vy(y) + Vz(z) + η[(px/?)σz− (pz/?)σx]
z)/(2m∗) + (eBzpy)/m∗+ (e2B2z2)/(2m∗) − (g/2)µBBσx
where g is the Land` e g-factor, µBis the Bohr magneton, Vy(y) and Vz(z) are
the confining potentials along the y- and z-directions, σ-s are the Pauli spin
matrices, and η is the strength of the Rashba spin-orbit interaction in the
in the z-direction)
to apply an electric
field to induce the
Fig. 1. Physical structure of the magnetic field sensor.
We will assume that the wire is narrow enough and the temperature is low
enough that only the lowest magneto-electric subband is occupied. In that
case, the Hamiltonian simplifies to 
H = ?2k2
x/(2m∗) + E0− βσx+ ηkxσz
where E0 is the energy of the lowest magneto-electric subband and β =
Diagonalizing this Hamiltonian in a truncated Hilbert space spanning the two
spin resolved states in the lowest subband yields the eigenenergies
and the corresponding eigenstates
where θkx= -(1/2)arctan[β/ηkx].
Note that if the magnetic flux density B = 0, so that β = 0, then the en-
ergy dispersion relations given in Equation (3) are parabolic, but more impor-
tantly, the eigenspinors given in Equation (4) are independent of kxbecause
θkxbecomes independent of kx. In fact, the eigenspinors become [1, 0] and
[0, 1], which are +z-polarized and -z-polarized states. Therefore, with B =
0, each of the spin resolved subbands will have a definite spin quantization
axis (+z-polarized and -z-polarized). Furthermore, these quantization axes are
anti-parallel since the eigenspinors are orthogonal. As a result, there are no
two states in the two spin-resolved subbands that can be coupled by any non-
magnetic scatterer, be it an impurity or a phonon, or anything else. Hence, if a
carrier is injected into the wire with its spin either +z-polarized or -z-polarized,
then the spin will not relax (or flip) no matter how frequently energy and mo-
mentum relaxing scattering events take place. The Elliott-Yafet mechanism of
spin relaxation  will also be completely suppressed since the eigenspinors
are momentum independent. Furthermore, there is no D’yakonov-Perel’ spin
relaxation in a quantum wire if only a single subband is occupied . There-
fore, spin transport will be ballistic when B = 0, even if charge transport is
not. In other words, the spin relaxation length would have been infinite were
it not for such effects as hyperfine interaction with nuclear spins which can
relax electron spin. Such relaxation can be made virtually non-existent by ap-
propriate choice of (isotopically pure) materials. Even in materials that have
isotopes with strong nuclear spin, the spin relaxation rate due to hyperfine
interaction is very small.
Now consider the situation when the magnetic field is non-zero (β ?= 0). Then
the eigenspinors given by Equation (4) are wavevector dependent. In this case,
neither subband has a definite spin quantization axis since the spin state in
either subband depends on the wavevector. Consequently, it is always possible
to find two states in the two subbands with non-orthogonal spins. Any non-
magnetic scatterer (impurity, phonon, etc.) can then couple these two states
and cause a spin-relaxing scattering event. In this case, no matter what spin
eigenstate the carrier is injected in, spin transport is non-ballistic. That is
to say, the spin relaxation length is much shorter compared to the case when
there is no magnetic field. Since phonon scattering can flip spin in the presence
of a magnetic field, and phonon scattering is quite strong in quantum wires
(because of the van Hove singularity in the density of states) , the spin
relaxation length can be rather small in the presence of a magnetic field.
Therefore, a magnetic field drastically reduces the spin relaxation length. This
is the basis of the magnetic field sensor.
The way the sensor works is as follows. Using a ferromagnetic spin injector
contact magnetized in the +z-direction, we will inject carriers into the wire
with +z-polarized spins. The ferromagnetic contact results in a magnetic field
directed mostly in the z-direction, but also with some small fringing compo-
nent along the x-direction. The z-directed component of the field does not
matter, since it does not extend along the wire, which is in the x-direction.
