proposal for a spintronic femto-tesla magnetic field sensor
TRANSCRIPT
ARTICLE IN PRESS
1386-9477/$ - se
doi:10.1016/j.ph
�CorrespondiE-mail addre
(S. Bandyopadh
Physica E 27 (2005) 98–103
www.elsevier.com/locate/physe
Proposal for a spintronic femto-Tesla magnetic field sensor
S. Bandyopadhyaya,�, M. Cahayb
aDepartment of Electrical and Computer Engineering, Virginia Commonwealth University,
601 W Main Street, Richmond, VA 23284, USAbDepartment of Electrical and Computer Engineering and Computer Science, University of Cincinnati, OH 45221, USA
Received 6 August 2004; accepted 22 October 2004
Available online 16 December 2004
Abstract
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=ffiffiffiffiffiffiffiHz
pat a temperature of 4.2K, and �80 fT=
ffiffiffiffiffiffiffiHz
pat room
temperature. Such sensors have commercial applications in magnetometry, quantum computing, solid-state nuclear
magnetic resonance, magneto-encephalography, and military applications in weapon detection.
r 2004 Elsevier B.V. All rights reserved.
PACS: 85.75.Hh; 72.25.Dc; 71.70.Ej
Keywords: Spintronics; Magnetic sensors; Spin–orbit interaction
There is considerable interest in devising mag-netic field sensors capable of detecting weak DCand AC magnetic fields. In this paper, we describea novel concept for realizing such a device utilizingspin–orbit coupling effects in a quantum wire.Consider a semiconductor quantum wire with
weak (or non-existent) Dresselhaus spin–orbitinteraction [1], but a strong Rashba spin–orbitinteraction [2] caused by an external transverseelectric field. The Dresselhaus interaction accrues
e front matter r 2004 Elsevier B.V. All rights reserve
yse.2004.10.012
ng author.
yay).
from bulk inversion asymmetry and is thereforevirtually non-existent in centro-symmetric crystals,whereas the Rashba interaction arises fromstructural inversion asymmetry and hence can bemade large by applying a high symmetry breakingtransverse electric field. We will assume that thewire is along the x-direction and the externalelectric field inducing the Rashba effect is alongthe y-direction (see Fig. 1).This device is now brought into contact with the
external magnetic field to be detected, and orientedsuch that the field is directed along the wire axis(i.e. x-axis). Assuming that the field has a magneticflux density B, the effective mass Hamiltonian for
d.
ARTICLE IN PRESS
Semiconductor wire
Ferromagneticcontact (magnetizedin the z-direction)
Schottky contactto apply an electric field to induce the Rashba effect
x
y
Fig. 1. Physical structure of the magnetic field sensor.
S. Bandyopadhyay, M. Cahay / Physica E 27 (2005) 98–103 99
the wire, in the Landau gauge A ¼ ð0;�Bz; 0Þ; canbe written as
H ¼ ðp2x þ p2y þ p2zÞ=ð2m�Þ þ ðeBzpyÞ=m�
þ ðe2B2z2Þ=ð2m�Þ � ðg=2ÞmBBsx
þ V yðyÞ þ V zðzÞ þ Z½ðpx=_Þsz � ðpz=_Þsx;
ð1Þ
where g is the Lande g-factor, mB is the Bohrmagneton, VyðyÞ and V zðzÞ are the confin-ing potentials along the y- and z-directions,s’-s are the Pauli spin matrices, and Z is thestrength of the Rashba spin–orbit interaction inthe wire.We will assume that the wire is narrow enough
and the temperature is low enough that only thelowest magneto-electric subband is occupied. Inthat case, the Hamiltonian simplifies to [3]
H ¼ _2k2x=ð2m�Þ þ E0 � bsx þ Zkxsz; (2)
where E0 is the energy of the lowest magneto-electric subband and b ¼ gmBB=2:Diagonalizing this Hamiltonian in a truncated
Hilbert space spanning the two spin resolved statesin the lowest subband yields the eigenenergies
E� ¼_2k2
x
2m�þ E0 �
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðZkxÞ
2þ b2
q(3)
and the corresponding eigenstates
CþðB;xÞ ¼cosðykx
Þ
sinðykxÞ
" #eikxx
C�ðB;xÞ ¼sinðykx
Þ
� cosðykxÞ
" #eikxx; ð4Þ
where ykx¼ �ð1=2Þ arctan½b=Zkx:
Note that if the magnetic flux density B ¼ 0; sothat b ¼ 0; then the energy dispersion relationsgiven in Eq. (3) are parabolic, but more impor-tantly, the eigenspinors given in Eq. (4) areindependent of kx because ykx
