Resonant charge transfer at dielectric surfaces

0.56 MB·English

by  Johannes Marbach

EPJ manuscript No.
(will be inserted by the editor)
Resonant charge transfer at dielectric surfaces
Electron capture and release due to impacting metastable nitrogen molecules
Johannes Marbach, Franz Xaver Bronold, and Holger Fehske
Institut fu¨r Physik, Ernst-Moritz-Arndt-Universit¨at Greifswald, 17489 Greifswald, Germany
2
1 Received: January 10, 2012/ Revised version: January 10, 2012
0
2 Abstract. We report on the theoretical description of secondary electron emission due to resonant charge
n transfer occurring during the collision of metastable N2(3Σu+) molecules with dielectric surfaces. The
a emission is described as a two step process consisting of electron capture to form an intermediate shape
J resonanceN−2(2Πg)andsubsequentelectronemissionbydecayofthision,eitherduetoitsnaturallifetime
9 or its interaction with the surface. The electron capture is modeled using the Keldysh Green’s function
technique and the negative ion decay is described by a combination of the Keldysh technique and a rate

] equationapproach.Wefindtheresonantcaptureofelectronstobeveryefficientandthenaturaldecaytobe
h
clearly dominating over the surface-induced decay. Secondary electron emission coefficients are calculated
p
for Al O , MgO, SiO , and diamond at several kinetic energies of the projectile. With the exception of
- 2 3 2
m MgOthecoefficientsturnouttobeoftheorderof10−1overthewholerangeofkineticenergies.Thisrather
large value is a direct consequence of the shape resonance acting as a relay state for electron emission.
s
a
l PACS. 34.35.+a Gas-surface interactions – 34.70.+e Charge transfer – 79.20.Hx Secondary electron
p
emission
.
s
c
si 1 Introduction Anotherapplicationareplasmadisplaypanels(PDPs),
y which are used in large size TV monitors. PDPs are usu-
h Secondary electron emission from dielectric surfaces due allybasedonACdrivenhighpressuremicrodischarges[4]
p to various projectiles is a fundamental process of great innoblegasmixtures[5]andoftenemployelectrodescoat-
[
technological significance. The associated electron yield is edwithAl2O3 orMgO[6].Thepanel’sfiringvoltage,that
1 usuallyrepresentedbythesecondaryelectronemissionco- is, the voltage required for discharging, is directly influ-
v efficientγe whichdenotestheaveragenumberofelectrons enced by secondary electron emission from the dielectric
4 releasedbyasinglecollisionofaparticulartype.Thus,γe electrodes [6], for instance, due to impacting metastable
2 is specific to the surface material, the projectile, and the species. In order to improve the performance of PDPs,
7 particular collision process under consideration. theoretical investigations of the employed materials, the
1
Toillustratetheimportanceofsecondaryelectronemis- overall discharge geometry, and the operation mode are
.
1 sion from dielectric materials we first consider an atmo- necessary[4].Thesestudiesagainrequiretheemissionco-
0 spheric pressure dielectric barrier discharge (DBD). Var- efficient γ as an input parameter.
e
2 ious technological processes like polymer surface treat-
1 ment, thin film coating, sterilization, gas flow control and The demand for reliable γe data for use in numerical
: simulations of PDPs, DBDs, and related gas discharges
v ozoneproduction[1,2]arebasedonthistypeofdischarge.
contrasts with the very sparse and often uncertain avail-
i All of these are sensitive to the discharge’s stability and
X able reference data. In many cases rules of the thumb
homogeneity. Secondary electron emission from the elec-
r trodesincreasestheDBD’sself-sustainandisthuscrucial have to be used [7]. Especially sparse is the data base for
a secondary electron emission due to metastable molecules.
to the discharge’s operation mode. In particular, there is
Theobjectiveofthepresentworkisthereforetocontribute
some evidence that the stability of DBDs is controlled by
to the filling of this particular gap. We aim to establish
secondary electron emission from plasma boundaries by
a model that allows for easy calculation of the secondary
impacting metastable species [3]. Although DBDs have
electron emission coefficient due to impacting metastable
been studied for more than 20 years, their basic mecha-
moleculesondielectricsurfaces,takingN (3Σ+)asanex-
nism is still not entirely understood [1]. To overcome this 2 u
ample.Otherprojectile-surfacecombinationscanbemod-
deficiencynumericalinvestigationsareinevitable[2,3].These
elled in the same spirit.
