Dangling-bond spin relaxation and magnetic 1/f noise from the amorphous-semiconductor/oxide interface: Theory

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by  Rogerio de Sousa

Dangling-bond spin relaxation and magnetic 1/f noise from the
amorphous-semiconductor/oxide interface: Theory
Rogerio de Sousa∗
Department of Physics, University of California, Berkeley, California 94720, USA
(Dated: February 1, 2008)
Weproposeamodelformagneticnoisebasedonspin-flips(notelectron-trapping)ofparamagnetic
dangling-bonds at the amorphous-semiconductor/oxide interface. A wide distribution of spin-flip
8 timesisderivedfromthesingle-phononcross-relaxation mechanismforadangling-bondinteracting
0 with the tunneling two-level systems of the amorphous interface. The temperature and frequency
0 dependence is sensitive to three energy scales: The dangling-bond spin Zeeman energy (δ), as well
2 astheminimum(Emin)andmaximum(Emax)valuesfortheenergysplittingsofthetunnelingtwo-
n level systems. At the highest temperatures, kBT ≫ Max(δ,Emax), the noise spectral density is
independent of temperature and has a 1/f frequency dependence. At intermediate temperatures,
a
J kanBdTp≪osseδssaensdaE1m/finp≪spekcBtrTal≪denEsimtya,x,witthhepno=ise1.2is p1ro.5p.orAtitonthael tloowaesptotwemerplearwatuinrest,emkBpTeraturδe,
9 − ≪
1 or kBT ≪ Emin, the magnetic noise is exponentially suppressed. We compare and fit our model
parameterstoarecentexperimentprobingspincoherenceofantimonydonorsimplantedinnuclear-

spin-free silicon [T. Schenkel et al., Appl. Phys. Lett. 88, 112101 (2006)], and conclude that a
]
l dangling-bond area density of the order of 1014 cm−2 is consistent with thedata. This enables the
l
a prediction of single spin qubit coherence times as a function of the distance from the interface and
h the dangling-bond area density in a real device structure. We apply our theory to calculations of
- magneticfluxnoise affectingSQUIDdevices duetotheirSi/SiO2 substrate. Ourexplicit estimates
s offluxnoiseinSQUIDsleadtoanoisespectraldensityoftheorderof10−12Φ2(Hz)−1 atf =1Hz.
e 0
m This value might explain the origin of flux noise in some SQUID devices. Finally, we consider the
suppression of these effects using surface passivation with hydrogen, and the residual nuclear-spin
.
t noise resulting from a perfect silicon-hydride surface.
a
m
PACSnumbers: 05.40.Ca;61.43.-j;76.30.-v;85.25.Dq.
-
d
n I. INTRODUCTION rentnoiseforinterfaceconductionelectrons.10 Neverthe-
o
less at low temperatures the area density for trapping-
c
center DBs is only a tiny fraction of the area density for
[ Ourphysicalunderstandingofspinrelaxationinsemi-
paramagnetic DBs. For example, at T = 5 K this frac-
conductors plays a crucial role in the current develop-
3 ment of spin-based electronics1 and spin-based quantum tion is only kBT/U 10−3 (Fig. 1). As a consequence,
v ∼
8 computation.2 Onequestionthatreceivedlittle orno at- the magnetic noise due to paramagnetic DBs is at least
a factor of U/k T 1 larger than magnetic noise gen-
8 tention so far is related to magnetic noise in semicon- B ≫
erated by electron trapping, provided the paramagnetic
0 ductor devices and nanostructures. Magnetic noise from
4 impurities and other defects at the interface may be the DBs have a non-zero spin-flip rate (Magnetic noise due
. to electron trapping is discussed in appendix A).
5 dominant source of spin phase relaxation (decoherence)
0 for implanted donor electrons3 or nuclear spins4 in iso- Thespinrelaxationratefordangling-bondtypedefects
7 topicallypurifiedsilicon. Moreover,becauseSi/SiO and depends crucially on the non-crystalline nature of amor-
2
0 other amorphous oxide interfaces are used as the sub- phous compounds.11–13 However, a detailed theoretical
v: strate for sensitive SQUID magnetometers,5–7 the spin study of the magnetic field and temperature dependence
i relaxationof magnetic impurities at the substrate might ofthiseffecthasnotbeendone. Inthisarticlewepresent
X
explaintheobservedmagneticfluxnoiseinthesedevices. a generaltheory of dangling-bond spin-lattice relaxation
ar One universal characteristic of silicon devices is the inamorphousmaterials,andshowthatthe noisecreated
presenceofaninsulatinginterface,usuallyanoxide,sep- bythe magneticdipolarfieldofanensembleofdangling-
arating the metallic gate from the semiconductor. It is bondshasthe1/f frequencydependenceathightemper-
known for a long time that these interfaces are rich in atures. We fitourtheoryto arecentexperimentprobing
dangling-bond type defects (also denoted “P centers”) spincoherenceofantimony donorsimplantedin nuclear-
b
which can be detected using spin resonance techniques. spin-free silicon3 in order to estimate our model param-
These studies have established a wide distribution of eters.
dangling-bond (DB) energy levels, spanning almost the We exploit the important relationship between phase
whole semiconductor energy gap, with each DB charac- coherenceofalocalized“probe”spin(e.g. theimplanted
terizedbyalargeon-siteCoulombenergyU 0.5eV.8,9 Sb spins in Ref. 3) and its environmentalmagnetic noise
∼
When the dangling-bond (DB) energy level falls within (Fig.2). Thecoherencedecayenvelopeofa“probe”spin
k T of the interface Fermi level, it acts as a trapping- measured by a class of pulse spin resonance sequences
B
center and leads to the well known 1/f charge and cur- is directly related to a frequency integral over magnetic
2
Energy
SiO Si Sample Interface Peak depth[nm] T1 [ms] T2 [ms]
2
120 KeV Si/SiO2 50 15 2 0.30 0.03
± ±
120 KeV Si-H 50 16 2 0.75 0.04
± ±
400 KeV Si/SiO2 150 16 1 1.5 0.1
Empty ± ±
400 KeV Si-H 150 14 1 2.1 0.1
± ±
Trapping center e – k T
F B TABLEI:Spinrelaxation data3 takenat 5.2 K for antimony
Paramagnetic donorelectronspinsimplantedinisotopicallypurifiedsilicon.