becomes indepen-dent of kx: In fact, the eigenspinors become [1, 0]and [0, 1], which are þz-polarized and �z-polarized states. Therefore, with B ¼ 0; each ofthe spin-resolved subbands will have a definite spinquantization axis (þz-polarized and �z-polarized).Furthermore, these quantization axes are anti-parallel since the eigenspinors are orthogonal. As aresult, 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 intothe wire with its spin either þz-polarized or�z-polarized, then the spin will not relax (or flip)no matter how frequently energy and momentumrelaxing scattering events take place. The Elliott–Yafet mechanism of spin relaxation [4] will also becompletely suppressed since the eigenspinors aremomentum independent. Furthermore, there is noD’yakonov–Perel’ spin relaxation in a quantumwire if only a single subband is occupied [5].Therefore, spin transport will be ballistic when B ¼
0; even if charge transport is not. In otherwords, the spin relaxation length would have beeninfinite were it not for such effects as hyperfineinteraction with nuclear spins which can relaxelectron spin. Such relaxation can be madevirtually non-existent by appropriate choice of(isotopically pure) materials. Even in materialsthat have isotopes with strong nuclear spin, thespin relaxation rate due to hyperfine interaction isvery small.Now consider the situation when the magnetic
field is non-zero ðba0Þ: Then the eigenspinorsgiven by Eq. (4) are wavevector dependent. In this
ARTICLE IN PRESS
S. Bandyopadhyay, M. Cahay / Physica E 27 (2005) 98–103100
case, neither subband has a definite spin quantiza-tion axis since the spin state in either subbanddepends on the wavevector. Consequently, it isalways possible to find two states in the twosubbands with non-orthogonal spins. Any non-magnetic scatterer (impurity, phonon, etc.) canthen couple these two states and cause a spin-relaxing scattering event. In this case, no matterwhat spin eigenstate the carrier is injected in, spintransport is non-ballistic. That is to say, the spinrelaxation length is much shorter compared to thecase when there is no magnetic field. Since phononscattering can flip spin in the presence of amagnetic field, and phonon scattering is quitestrong in quantum wires (because of the van Hovesingularity in the density of states) [6], the spinrelaxation length can be rather small in thepresence of a magnetic field. Therefore, a magneticfield 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 inthe þz-direction, we will inject carriers into thewire with þz-polarized spins. The ferromagneticcontact results in a magnetic field directed mostlyin the z-direction, but also with some smallfringing component along the x-direction. The z-directed component of the field does not matter,since it does not extend along the wire, which is inthe x-direction. However, any x-directed compo-nent will matter. Fortunately, the fringing fielddecays 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 magnetic field, an injected electron willarrive with its spin polarization practically intactat the other end where another ferromagneticcontact (magnetized in the þz-direction) is placed.This second contact will then transmit the carriercompletely and the spin polarized current will behigh so that the device resistance will be low.However, if there is an external x-directedmagnetic field, then the injected spin is no longeran eigenstate and therefore can flip in the channel[8] and arrive at the second contact with arbitrarypolarization. In fact, if the wire is long enough(much longer than the spin relaxation length), thenthe spin polarization of the current arriving at the
second contact would have essentially decayed tozero, so that there is equal probability of anelectron being transmitted or reflected. In this case,the device conductance can decrease by �50%compared to the case when there is no magneticfield. This of course assumes that every carrier wasinitially injected with its spin completely polarizedin the þz-direction. In other words, the spininjection efficiency is 100%. A more realisticscenario is to assume, say, a 32% injectionefficiency since it has already been demonstratedin an Fe/GaAs heterostructure [9]. For 32%injection efficiency, the conductance will decreaseby 24% in a magnetic field. We will err on the sideof caution and assume conservatively that thedevice conductance will decrease by only 10% in amagnetic field. A recent self-consistent drift diffu-sion simulation, carried out for a similar type ofdevice, has shown that the conductance decreasesby 20–30% because of spin flip scatterings [10].At this point, we mention that the idea of