studiesrequireγ asaninputparameter,whicheitherhas
e
to be measured experimentally or calculated from first The model we propose is an effective one, stripped of
principles. some of the microscopic details of the system and charac-
2 Johannes Marbach et al.: Resonant charge transfer at dielectric surfaces
E E reaction(2)whichwasalsofoundtobeveryefficientwhen
vac. level vac. level
N (3Σ+) collides with a metallic surface [9,10]. In order
−εα Al2O3 z −εα ε Diamond z to2descuribethisprocess,wewilluseageneralizationofthe
g model presented in [8] and apply it to dielectric surfaces.
ε
g Since the adaption is straightforward we only give a brief
sketch of the model and refer the reader to [8] for details.
The solid surface is treated as a potential well with a
depth
∆εV ε (z) ∆εV ε (z)
0 0
V = ε ε ∆ε , (3)
0 α g V
− − −
filled up to the top of the valence band
ε = ε ε . (4)
V α g
− −
E E
vac. level vac. level
−εα MgO z −εα SiO z Hereεα,εg and∆εV denotethesolid’selectronaffinity,its
2 energygapandthewidthofitsvalenceband,respectively.
εg εg The wave functions of such a potential well are easily cal-
culatedandmaybefoundin[8].Theparticularelectronic
parameters of the dielectric materials investigated in this
work[11,12,13,14,15,16,17,18,19,20,21]aresummarizedin
∆ε Table 1.
V
ε (z) ε (z)
0 0 The molecule is characterized by a two level system
∆εV consisting of an initially empty ground state level 0 and
aninitiallyfilledexcitedlevel1.Themolecularwavefunc-
tions associated with these two levels are calculated from
Fig.1.Energeticstructureofthedifferentdielectricmate- a LCAO combination of atomic nitrogen wave functions,
rials considered inthis work characterized by the electron identifying 0 and 1 with the 2π and 2π molecular or-
u g
affinityεα,thesizeoftheenergygapεg,andthewidthof bitals [8]. The ionization energies ε∞ of the two lev-
0/1
thevalenceband∆ε .ForSiO theenergeticallyblocked
V 2 els [22,23] are listed in Table 2 for the unperturbed mol-
upper valence band is marked in shaded grey. Also in-
ecule. As will be shown in Sec. 2.1, both of these levels
dicated is the variation of the molecular hole level with
shift downwards upon approaching the surface, provided
distance. For an explanation of the molecular image shift
that only the RCT channel (2) is considered.
see Sec. 2.1. Note that the figure is to scale in terms of
energy units. We will describe the resonant reaction (2) as a two-
step process. First, we will employ the Keldysh Green’s
functiontechniquetomodeltheresonantelectroncapture
terized by a small number of material specific parameters into the initially vacant ground state level 0. This step of
instead. In a previous work [8] we presented such a model the process is depicted in Fig. 1. The second step consists
for the case of Auger de-excitation of N2(3Σu+) on alu- ofresonantelectronemissionbydecayoftheintermediate
minum, corresponding to the reaction negativeion.Thedecaycanbeduetotheion-surfacecou-
pling, which is mediated by the image interaction, or due
N2(3Σu+)+ek −A−u−g−e→r N2(1Σg+)+eq , (1) tCoonthsiedeiorinnsgnaatNu−ra(l2Πlife)tiiomnec,otlhliadtinigs,wdiuthe tthoeasuutrof-adceecawye.
with e and e denoting an electron within the surface 2 g
k q will model the surface-induced decay again by a Keldysh
and a free electron, respectively. Another important pro-
approach,theauto-decay,however,byarateequationap-
cess capable of generating secondary electrons by impact
proach. Electron capture and release we will be coupled
of N (3Σ+) is the combination of electron capture by res-
2 u by a rate equation scheme that is based on a conditional
onant charge transfer (RCT) to form the N−(2Π ) shape
2 g decay reaction.
resonance and subsequent decay of the intermediate ion,
corresponding to Theoutlineoftheremainderofthisarticleisasfollows.