e - U T1wasmeasuredusinginversionrecoveryESR,whileT2isthe
F 1/edecayofHahnecho. Foreachsample,datawastakenfor
Doubly occup. the untreated oxidized surface (SiO2) and for the passivated
surface, treated with hydrofluoric acid in order to obtain a
hydrogenterminatedsurface. Thedataclearly indicatesthat
(1)donorsclosetothesurfacehavelowerspincoherencetimes
T2butthesamespin-fliptimeT1;(2)Surfacepassivationleads
to a sizable increment in T2, but nochange in T1.
z (position)
genericallydenoted“P centers”withchemicalstructure
FIG.1: (Color online) Band diagram for aSi/SiO2 interface. representedby Si Sbi.8,9 There is yet no experimental
Dangling bondswith energy much larger than ǫF are empty; or theoretical stu3di≡es of· spin relaxation times (TDB) for
DBs with energy in the interval (ǫF kBT,ǫF +kBT) are 1
− DBs at the Si/SiO interface. Nevertheless a systematic
trapping-centers for interface conduction electrons, responsi- 2
bleforcharge,current,andmagneticnoise. DBswithenergy study of DB spin relaxation in bulk amorphous silicon
intheinterval(ǫF U,ǫF kBT)aresinglyoccupied(param- wascarriedoutinthe1980’s.11,13 ThemeasuredDBspin
− −
agnetic), and hencecontributeexclusively to magnetic noise. relaxation rate was found to increase as a power law on
dDoBnsowtictohnetrniebrugtyelteossanthyakninǫdFo−f nUoiasree. doubly occupied and tnem=p2erat4udree,pe1n/dTe1DnBt o∝nTthnewsiatmhpalne panreopmaaralotuiosnemxpeotnheondt.
At T =−5 K and B = 0.3 T the typical TDB was in the
1
range 0.1 1 ms.13
noisetimesafilterfunction.14 Thisallowsustointerpret At first−it seems puzzling that the dangling-bond spin
pulse spin resonance experiments of localized spins as wouldrelax in sucha shorttime scale at the lowesttem-
sensitive detectors of magnetic noise in nanostructures. peratures. The typical T of localized electron spins
1
The spin qubit phase coherence is a local probe of low in crystalline silicon (e.g. phosphorus donor impuri-
frequency magnetic noise. The same ideas apply equally ties) is almost a thousand seconds in the same regime.17
well to experiments probing the coherent dynamics of This happens due to the weak spin-orbit coupling in
superconducting devices.15,16 bulkcrystallinesilicon. However,dangling-bondsinnon-
An important step towards this characterization was crystalline silicon are coupled to unstable structural de-
givenrecently,bythereportofthethefirstmeasurements fects, and this fact seems to explain their short T .11,13
1
of spin echo decay in siliconimplanted with an ultra-low These structural defects behave as tunneling two level
dose of antimony donors ( 1011 cm−2).3 Two samples systemsstronglycoupledtolatticevibrations(phonons).
∼
werereported,120KeVand400KeV,withlowandhigh Each time a tunneling two level system (TTLS) under-
implantenergyrespectively. Theformerleadstoadonor goes a phonon-induced transition, the DB spin feels a
distribution closer to the interface, see Table I. sudden shift in its local spin-orbit interaction, which
Table I provides experimental evidence that the sur- may be quite large because the TTLS is associated with
faceleadstoadditionalmechanismsfordonorspinphase a local reordering of the atomic positions of the non-
fluctuation andmagnetic noise. These mechanisms seem crystallinematerial. As aconsequence,the DB spinmay
to contribute exclusively to the phase coherence time flip each time the TTLS switches. Remarkably, this
(T ) but not to the spin-flip time (T ) of the Sb donors, cross-relaxation process remains effective even at zero
2 1
therefore the associated noise spectrum should be low magneticfieldbecauseitdoesnotinvolveaKramerscon-
frequency in nature (with a high frequency cut-off much jugate pair (in contrast to spin-flips without a simulta-
smaller than the spin resonance frequency). neous TTLS switch).
Here we consider the mechanisms of magnetic noise We developthis theory further in orderto incorporate
that might be playing a role in these experiments. For theexponentiallywideTTLSparameterdistributiontyp-
a Si/SiO interface we show that dangling-bond spin- icalofamorphousmaterials. Asaresult,wefindthatthe
2
flips play a dominant role. A dangling-bond (DB) is a magnetization of an initially polarized ensemble of DB
paramagnetic defect usually associated with an oxygen spins will undergo non-exponential relaxation in time.
vacancyintheSi/SiO interface. Thesepointdefectsare Our theory of dangling-bond spin-lattice relaxation and
2
3
magnetic noise is based on an effective Hamiltonian ap- Sb, γ 1.76 107 (sG)−1 is close to the free electron
e
proach,allowingustodrawgenericconclusionsaboutthe value].≈Note t×hat Eq. (1) was divided by ~ so that en-
frequency, temperature, and magnetic field dependence ergy is measured in units of frequency. Each component
of spin-noise in a variety of amorphous materials. For ofthe vectorηˆ=(ηˆ ,ηˆ ,ηˆ )isanoperatormodeling the
x y z
example, our results apply equally well to the magnetic magnetic environment (the DB or other impurity spins)
noise produced by E′ centers in bulk SiO , another well surroundingthedonorspin. Thesimplestwaytodescribe
2
studied dangling-bond. Other materials of relevance to the time evolution of the spin’s magnetization σ is the
h i
our work are the bulk Al O (sapphire), and Al/Al O Bloch-Wangsness-Redfield approach, which assumes σ
2 3 2 3
h i
and Si/Si N interfaces, whose paramagnetic dangling- satisfies a first order differential equation in time. The
3 4
bonds/magnetic impurities are yet to be characterized decay rate for σ is then given by
z
h i
experimentally.