However, any x-directed component will matter. Fortunately, the fringing field
decays very quickly and if the wire is long enough, we can neglect the effect
of the fringing field. Therefore, in the absence of any external x-directed mag-
netic field, an injected electron will arrive with its spin polarization practically
intact at the other end where another ferromagnetic contact (magnetized in
the +z-direction) is placed. This second contact will then transmit the carrier
completely and the spin polarized current will be high so that the device re-
sistance will be low. However, if there is an external x-directed magnetic field,
then the injected spin is no longer an eigenstate and therefore can flip in the
channel  and arrive at the second contact with arbitrary polarization. In
fact, if the wire is long enough (much longer than the spin relaxation length),
then the spin polarization of the current arriving at the second contact would
have essentially decayed to zero, so that there is equal probability of an elec-
tron being transmitted or reflected. In this case, the device conductance can
decrease by ∼ 50% compared to the case when there is no magnetic field.
This of course assumes that every carrier was initially injected with its spin
completely polarized in the +z-direction. In other words, the spin injection
efficiency is 100%. A more realistic scenario is to assume, say, a 32% injection
efficiency since it has already been demonstrated in an Fe/GaAs heterostruc-
ture . For 32% injection efficiency, the conductance will decrease by 24% in
a magnetic field. We will err on the side of caution and assume conservatively
that the device conductance will decrease by only 10% in a magnetic field. A
recent self-consistent drift diffusion simulation, carried out for a similar type
of device, has shown that the conductance decreases by 20 - 30% because of
spin flip scatterings .
At this point, we mention that the idea of changing device resistance by modu-
lating spin flip scattering was the basis of a recently proposed “spin field effect
transistor” . Unfortunately, the small conductance modulation achievable
by this technique (maximum 50%) is insufficient for a “transistor” which is an
active device requiring a conductance modulation by three orders of magni-
tude to be useful. In contrast, the device proposed here is a passive sensor that
does not have the stringent requirements of a transistor and, as we show below,
a 10% conductance modulation is adequate to provide excellent sensitivity.
Let us now estimate how much magnetic field can decrease the device conduc-
tance by the assumed 10%. Roughly speaking, this could correspond to the
where kF is the Fermi wavevector in the channel. For a linear carrier concen-
tration of 5×104/cm, kF ≈ 8×106/m. The measured value of η in materials
such as InAs is of the order of 10−11eV-m . For most materials (with weaker
Rashba effect), the value of η may be two orders of magnitude smaller. Ac-
tually, this value can be tuned with an external electric field. Therefore, it is
reasonable to estimate that ηkF ∼ 1 µeV. The magnetic field that can cause
the 10% decrease in the conductance of the device is then found from Equation
(5) to be ∼ 10 Oe, if we assume |g| ≈ 15.
We now carry out a standard sensitivity analysis following ref. . The rms
noise current for a single wire is taken to be the Johnson noise current given by
In=√4kTG∆f = C√G, where k is the Boltzmann constant, T is the absolute
temperature, G is the device conductance, ∆f is the noise bandwidth, and C
=√4kT∆f. In reality, the noise current can be suppressed in a quantum wire
by two orders of magnitude because of phonon confinement . Therefore, In
= C√Gζ, where ζ is the noise suppression factor which is about 0.01. There
may be also other sources of noise, such as 1/f noise, but this too is suppressed
in quantum wires. 1/f noise can be reduced by operating the sensor under an
ac bias with high frequency.
The change in current through a wire in a magnetic field H is the signal
current Isand is expressed as SH, where S is the sensitivity. We will assume
that the current drops linearly in a magnetic field, so that S is independent of
H. In reality, the current is more likely to drop superlinearly with magnetic
field so that S will be larger at smaller magnetic field. This is favorable for
detecting small magnetic fields, but since we intend to remain conservative,
we will assume that S is independent of magnetic field. Therefore, the signal
to noise ratio for a single wire is
(S : N)1=SH
The noise current of N wires in parallel is C√NGζ, whereas the signal current
is NSH. Therefore, the signal-to-noise ratio of N wires in parallel is√N times
that of a single wire.
For a carrier concentration of 5×104/cm as we have assumed, and for a mo-
bility of only 25 cm2/V-sec (at 4.2 K), a single wire of length 10 µm has a
conductance G ≈ 2 × 10−10Siemens. At a 100 mV bias, the current through
a single wire is therefore 20 pA. Consequently, a 10% change in conductance
produces a change of current by 2 pA. Since the 10% change in conductance
is produced at 10 Oe, the sensitivity S = 0.2 pA/Oe. At 1 fT (= 10−11Oe),
the signal current Isfor a single wire = SH = 2 × 10−24A, whereas the noise
current at a temperature of 4.2 K is 2 × 10−18A/√Hz. Therefore the signal
to noise ratio for a single wire is 10−6:1/√Hz and that for a parallel array
of 1012wires is 1:1/√Hz. Consequently, the optimum sensitivity is about 1
fT/√Hz at 4.2 K. If we repeat the calculation for 300 K (assuming that the
mobility is reduced by a factor of 100 from its value at 4.2 K), the optimum
sensitivity is about 80 fT/√Hz.