changing device resistance by modulating spin flipscattering was the basis of a recently proposed‘‘spin field effect transistor’’ [11]. Unfortunately,the small conductance modulation achievable bythis technique (maximum 50%) is insufficient for a‘‘transistor’’ which is an active device requiring aconductance modulation by three orders ofmagnitude to be useful. In contrast, the deviceproposed here is a passive sensor that does nothave the stringent requirements of a transistorand, as we show below, a 10% conductancemodulation is adequate to provide excellentsensitivity.Let us now estimate how much magnetic field
can decrease the device conductance by theassumed 10%. Roughly speaking, this couldcorrespond to the situation
bZkF
X0:1; (5)
where kF is the Fermi wavevector in the channel.For a linear carrier concentration of 5� 104=cm;kF 8� 106=m: The measured value of Z inmaterials such as InAs is of the order of10�11 eV-m [7]. For most materials (with weakerRashba effect), the value of Zmay be two orders ofmagnitude smaller. Actually, this value can be
ARTICLE IN PRESS
S. Bandyopadhyay, M. Cahay / Physica E 27 (2005) 98–103 101
tuned with an external electric field. Therefore, it isreasonable to estimate that ZkF �1meV: Themagnetic field that can cause the 10% decreasein the conductance of the device is then foundfrom Eq. (5) to be �10Oe if we assume jgj ¼ 15:We now carry out a standard sensitivity analysis
following Ref. [12]. The rms noise current for asingle wire is taken to be the Johnson noise currentgiven by In ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi4kTGDf
p¼ C
ffiffiffiffiG
p; where k is the
Boltzmann constant, T is the absolute tempera-ture, G is the device conductance, Df is the noisebandwidth, and C ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi4kTDf
p: In reality, the
noise current can be suppressed in a quantumwire by two orders of magnitude because ofphonon confinement [6]. Therefore, In ¼ C
ffiffiffiffiG
pz;
where z is the noise suppression factor which isabout 0.01. There may be also other sources ofnoise, such as 1=f noise, but this too is suppressedin quantum wires. The 1=f noise can be reduced byoperating the sensor under an AC bias with highfrequency.The change in current through a wire in a
magnetic field H is the signal current I s and isexpressed as SH, where S is the sensitivity. We willassume that the current drops linearly in amagnetic field, so that S is independent of H. Inreality, the current is more likely to drop super-linearly with magnetic field so that S will be largerat smaller magnetic field. This is favorable fordetecting small magnetic fields, but since we intendto remain conservative, we will assume that S isindependent of magnetic field. Therefore, thesignal to noise ratio for a single wire is
ðS : NÞ1 ¼SH
In¼
SH
CzffiffiffiffiG
p : (6)
The noise current of N wires in parallel is CffiffiffiffiffiffiffiffiNG
pz;
whereas the signal current is NSH. Therefore, thesignal-to-noise ratio of N wires in parallel is
ffiffiffiffiffiN
p
times that of a single wire.For a carrier concentration of 5� 104=cm as we
have assumed, and for a mobility of only25 cm2=V-s (at 4.2K), a single wire of length10mm has a conductance G 2� 10�10 S. At a100mV bias, the current through a single wire istherefore 20 pA. Consequently, a 10% change inconductance produces a change of current by2 pA. Since the 10% change in conductance is
produced at 10Oe, the sensitivity S ¼ 0:2 pA=Oe:At 1 fT ð¼ 10�11 OeÞ; the signal current I s for asingle wire ¼ SH ¼ 2� 10�24 A; whereas the noisecurrent at a temperature of 4.2K is 2�10�18 A=
ffiffiffiffiffiffiffiHz
p: Therefore the signal to noise ratio
for a single wire is 10�6 : 1=ffiffiffiffiffiffiffiHz
pand that for a