1 π
Our results are of particular importance to magnetic = S˜ (+γ B)+S˜ ( γ B) , (2)
q e q e
flux noise in SQUID devices, whose microscopic originis T1 2 −
a longstanding puzzle (for a review see section IV-G of qX=x,yh i
Ref. 18). In section VII we apply our results to calcula- with the environmental noise spectrum defined by
tionsoffluxnoiseduetoDBswithintheareaenclosedby
the SQUID loop, and show that this contribution might S˜ (ω)= 1 ∞ eiωt ηˆ (t)ηˆ (0) dt. (3)
explain some of the available flux noise measurements. q 2π h q q i
Z−∞
It is possible to considerably reduce the dangling-
bond area density using a surface passivation technique. Note that the energy relaxation time T1 for the donor
For example, the application of hydrofluoric acid to the spinisdeterminedbythenoiseatω = γeB,thatisjust
±
Si/SiO surface removes dangling-bonds by covering the a statement of energy conservation. Within the Bloch-
2
surface with a monolayer of hydrogen atoms. Recently, Wangsness-Redfield theory the spin’s transverse magne-
Kaneandcollaboratorsfabricatedafield-effect-transistor tization( σ+ = σx+iσy /2)decaysexponentiallywith
h i h i
using a passivated Si(111)H surface, and demonstrated the rate
record high electron mobility.19 Nevertheless, the large
1 1
density of hydrogen nuclear spins might be an impor- = +πS˜ (0), (4)
T∗ 2T z
tant source of magnetic noise. The nuclear spins are 2 1
constantly fluctuating due to their mutual dipolar cou-
where we added a to emphasize this rate refers to
pling. In section VIII we consider calculations of mag- ∗
a free induction decay (FID) experiment. The Bloch-
netic noise due to a hydrogen terminated Si(100)H sur-
Wangsness-Redfieldapproachleadstoasimple exponen-
face. We use the same theory previously developed for
tial time dependence for all spin observables. Actually
Hahnechodecayofaphosphorusimpurityinbulkdoped
this is not true in many cases of interest, including the
natural silicon.14,20 We show that the Hahn echo decay
caseofagroupVdonorinbulksiliconwherethisapprox-
in a Si(100)H surface has many peculiarities, including a
imationfailscompletely(forSi:PtheobservedHahnecho
special crystal orientation dependence for the donor T2 decayfitswelltoe−τ2.3 inmanyregimes).20,34 Theprob-
times that may be used as the fingerprint for detecting
lem lies in the fact that the Bloch-Wangsness-Redfield
this source of noise experimentally.
theory is based on an infinite time limit approximation,
thataveragesoutfinitefrequencyfluctuations. Notethat
T∗ differs from T only via static noise, S˜ (0) in Eq. (4).
II. RELATIONSHIP BETWEEN MAGNETIC 2 1 z
A large number of spin resonance sequences, most no-
NOISE AND PHASE RELAXATION IN PULSE
tablytheHahnechoareabletoremovestaticnoisecom-
SPIN RESONANCE EXPERIMENTS:
pletely.
ELECTRON SPIN AS A LOCAL PROBE OF
MAGNETIC NOISE We may develop a theory for spin decoherence that
takes into account low frequency fluctuations in the
semiclassical regime ~ω k T, when S˜ ( ω) =
terCaoctnisoindeorftahelofcoallliozwedinsgpimnowdietlhHaanmoiilstyoneinavnirfoonrmtheenti,n- e−~ω/kBTS˜z(ω) S˜z(ω). ≪The sBpin coherenceze−nvelope
≈
maybecalculatedinthepuredephasinglimit(ηˆ =ηˆ =
x y
1 0), with the assumption that ηˆz ηz is distributed ac-
= γ Bσ +ηˆ(t) σ. (1) →
H 2 e z · cording to Gaussian statistics. For derivations and dis-
cussions on the applicability of this theory, we refer to
Here σ = (σx,σy,σz) is the vector of Pauli matrices de- Ref. 14. A similar method in the context of supercon-
noting the state of the electron spin being probed by a ducting qubits was proposed in Ref. 16. The final re-
pulse spin resonance experiment (henceforth called the sultisadirectrelationshipbetweenphasecoherenceand
donor spin - e.g. the Sb spins in Ref. 3), γeB is the spin magnetic noise according to
Zeeman frequency in an applied external magnetic field
∞
B, and γe = ge/(2mec) is a gyromagnetic ratio for the σ (t) =exp dω S˜ (ω) (t,ω) , (5)
electronspin [for a groupV donor impurity such as P or |h + i| − z F
(cid:20) Z−∞ (cid:21)
4
Sb one and a hundred milliseconds at the lowest tempera-
tures (T =0.3 4 K).13 The proposedtheoreticalexpla-
DBs nation was tha−t DB spin relaxation happens due to its
SiO coupling to phonon-induced transitions of tunneling two
2 level systems (TTLS) in the amorphous material.11 The
d Si TTLSs are thought to be structural rearrangements be-
tweengroupsofatoms,thatcanbe modeledby a double
well potential [see Fig. 3(a)]. The TTLS assumption is
Sb able to explain several special properties of amorphous
materials at low temperatures.21 The DB spin couples
to the TTLSs either through spin-orbit or hyperfine in-
teraction, both of which are modulated by the TTLS
FIG.2: (Coloronline)Howtodetectlowfrequencymagnetic transition. Note that the presence of a TTLS breaks the
noise using electron spin resonance. A low density of anti- crystal inversion symmetry.
mony (Sb) donor impurities is implanted in a Si/SiO2 sam- We start by developing the theory of phonon-induced
ple using an ion gun, and the distribution of Sb donors is transitions for the TTLS,22 and the associated cross-
determined using secondary ion mass spectroscopy. Next, a
relaxationof the DB spin. The Hamiltonian for a TTLS
Hahn echo decay experiment is performed on the Sb spins.3
reads
TheHahnechodecayenvelopeisdirectlyrelatedtomagnetic
noise produced by e.g. dangling bonds at the interface, see
1 ǫ ∆
Eq. (5). ′ = . (8)
HTTLS 2 ∆ ǫ!
−
The energy scale ǫ is a double well asymmetry, while
with (t,ω)a filter function that depends onthe partic-
F ∆=∆ e−λ is the tunneling matrix element between the
ular pulse spin resonance sequence. For a free induction 0
states [λ is related to the barrier height and its thick-
decay experiment (π/2 t measure) we have
− − ness, see Fig. 3(a)]. After diagonalizing Eq. (8) we ob-
1sin2(ωt/2) tain TTLS = diag E/2, E/2 , with E = √ǫ2+∆2
FFID(t,ω)= 2 (ω/2)2 , (6) (for nHotational clarit{y we p−rime t}he Hamiltonians in the
non-diagonalbasis). Thecouplingtophononscanbeob-
tainedbyexpandingthe parameterǫ tofirstorderinthe
while for the Hahn echo (π/2 τ π τ measure) the
− − − − phonon strain operator,
filter function becomes
1sin4(ωτ/2) ~
FHahn(2τ,ω)= 2 (ω/4)2 . (7) uˆ=i q s2ρVωq|q| aqeiq·r+a†qe−iq·r , (9)
X (cid:0) (cid:1)
Note that in the limit t Eq. (6) becomes πδ(ω)t, leading to ǫ ǫ +ǫ′uˆ. Below we average over TTLS
→ ∞ →
recovering the Bloch-Wangsness-Redfield result Eq. (4). parameters with ǫ ∆, so to be consistent we must
The Hahn echo filter function satisfies (2τ,0)= 0, assumethedeformat≫ionparameter∆′ =0. Applyingthis
Hahn
F
showing that it filters out terms proportional to S˜ (0) expansion to Eq. (8) and transforming to the diagonal
z
in spin evolution. This is equivalent to the well known basis we get
removal of inhomogeneous broadening by the spin echo.