We now calculate the power consumption of the sensor. The power dissipated
in a single wire is 20 pA × 100 mV = 2 pW and for an array of 1012wires, the
total power dissipation is 2 W at 4.2 K. At room temperature, the conductance
of each wire is reduced by approximately a factor of 100 because of mobility
degradation, so that the power dissipation is reduced to 20 mW for the array.
Therefore, the sensor we have designed is a low power sensor.
It should be obvious from the foregoing analysis that we can increase the
sensitivity, without concomitantly increasing power dissipation, if we decrease
the carrier concentration in the wire, but increase the mobility or decrease the
wire length, to keep the conductance the same. This could allow sub-femto-
Tesla field detection.
At a bias of 100 mV, the average electric field over a 10 µm long wire is 100
V/cm. In the past, Monte Carlo simulation of carrier transport has shown
that even at a much higher field than this, transport remains primarily sin-
gle channeled in a quantum wire because of the extremely efficient acoustic
phonon mediated energy relaxation . Therefore, the assumption of single
channeled transport is mostly valid.
Finally, one needs to identify a realistic route to fabricating an array of 1012
quantum wires. We propose using a porous alumina template technique for
this purpose. The required sequence of steps is shown in Fig. 2. A thin foil of
aluminum is electropolished and then anodized in 15% sulfuric acid at 25 V dc
and at a temperature of 0◦C to produce a porous alumina film containing an
ordered array of pores with diameter ∼ 25 nm . A semiconductor is then
selectively electrodeposited within the pores [15,16] to create an array of ver-
tically standing nanowires of 25 nm diameter. The density of wires is typically
1011/cm2, so that 1012wires require an area of 10 cm2. Next a ferromagnetic
metal is electrodeposited within the pores  on top of the semiconductor.
The pores are slightly overfilled so that the ferromagnetic metal makes a two-
dimensional layer on top. This is then covered with an organic layer to provide
mechanical stability during the later steps.
The aluminum foil is then dissolved in HgCl2, and the alumina barrier layer
at the bottom of the pores is etched in phosphoric acid to open up the pores
from the bottom. Another ferromagnet is then electrodeposited on the exposed
tips of the semiconductor wires through a mask. The organic layer is then
dissolved in acetone and the multilayered film is harvested and placed on top
of a conducting substrate covered with silver paste. Two wires are attached to
two remote diffused Schottky contacts in order to apply a transverse electric
field that induces the Rashba effect. Finally wires are attached to top and
bottom to provide ohmic electrical contacts. This completes the fabrication of
In conclusion, we have proposed a highly sensitive solid-state magnetic field
detector based on spin orbit interaction. Such sensors can operate at room
temperature since the energy separation between subbands can exceed the
room temperature thermal energy in 25-nm diameter wires. The sensitivity
of these sensors could be comparable to those of superconducting quantum
interference devices [18,19,20] and therefore may be appropriate for magneto-
encephalography where one directly senses the very small magnetic fields pro-
Fig. 2. Fabrication steps. (a) Creation of a porous alumina film with 25 nm diam-
eter pores. This is produced by anodizing an aluminum foil at 25 V dc in sulfuric
acid at 0◦C. (b) Electrodepositing a semiconductor selectively within the pores. (c)
Electrodepositing a ferromagnetic layer on top. This could be either a metallic fer-
romagnet, or a semiconducting ferromagnet such as ZnMnSe. (d) Pasting a thick
organic layer on top for mechanical stability of the structure. (e) Dissolving out the
Al foil in HgCl2. (f) Etching the alumina in phosphoric acid at room temperature
to expose the tips of the semiconductor wires. (g) Evaporating a ferromagnet on the
tips of the semiconductor wires through a mask. (h) Dissolving the organic layer in
ethanol, harvesting the film and placing it on top of a conducting substrate covered
with silver paste. (i) Making Schottky contacts on the sides and ohmic contacts at
top and bottom.
duced by neural activity in human brains.
The work of S. B. is supported by the Air Force Office of Scientific Research
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