parallel array of 1012 wires is 1 : 1=ffiffiffiffiffiffiffiHz
p: Conse-
quently, the optimum sensitivity is about1 fT=
ffiffiffiffiffiffiffiHz
pat 4.2K. If we repeat the calculation
for 300K (assuming that the mobility is reducedby a factor of 100 from its value at 4.2K), theoptimum sensitivity is about 80 fT=
ffiffiffiffiffiffiffiHz
p:
We now calculate the power consumption of thesensor. The power dissipated in a single wire is20 pA� 100mV ¼ 2 pW and for an array of 1012
wires, the total power dissipation is 2W at 4.2K.At room temperature, the conductance of eachwire is reduced by approximately a factor of 100because of mobility degradation, so that the powerdissipation is reduced to 20mW for the array.Therefore, the sensor we have designed is a lowpower sensor.At a bias of 100mV, the average electric field
over a 10mm long wire is 100V/cm. In the past,Monte Carlo simulation of carrier transport hasshown that even at a much higher field than this,transport remains primarily single channeled in aquantum wire because of the extremely efficientacoustic phonon mediated energy relaxation [13].Therefore, the assumption of single channeledtransport is mostly valid.It should be obvious from the foregoing analysis
that we can increase the sensitivity, withoutconcomitantly increasing power dissipation, if wedecrease the carrier concentration in the wire, butincrease the mobility or decrease the wire length tokeep the conductance the same.Finally, one needs to identify a realistic route to
fabricating an array of 1012 quantum wires. Wepropose using a porous alumina template techni-que for this purpose. The required sequence ofsteps is shown in Fig. 2. A thin foil of aluminum iselectropolished and then anodized in 15% sulfuricacid at 25V DC and at a temperature of 0 �C toproduce a porous alumina film containing anordered array of pores with diameter �25 nm; seefor example Ref. [14] and references therein. Asemiconductor is then selectively electrodeposited
ARTICLE IN PRESS
Fig. 2. Fabrication steps. (a) Creation of a porous alumina film with 25 nm diameter pores. This is produced by anodizing an
aluminum foil at 25V 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 ferromagnet, 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.
S. Bandyopadhyay, M. Cahay / Physica E 27 (2005) 98–103102
within the pores [15,16] to create an array ofvertically standing nanowires of 25 nm diameter.The density of wires is typically 1011=cm2; so that1012 wires require an area of 10 cm2: Next aferromagnetic metal is electrodeposited within thepores [17] on top of the semiconductor. The poresare slightly overfilled so that the ferromagneticmetal makes a two-dimensional layer on top. Thisis then covered with an organic layer to providemechanical stability during the later steps.The aluminum foil is then dissolved in HgCl2;
and the alumina barrier layer at the bottom of thepores is etched in phosphoric acid to open up thepores from the bottom. Another ferromagnet isthen electrodeposited on the exposed tips of thesemiconductor wires through a mask. The organiclayer is then dissolved in acetone and the multi-layered film is harvested and placed on top of a
conducting substrate covered with silver paste.Two wires are attached to two remote diffusedSchottky contacts in order to apply a transverseelectric field that induces the Rashba effect. Finallywires are attached to top and bottom to provideohmic electrical contacts. This completes thefabrication of the sensor.In conclusion, we have proposed a highly
sensitive solid-state magnetic field detector basedon spin–orbit interaction. Such sensors can oper-ate at room temperature since the energy separa-tion between subbands can exceed the roomtemperature thermal energy in 25-nm diameterwires produced by this method. The sensitivity ofthese sensors could be comparable to those ofsuperconducting quantum interference devices[18–20] and therefore may be appropriate formagneto-encephalography where one directly
ARTICLE IN PRESS
S. Bandyopadhyay, M. Cahay / Physica E 27 (2005) 98–103 103
senses the very small magnetic fields produced byneural activity in human brains.