Any pulse spin resonance sequence containing instanta- ǫ′uˆ ǫ ∆
TTLS−ph = − . (10)
neous π/2 or π-pulses can be described by Eq. (5). An- H 2E ∆ ǫ !
− −
other important example is the class of Carr-Purcell se-
quences used for coherence control (π/2 [τ π τ Using Fermi’s golden rule for dissipation into a phonon
echo]repeat). − − − − bath Hph = q~ωqa†qaq, we find that the transitions
from +E/2 to E/2 and vice-versa are given by
P−
r = aE∆2[n (E)+1], (11)
III. DANGLING-BOND SPIN RELAXATION: + ph
DIRECT VS. CROSS-RELAXATION r = aE∆2n (E), (12)
− ph
with phonon occupation number
The presence of an inversion center in crystalline Si
leadstoweakspin-orbitcouplingandextremelylongspin 1
relaxation times. The T1 for localized donor electrons in nph(E)= eE/kBT 1. (13)
crystalline silicon can reach thousands of seconds at low −
temperatures.17 This is in contrast to spin-lattice relax- In Eqs. (11), (12) the parameter a depends on the ma-
ation of dangling-bonds in various forms of amorphous terial density ρ,sound velocity s, and deformation po-
silicon where instead TDB was found to range between tential ǫ′ [a = (8π ǫ′ 2~4ρs5)−1]. The DB spin Zeeman
1 | |
5
tehneerngyotaistiodnenwoteeddebfiyneHδDB =~γ~BγeaBsStzDhBe.DBTospsiinmpZleifey- (a) D = D 0e- l (b) + +
e
≡
man energy. The coupling of the DB spin to the TTLS
may be derived directly from the spin-orbit interaction E -
= αSDB (E p), where SDB is the DB spin op- - d
so
H · ×
erator, p is the DB orbital momentum, and E a local E
electric field. After averagingover the coordinate states, e
E ≫d
theresultingeffectiveHamiltonianbecomesdirectlypro-
portional to the magnetic field, a consequence of time
reversal symmetry.23 For simplicity, we assume that E (c) - + (d)
is perpendicular to the interface,24 and that the spin- x
(cid:3)j
orbit energy fluctuates by a certain amount A δ when d y r DB
the TTLS switches. This leads to the following×effective + i
Hamiltonian in the non-diagonal basis E - y (cid:3)
i B
′ = Aδ SDB+SDB 1 0 , (14) E ≪d donor at z=d
HTTLS−DB 2 + − 0 1! z
−
(cid:0) (cid:1)
FIG. 3: (a) Effective double well potential for the tunneling
whereSDB areraisingandloweringoperatorsfortheDB
± two level system (TTLS). (b,c) Energy level structure for a
spin. The dimensionless constant A will play the role of
dangling-bond spin (DB) coupled to a TTLS, for (b) E δ
a small parameter in our theory. Transforming to the ≫
and(c)E δ. (d)Coordinatesystemfortheinteractionofa
diagonal basis we get dangling-b≪ond located at ri with the donor spin. ψi denotes
the angle formed by the donor-DB vector (dashed) and the
Aδ +ǫ ∆ external B field.
= SDB+SDB . (15)
HTTLS−DB 2E + − ∆ ǫ !
−
(cid:0) (cid:1)
cancellation”,23givingasimpleexplanationofwhydirect
As a result of Eq. (15), the DB-TTLS eigenstates are
spin-flip rates are generally weak]. Moreover, Eq. (17)
admixturesbetweenspinupanddown. Wemaystilllabel
vanishes at B =0 in accordancewith time reversalsym-
the eigenstates by their spin quantum number, provided
metry (the direct process couples a Kramers pair).
we think of ( ) as having a large projection onto the
↑ ↓ The “cross”-relaxation rates, whereby the DB spin
pure spin up (down) state. The four level structure is
flips simultaneously with a TTLS switch are given by
shown in Fig. 3(b) in the limit E δ and in Fig. 3(c)
≫
for E δ. Γ = a M 2(E+δ)n (E+δ), (18)
The≪total Hamiltonian is given by −↓ | +| ph
Γ = a M 2(E+δ)[n (E+δ)+1], (19)
+↑ + ph
| |
H=HTTLS+HDB+HTTLS−DB+Hph+HTTLS−ph. (16) Γ−↑ = a|M−|2(E−δ)nph(E−δ), (20)
Note that the first three contributions denote the dis- Γ+↓ = a M− 2(E δ)[nph(E δ)+1], (21)
| | − −
crete TTLS-DB states (a four-level system), the fourth
where the sub-indexes label the level that the system is
is the energy bath (a continuum of phonon states) and
exiting, for example Γ Γ . Note that the final
the fifth is the coupling between the TTLS-DB to the +↑ ≡ +↑→−↓
state is obtained from the initial state by changing the
phononbath. The eigenstatesofthe firstthreecontribu-
sign of the TTLS and flipping the DB spin. The matrix
tions may be calculated using perturbation theory, and
element M is defined by
the transitionratesarestraightforwardtocompute. The ±
“direct” relaxationrate, corresponding to a DB spin-flip
Aǫ∆
with the TTLS state unchanged is given by M± = E2 [|E±δ|+δ]. (22)
a ∆4A2 Remarkably,thiscross-relaxationprocessisnotatran-
D = δ5[n (δ)+1], (17)
±↑→±↓ 4E2(E2 δ2)2 ph sition between Kramers conjugate states. Asaresult,the
− rates are qualitatively different from the direct process,
with [n (δ)+1] n (δ) for the reverserate D . particularlyduetotheirmagneticfield(δ)andTTLSen-
ph ph ±↓→±↑
Note that Eq. (→17) is proportional to ∆4, reflecting ergy (E) dependence. At low temperatures (k T δ),
B
≪
the fact that a direct spin-flip may only occur together the direct rate always scales as D δ5.17,23 In con-
with a virtual transition to an excited orbital state.17,23 trast, the cross-relaxation rate has tw∝o distinct behav-
In our case this virtual transition is a “double-switch” iors, depending whether E δ, or E δ. For E δ,
of the TTLS, hence D ∆4 [terms independent of M Aǫ∆/E, and the Γ’s≫are indepen≪dent of mag≫netic
±
∆ in Eq. (17) cancel exa∝ctly. This general feature of field.≈For E δ we get instead M 2δAǫ∆/E2, and
±
a direct spin-flip process is referred to as “van Vleck Γ δ3 in con≪trast to the δ5 scaling of≈the direct rate.
∝
6
Of extreme importance to our theory is to note that h ( 1, 1,+1,+1). In Appendix B we prove the con-
dip
− −
whenever the energy scales E and δ are well separated, venient identity
the direct rates are much smaller than the cross relax-
ation rates. For E δ we have D/Γ (∆/ǫ)2(δ/E)5, Sz(t)= [ηz(t) η¯z][ηz(0) η¯z] =x p(t) xw. (26)
≫ ∼ h − − i · ·
whileforE δwehaveD/Γ (E/ǫ)2(∆/δ)2. Thetypi-
≪ ∼ Here x = (x w ,x w ,...), with w the equilibrium
calassumptionforamorphoussemiconductorsis∆ ǫ,δ w 1 1 2 2 i
and E ǫ.10 In this regime the direct rates are sub≪stan- probabilitiesforthei-thleveloftheDB+TTLSnetwork.
≈ The matrix p(t) = e−Λt describes the occupation prob-
tially weaker than the cross relaxation rates, except at
ability for each level, and decays according to the relax-
the resonance point E = δ. It is useful to list simple
ation tensor Λ. Below we discuss the important analytic
expressionsforthecross-relaxationratesinthe twomost
solutions for S (t) in the limit of small spin-orbit cou-
physically relevant regimes considered in this work. For z
pling, A 1.
low magnetic field δ kBT, E δ but with E/kBT ≪
≪ ≫
arbitrary we have simply
Γ =Γ Γ A2r . (23) A. Case E δ, δ kBT, E/kBT arbitrary
±↑ ±↓ ± ± ≫ ≪
≡ ≈
Hence when the spin-orbit coupling parameter satisfies InthisregimetheTTLSandcross-relaxationratesare
A 1, the cross-relaxation spin-flips are much less fre- simplyrelatedbyEq.(23). Thetimecorrelationfunction
≪
quent than the spin-preserving TTLS switching events. fortheDBspinmaybecalculatedexactlyfromEq.(26),
TheoppositehighmagneticfieldregimewithE δ and but for simplicity we show the result to lowest order in
≪
E k T with δ/k T arbitrary leads to powers of A:
B B
≪
δ3A2∆2 S (t) h2 Ψe−(r++r−−Γ¯)t+(1 Ψ)e−Γ¯t , (27)
Γ+↑ ≈Γ−↑ ≡Γ↑ ≈4a E2 [nph(δ)+1], (24) z ≈ dip −
h i
with a visibility loss given by
with the reverse rate Γ given by [n (δ)+1] n (δ).
↓ ph ph
→
Note that these Γ↑↓ rates are still much larger than the tanh2(E/2k T)
direct rates, since D/Γ (∆/δ)2. Ψ= B A4, (28)
↑↓ ∼ cosh2(E/2k T)
Finally, we discuss how the cross-relaxation rates are B
affected by the presence of phonon broadening in a non-
and a thermalized DB spin relaxation rate given by
crystallinematerial. Inthiscasewegeneralizeourtheory
byincludingacomplexparttothephononspectra,ω = 2r 2r
q Γ¯ = − Γ + + Γ
sq+iγph. The modified Eq. (22) becomes r++r− + r++r− −
E
Aǫ∆ δ2+γ2 /2 2aA2∆2 , (29)
M± = E2 " δ2+γphp2h q(E±δ)2+γp2h ≈ sinh(E/kBT)
where we used ǫ E. Interestingly, Eq. (27) shows that
δ (E δ)2+γ2 /2 DB spin relaxati≈on happens in two stages: In the first
+ ± ph . (25)
|E±δ| (E±δ)2+γp2h slotassgeΨt,hewiDthBaspriantedesceatysbyabtrhueptTlyTLtoSaswsmitcahll.viDsiubriliintyg
q  this first stage the TTLS levels E/2 achieve thermal
For amorphous Si we estimate γph 0.01sq.25 For E equilibrium. In the second stag±e the DB spin relaxes
∼ ≫
δ, γph 0.01E may be comparableto δ, and we see that fully with a much slower “thermalized” cross-relaxation
∼
M± is reduced by a factor of two, and an additional B rateΓ¯. ForA 1wemaydroptheΨ A4 contribution
field dependence results. to Eq. (27). ≪ ∝
The theory developed above can be generalized to a
single DB coupled to an ensemble of TTLSs, provided
IV. DANGLING-BOND SPIN RELAXATION: the TTLSs are not coupled to each other. In this case
ENSEMBLE AVERAGE
the rate Eqs. (29) and (37) are generalized to a sum of
rates Γ relating to the i-th TTLS. Each exponential in
i
In order to evaluate the ensemble averagesoverTTLS Eq. (27) becomes e−PiΓit. This happens whenever
parameters we must first determine the time-dependent the DB+TTLS ne∼twork can be separated into discon-
correlationfunction for the four-levelrelaxationnetwork nected four level subspaces as in Fig. 3(b,c).
described in Fig. 3(b,c). Using the notation of Eq. (3), We now proceed to average over disorder realizations
the magnetic dipolar field produced by a single DB spin of the amorphous material. We assume the following
maps into a c-number ηˆz = 2hdipSzDB → hdipsi, with two-parameter distribution
s = +1 (DB spin up) or s = 1 (DB spin down).
i i
In the four-level system notation −(+ , ,+ , ) P¯v E α
↓ − ↓ ↑ − ↑ P(λ,E)= , (30)
the vector x of dipolar fields assumes the values x = λ E
max (cid:18) max(cid:19)
7
for λ [0,λ ], and E [E ,E ]; P(λ,E) = 0 showing that the DB spin relaxation rate will scale pro-
max min max
otherw∈ise. Note that the u∈niform distribution in λ leads portional to T2+α.
to a broad distribution of TTLS tunneling parameters Attheverylowesttemperaturesk T E thereare
B min
∆ = ∆ e−λ. To our knowledge there are no estimates no thermally activated TTLSs, therefor≪e the mechanism
0
available for P¯,E ,E close to an interface, only of DB cross-relaxationis exponentially suppressed. Here
max min
for bulk SiO . For the latter material the energy den- other sources of DB spin relaxation may dominate [e.g.