The work of S.B. is supported by the Air ForceOffice of Scientific Research under Grant FA9550-04-1-0261.
References
[1] G. Dresselhaus, Phys. Rev. 100 (1955) 580.
[2] E.I. Rashba, Sov. Phys. Semicond. 2 (1960) 1109;
Y.A. Bychkov, E.I. Rashba, J. Phys. C 17 (1984) 6039.
[3] S. Bandyopadhyay, S. Pramanik, M. Cahay, Superlat.
Microstruct. 35 (2004) 67.
[4] R.J. Elliott, Phys. Rev. 96 (1954) 266.
[5] S. Pramanik, S. Bandyopadhyay, M. Cahay, www.arXi-
v.org/cond-mat/0403021 (also to appear in IEEE Trans.
Nanotech.).
[6] A. Svizhenko, A. Balandin, S. Bandyopadhyay, M.A.
Stroscio, Phys. Rev. B 57 (1998) 4687.
[7] J. Nitta, T. Akazaki, H. Takayanagi, T. Enoki, Phys. Rev.
Lett. 78 (1997) 1335.
[8] M. Cahay, S. Bandyopadhyay, Phys. Rev. B 69 (2004)
045303.
[9] A.T. Hanbicki, et al., Appl. Phys. Lett. 80 (2002) 1240.
This experiment pertains to spin injection in bulk
semiconductors. Spin injection in quantum wires may be
even more efficient. In fact, our recent theoretical work
shows that nearly 100% spin injection efficiency is possible
at certain injection energies, even if the ferromagnetic
contact has much less than 100% spin polarization (M.
Cahay and S. Bandyopadhyay, unpublished).
[10] E. Shafir, M. Shen, S. Saikin, www.arXiv.org/cond-mat/
0407416.
[11] J. Schliemann, J.C. Egues, D. Loss, Phys. Rev. Lett. 90
(2003) 14801;
X. Cartoixa, D.Z.-Y. Ting, Y.-C. Chang, Appl. Phys. Lett.
83 (2003) 1462.
[12] M. Tondra, J.M. Daughton, D. Wang, R.S. Beech, A.
Fink, J.A. Taylor, J. Appl. Phys. 83 (1998) 6688.
[13] A. Svizhenko, S. Bandyopadhyay, M.A. Stroscio, J. Phys.:
Condens. Matter 11 (1999) 3697.
[14] K. Nielsch, J. Choi, K. Schwirn, R.B. Wehrspohn, U.
Gosele, Nano Lett. 2 (2002) 677.
[15] N. Kouklin, L. Menon, A.Z. Wong, D.W. Thompson, J.A.
Woollam, P.F. Williams, S. Bandyopadhyay, Appl. Phys.
Lett. 79 (2001) 4423.
[16] S. Bandyopadhyay, L. Menon, N. Kouklin, P.F. Williams,
N.J. Ianno, Smart Mater. Struct. 11 (2002) 761.
[17] L. Menon, M. Zheng, H. Zeng, S. Bandyopadhyay, D.J.
Sellmyer, J. Elec. Mater. 29 (2000) 510.
[18] J. Gallop, Supercond. Sci. Technol. 16 (2003) 1575.
[19] H.J. Barthelmess, et al., IEEE Trans. Appl. Supercond. 11
(2001) 657.
[20] M. Pannetier, C. Fermon, G. Le Goff, J. Simola, E. Kerr,
Science 304 (2004) 1648.