2
sity of TTLSs per unit volume P¯ has been estimated direct relaxation as in Eq. (17)], or the DB spin may
as P¯ = 1020 1021eV−1 cm−3, and typical values for notrelaxatallwithinthe characteristictimescaleofthe
−
the TTLS energy range are E /k 0.1 K, and experiment.
min B
E /k 10 K.21 Here we introduce a∼new parame- Askew et al. measured average DB relaxation rates
max B
∼
ter v with units of volume, denoting the effective range in bulk amorphous silicon at low temperatures (T =
for TTLSs to couple to a each DB spin (for a SiO layer 0.3 5 K).13 Two different preparation methods, sili-
2
of 10 nm we estimate v 103 nm3). The exponent α is con−implanted with 28Si, and silicon sputtered in a sub-
material dependent: Wh∼ile α 0 seems to be appropri- strate,ledtotheexperimentalfit Γ¯ T2.35. Twoother
ate for bulk SiO ,21 it was fo≈und that bulk amorphous preparation methods, silicon imphlanit∝ed with 20Ne, and
2
Si can be described by α = 0.1 0.4 or α = 1.2 1.5 siliconevaporatedonasubstrateledtoT3.3 andT3.5 fits
− −
depending on sample preparation method (See Ref. 13 respectively. Two different values of the magnetic field
and section IVC below). were studied (0.3 and 0.5 T), and no magnetic field de-
TheaveragenumberofTTLSscoupledtoeachDBspin pendence could be detected. The T and B dependence
is given by predictedby our modelagreeswith experimentprovided
α = 0.35 for the 28Si implanted and sputtered samples,
P¯vE
= dλ dE P(λ,E) max. (31) and α = 1.3,1.5 for the 20Ne implanted and the evapo-
N ≈ α+1
rated samples. It’s perhaps expected that α is different
Z Z
This is also the number of thermally activated TTLSs for each of these because the density of TTLSs should
at high temperatures, k T E . For lower tempera- depend on the way they were created. At high temper-
B max
turessatisfyingE k T≫ E ,Eq.(31)isdivided atures, the linear in T behavior has been observed in
min B max
by cosh2(E/2k T), l≪eading t≪o amorphous silicon grown by evaporation.27
B
1+α
2k T
P¯vE B . (32)
NT ≈ max(cid:18)Emax(cid:19) B. Case E ≪δ, E ≪kBT, δ/kBT arbitrary
This is the number of thermally activated TTLSs inter-
From Eq. (24) and Eq. (26) we get
acting with each DB spin. For extremely low temper-
atures k T E this number will be exponentially
B ≪ min h2
small. S (t) dip e−(Γ↑+Γ↓)t. (36)
Wenowturntocomputationsoftheensembleaveraged z ≈ cosh2(δ/2kBT)
DB spin relaxationrate, Γ¯ . At shorter times satisfying
Γ¯ t 1, the DB spinh miagnetization S (t) decays For E δ the DB relaxation rate becomes
Max z ≪
linearly≪intime.26 Therateforthislineardhecayiisequiv-
∆2 δ
alent to the 1/TDB rate measuredfor bulk amorphous Γ +Γ 4aA2 δ3coth . (37)
1 ↑ ↓ ≈ E2 2k T
silicon samples in Ref. 13. This is given by (cid:18) B (cid:19)
(cid:10) (cid:11)
1 Its ensemble average is given by
Γ¯ = = dλ dEP(λ,ǫ)Γ¯(λ,E). (33)
TDB
(cid:28) 1 (cid:29) Z Z 2aA2 ∆2 E α
(cid:10) (cid:11) Γ +Γ 0 max
leaAdtthoightemperatureskBT ≫Emax Eqs.(33)and(29) h ↑ ↓i ≈ λmax (cid:18)EmaxEmin(cid:19)(cid:18)Emin(cid:19)
1+α δ
δ3coth , (38)
Γ¯ =aA2∆20kBTλN . (34) ×(cid:18)1−α(cid:19)N (cid:18)2kBT(cid:19)
max
(cid:10) (cid:11) where we assumed α < 1. For α 1, the pref-
TheaverageDB spinrelaxationscaleslinearlywithtem- ≥
actor in Eq. (38) is modified, but the scaling
perature times the number of TTLSs surrounding the ∝
δ3coth(δ/2k T) remains.
DB. N B
AtlowertemperaturessatisfyingE k T E
min B max
≪ ≪
we have instead
C. Comparison to Ref. 13
aA2∆2P¯v ∞ xα
Γ¯ = 0 (k T)2+α dx
λmaxEmαax B Z0 sinh2x We now compare our results to the theoretical model
(cid:10) (cid:11) 3aA2∆2k T proposed by Askew et al..13 In their Eq. (5) the authors
= λ 201+Bα NT, (35) wrote the expression for Γ¯ in the E δ regime using
max ≫
8
free parameters D, M, C, N. In our work these are ex- spiniscoupledtoonly oneTTLSonaverage[i.e., 1
plicitly related to microscopic parameters: D = ǫ′ǫ/E, or 1, see Eqs. (31), (32)] we have N ∼
T
M = ǫ′∆/E, C = ǫ/(2E), N = ∆/(2E). In Ref. 13 it N ∼
is clai−med that when the inequality ND/E CM/δ dλ dEP(λ,E)δ Γ¯(λ,E) Γ′
≫ − P(Γ′)= − . (40)
is satisfied, the average DB relaxation rate scales as
dλ dEP(λ,E)
TDB −1 T2+αδ0(thesocalledLyoandOrbachregime R R (cid:0) (cid:1)
h 1 i ∝
after Ref. 12). When this inequality is reversed, they R R
Note that this is normalized to one according to
obtained hT1DBi−1 ∝ T4+αδ−2 (Kurtz and Stapleton dΓ′P(Γ′)=1. It is straightforward to extend Eq. (40)
regime, Ref. 11). Nevertheless, our result shows that
to a largernumber of TTLSs E ,E ,..., but the explicit
1 2
these parameters are related by ND = CM > 0, cRalculation of P(Γ′) becomes difficult. Below we will de-
−
so this inequality is equivalent to δ E. Because
rive explicit results for the case of a DB spin coupled to
≫
Eqs. (7) and (8) of Ref. 13 are based on two conflicting
a single TTLS on average.
approximations, δ E for the matrix element squared
Using Eqs. (29), (30), and (40) we may evaluate the
≫
and δ E for the phonon density of states, their result
integral over λ explicitly:
≪
needs to be corrected. We showed above that the av-
erage DB relaxation scales instead as δ3coth(δ/2k T)
for δ E and T2+αδ0 for δ E (the latter hBolds P(Γ′) = 1 dEP(0,E) dλδ[λ−λ0(E)]
≫ ≪ dΓ¯
for Emin kBT Emax. For high temperatures N Z Z |dλ|λ=λ0(E)
k T E≪ we ge≪t Tδ0). The corrected results are 1 1
B ≫ max = dEP(0,E)
in excellent qualitative agreementwith the experimental 2Γ′
N Z
data in Ref. 13. 2aA2E
Ref. 13 assumes ǫ = ∆ = E/√2 and averages E ac- θ Γ′ . (41)
× sinh(E/k T) −
cording to a density Eα. This is in contrast to our (cid:20) B (cid:21)
averagingprescription∼thatassumesinstead∆=∆0e−λ, Here λ (E) is the solution of Γ¯(λ ,E) = Γ′. The step
with ∆ < ǫ and as a consequence ǫ E. We as- 0 0
0 min ≈ function results from the fact that the delta function
sume λ is uniformly distributed and the ǫ density varies
will “click” only when λ (E) [0,λ ], or simply
basut∼ioǫnαs.oTfThiTsLasSsuremlapxtaiotnionisrmatoetsivoabtseedrvbeydtihnegwlaidsseeds,isatnrid- Γ′ ≤2aA2E/sinh(E/kBT)0. ∈ max
is usually employed to explain charge and current noise
in semiconductors.10 As we show below,the broaderdis-
tribution of DB relaxation times leads to 1/f magnetic 1. High temperature, kBT Emax
≫
noise and non-exponential relaxation for an ensemble of
DBs. In this case the theta function in Eq. (41) is always
one for Γ′ [Γ¯ ,Γ¯ ], with Γ¯ = 2aA2k T and
min max max B
Γ¯min =e−2λ∈maxΓ¯max. Therefore we have simply
V. MAGNETIC NOISE
1
P(Γ′)= , (42)
The totalnoisepowerforeachDB spinis independent 2λmaxΓ′
of the specific relaxation process and may be calculated
for Γ′ [Γ¯ ,Γ¯ ], and P(Γ′) = 0 otherwise. As a
exactly using elementary Boltzman statistics. The noise min max
must satisfy the following sum rule: check, n∈ote that dΓ′P(Γ′)=1 implies the relationship
∞ h2 λmax = 21ln Γ¯Γ¯mmainxR , as expected.
S˜ (ω)dω = η2 η 2 = h dipi . (39) The magn(cid:16)etic n(cid:17)oise is given by
z h zi−h zi cosh2(δ/2k T)
Z−∞ B
Γ/π
Thisshowsthatthenoisespectrumisexponentiallysmall S˜(ω) = hh2dipi dΓP(Γ)ω2+Γ2
inthehighmagneticfieldregimeδ k T. Fortheoppo- Z
siteregimeδ ≪kBT thetotalnoise≫powBerisindependent = hh2dipi 1 , (43)
oftemperature. However,asweshowbelow,thespectral 4λ ω
max
density S˜ (ω) may be temperature dependent when its | |
z
upper frequency cut-off is temperature dependent. for Γ¯ < ω < Γ¯ , and S˜(ω) = 0 for ω > Γ¯ . For
min max max
ω < Γ¯ it saturates at S˜(Γ¯ ). Hence at the highest
min min
temperatures we have temperature independent magnetic
A. Case E δ, δ kBT, E/kBT arbitrary 1/f noise.
≫ ≪
The1/f frequencydependence showsthattheaverage
Inordertodeterminethenoisespectrum,wemustfirst magnetizationofanensembleofDB spins outofequilib-
extract the distribution of relaxation rates P(Γ¯) from rium will decay non-exponentially with time t. At inter-
Eqs. (29) and (30). Under the assumption that each DB mediatetimessatisfyingΓ¯−1 t Γ¯−1 ,wemayshow
max ≪ ≪ min
9

that the time correlation function (or equivalently the 10-1
ensemble average of the DB z-magnetization) satisfies10 1/f noise
10-2 a =0
SDB(t) C +ln(Γ¯ t) )
(cid:10)hSzDzB(0)(cid:11)i ≈1− E 2λmaxmax . (44) > (s10-3 aa ==01..355
Th(1is/Γ¯mexaxptr)e.ssHioenre CisE =va0l.i5d772afitserthenEeuglleecr-tMingaschteerromnsi 2<hdip10-4
O
constant. )/10-5
w(
~S
10-6
2. Intermediate temperature, Emin kBT Emax
≪ ≪
10-7
In this case Eq. (41) becomes 10-1 100 101 102 103 104 105
1 1+α k T 1+α w (rad/s)
P(Γ′) = B
2Γ′ λ E
max (cid:18) max(cid:19)
xmaxdxxαθ Γ¯max x Γ′ . (45) FδIaGn.d4:EMmaingnetickBnoTiseatEinmtaexrm, efodriaαte=tem0p,e0r.3a5tu,1re.5sk(BTTTL≫S
× sinhx − ≪ ≪
Z0 (cid:16) (cid:17) energy density exponents). The distribution of relaxation
The upper limit of the integral is determined from rates [Eq. (46)] contains a logarithmic correction, leading to
x = Γ′/Γ¯ . We solved this equation numerically, S˜(ω) 1/fp, with p=1.2 1.5.
sinhx max ∝ −
and showed that the result is well approximated by the
analytic expression x 3 ln Γ′ . Using this
approximation we getmax ≈ 2(cid:12) (cid:16)2Γ¯max(cid:17)(cid:12) 4. Calculation of hh2dipi
(cid:12) (cid:12)
1 k T 1+(cid:12)α 3 Γ(cid:12)′ 1+α Finally,wecalculatethetotalnoisepowerbyaveraging
P(Γ′)= B ln .
2λ Γ′ E 2 Γ¯ the DB distribution over the interface plane. We choose
max (cid:18) max(cid:19) (cid:12)(cid:12) (cid:18) max(cid:19)(cid:12)(cid:12) (46) a coordinate system with originat the interface immedi-
Thedistributionofrelaxationrates(cid:12)(cid:12)hasthesame(cid:12)(cid:12)temper- ately abovethe donorspin. Defined asthe donordepth,
ature dependence as the number of thermally activated and ri, φi the coordinates of the ith DB with respect to
TTLSs [see Eq. (32)], and possesses an interesting loga- the interface [see Fig. 3(d)]. The dipolar frequency shift
rithmic correction with respect to the usual 1/Γ′ behav- produced by a DB spin aligned along the same direction
ior. as the donor spin is given by
The logarithm correction in Eq. (46) increases the
weight for smaller rates Γ′, at the expense of decreas- (h ) = γe2~ 1 3cos2ψ . (48)
ing the weight for higher rates. As a result the noise dip i 4(d2+r2)3/2 − i
i
spectrum is better described by a 1/fp relation, with (cid:0) (cid:1)
p > 1. Fig. 4 shows numerical calculations of S˜(ω) hdip is sensitive to the orientation of the external mag-
for α = 0,0.35,1.5 (we assumed Γ¯ = 1 s−1, and netic field B = (sinθ,0,cosθ)B with respect to the in-
min
Γ¯ = 104 s−1). For α = 0, the noise is described terface. This enters through
max
by a 1/f1.2 fit, while for α = 1.5 a fit of 1/f1.5 is more
appropriate. Therefore at intermediate temperatures we cos2ψ = (dcosθ+ricosφisinθ)2. (49)
have i d2+r2
i
h2 k T 1+α 1
S˜(ω)= h dipi B . (47) For θ = 0, the average h2 over an uniform DB area
4λ′ E ω p dip
max (cid:18) max(cid:19) | | density σDB is given by
Note that λ′ is determined from the normalization
max 2π ∞
condition dωS˜(ω)=hh2dipi for given Γ¯max/Γ¯min. hh2dipi = σDB dφ rdrh2dip(r,φ)
Z0 Z0
R 3π γ4~2
3. Extremely low temperature, kBT ≪Emin = 64σDB ed4 . (50)
In this case Γ¯(λ,E) is exponentially suppressed, and
there will be no magnetic noise due to the DB+TTLS VI. HAHN ECHO DECAY DUE TO 1/f NOISE:
mechanism. Ifspinrelaxationisdominatedbythe direct COMPARISON TO EXPERIMENT
process [Eq. (17)], the noise spectra may still have the
1/f dependence. Otherwise paramagnetic DBs may not The discussion above concluded that the following
contribute to magnetic noise at all. model for the noise spectrum is valid at high temper-
10
atures (kBT ≫δ and kBT ≫Emax): 1.0 x =0.001 -1nm)15120 KeV impla nt (annealed)
C/ω 0 ω <ω 0.8 -31010 SSImMoSo tdhaetda
The prefSa˜c(ωto)r=Cis gCiv/0e|mnωi|nbyωωmmin≤ax≤|≤||ω|ω|<|<mωim∞nax . (51) hn echo00..46 x =x =00.1.01 Prob. density ( 050 D50on10o0r 1d5e0p20th0 5(n00m)1000
a
H
h2 3π σ γ4~2 0.2
C = h dipi DB e . (52) x =0.2
4λ ≈ 256λ d4
max max x =1
0.0
We may calculate the Hahn echo response due to 1/f
0 100 200 300 400 500 600
noise using Eq. (5) with the filter function Eq. (7). If
the inter-pulse time τ is neither too long (so that c = t (m s)
min
πτω /2<1)nortooshort(sothatc =πτω /2>
min max max
1) we get FIG. 5: Theoretical calculations (solid lines) and experimen-
tal data (circles)3 for Hahn echo decay of Sb donors in the
σ (2τ) = exp Cτ2 4ln2 2c2 1 120 KeV implanted sample with the Si/SiO2 surface. The
h + i − − 3 min − 4c2 theory is in reasonable agreement with the data when the
(cid:26) (cid:20)(cid:18) (cid:19) max
theoretical parameter ξ 0.2 [Eq. (57)]. The inset shows
≈
3 4cos(2c )+cos(4c ) (,53) the Sb donor distribution measured by Secondary Ion Mass
max max
× − Spectroscopy (SIMS).
(cid:18) (cid:19)(cid:21)(cid:27)
after neglecting terms of order c3 and 1/c3 . When
cmin . 0.1 and cmax & 10 the emchino envelopmeasxaturates 1.0 -1m) 45400 KeV impla nt (annealed)
and is well approximated by the simpler expression x =0.01 -310n 3 SSImMoSothed
0.8 y ( 2
hσ+(2τ)i≈exp −4ln(2)Cτ2 , (54) ensit 1
o d 0
that is independent of the(cid:2) low and h(cid:3)igh frequency h0.6 x =0.2 b. 0 200 400 600 800 1000
c o
plateaus assumed in Eq. (51).28,29 n e Pr Donor depth (nm)
In the experiment of Ref. 3, each implanted Sb donor ah0.4
is a probe of magnetic noise from the interface. Because H x =1
theimplantedprofileisinhomogeneous,theparameterC
0.2
isdifferentforeachlayerofdonorsadistancedbelowthe
interface. The experimental data was taken at δ/k T =
B
0.3/5=0.06 1. From Eq. (52) we obtain 0.0
≪ 0 1000 2000 3000 4000
σ+(2τ) e−ξ[χ2(τd)]2, (55) t (m s)
h i≈
FIG. 6: Same as Fig. 5 for the 400 KeV implanted sample,
χ(d)= 6.25nmd2, (56) withtheSi/SiO2 surface. AspointedoutinRef.3,theexper-
γ2~ imentaldatasufferedfromexternalfieldnoiseforτ >500µs.
e
σ (nm)2
DB than the experimental data, while at longer time inter-
ξ = × . (57)
λmax valsthe theoryseemsto decayfaster. This lackofagree-
mentmay be due to deviations fromthe measuredSIMS
In this approximation we may fit the experimental data
distribution. The ultra-low donor densities were at the
using a single dimensionless parameter ξ, provided the
sensitivity threshold for the SIMS technique, hence the
distribution of Sb donors is well known.
donor distribution is quite noisy [see insets of Figs. 5, 6
We usedEq.(55)togetherwith the donordistribution
- we used a numerically smoothed version of the SIMS-
measuredbySecondaryIonMassSpectroscopy(SIMS)to
annealed data of Figs. 1(a), 1(b) of Ref. 3]. A higher
obtaintheoreticalestimates of Hahn echo decayrelevant
probability density near the interface could in principle
to the experiment of Ref. 3. Figs. 5 and 6 compares the
explain the faster decay at shorter times, while a deeper
theory with the 120 KeV and 400 KeV implanted sam-
tailinthedistributioncouldberesponsiblefortheslower
ples respectively, both with a Si/SiO surface. A value
2 decay at longer times.
of ξ 0.2 for the theoretical curves seems to be consis-
tent≈with the experimental data. However, in the short The value for λmax may be estimated from λmax =
time range the theoretical curve seems to decay slower 12ln Γ¯Γ¯mmainx ∼ 21ln 1100−61 ∼ 10. Combining this with
(cid:16) (cid:17) (cid:16) (cid:17)