In Sec. 2 we will setup the quantum mechanical model of
N (3Σ+)+e RCT N−(2Π ) RCT N (1Σ+)+e . (2) the molecular electron-capture and conduct a quantum-
2 u k −−−→ 2 g −−−→ 2 g q kinetic calculation on top of it. The treatment of surface-
Exceptfordiamond,theAugerchannel(1)isenergetically induced decay and auto-decay as well as their coupling to
suppressed for all of the dielectric materials considered in the initial electron capture will be presented in Section 3.
thisworkwhereastheRCTchannel(2)isenergeticallyal- Finally,weshowanddiscussnumericalresultsforselected
wayspossible(seeFig.1).Hence,wefocusontheresonant surfaces in Sec. 4 and give a conclusion in Sec. 5.
Johannes Marbach et al.: Resonant charge transfer at dielectric surfaces 3
εεεααα [[[eeeVVV]]] εεεggg [[[eeeVVV]]] ∆∆∆εεεVVV [[[eeeVVV]]] εεεbrbrbr NNN222(((111ΣΣΣggg+++)))[[[eeeVVV]]] NNN222(((333ΣΣΣuuu+++)))[[[eeeVVV]]] NNN−2−2−2(((222ΠΠΠggg)))[[[eeeVVV]]]
Al2O3 1.0 [11] 8.7 [12] 11.8 [12] 9.0 [13] ε∞0 −17.25 [22] – −14.49 [23]
diamond 0.52 [14] 5.48 [15] 21.0 [15] 5.66 [16] ε∞1 – −9.67 [22] 1.18 [23]
MgO 0.925 [17] 7.6 [18] 8.5 [18] 9.83 [19] Table 2. Ionization energies of the molecular ground
SiO 0.9 [20] 13.8 [21] 5.3 [21] 3.9 [13] state level 0 and the excited level 1 for a free nitrogen
2
Table 1. Electronic parameters for the dielectric mate- molecule in its ground state N2(1Σg+) and its metastable
rials considered in this work: electron affinity εα, energy state N2(3Σu+) as well as for the negative ion shape reso-
gap εg, valence band width ∆εV, and static relative di- nance N−2(2Πg).
electric constant εb. The electron affinity of diamond and
r
MgO is calculated from a mean value due to lack of con-
sistentdata.Note,forSiO onlythelowervalencebandis
2
considered. The upper valence band starts at 8.9eV and Sincethisprocedureisindependentofthespecificmolec-
has a width of 2.7eV [21]. It is thus energetically well ularlevel,weconcludethatuponapproachingthesurface
separated from the molecular vacancy level ε and does each and any of the negative ion’s ionization levels shifts
0
therefore not contribute to the electron capture. downwards. The reader may note that the decrease of the
ionization energies does not correspond to an actual en-
ergy change of the negative ion. The ionization levels are
defined solely in terms of the removal of an electron from
2 Resonant electron capture themoleculeand,thus,donotimplyanenergychangefor
a bound electron.
2.1 Model Employingtheusualtrajectoryapproximation[25]to-
gether with the trajectory
Themolecule-surfacesystemisdescribedbythefollowing
Hamiltonian R(t)=zR(t)ez =(v0 t +z0)ez , (8)
| |
H(t)= ε c† c +ε (t)c†c the relation (7b) can also be understood as ε0(t).
k k k 0 0 0
GeneralizingGadzuk’s[26]treatmentofmono-nuclear
k
(cid:88) (5) projectiles to bi-nuclear projectiles, the tunnelling matrix
+ Vk(t)c†kc0+Vk∗(t)ckc†0 , element Vk is given by
(cid:88)k (cid:16) (cid:17)
wherekdenoteselectronicstateswithinthesolid’svalence Vk(t)= dr Ψk∗(r) VM(r,t)+VSM(z) Ψ0(rϕ(t)), (9)
bandand0labelsthemolecule’sgroundstatelevel,which (cid:90)
(cid:2) (cid:3)
is initially empty. Since we are considering the capture of wherethetimedependencearisesfromthemolecule’smo-
an additional electron by the metastable state N2(3Σu+), tionandrϕ equalsthevectorrasseenfromthemolecular
the level 0 equals the lower ionization level of N−2(2Πg). reference frame, that is,
To understand the variation of ε with time we fol-
0
low [24] and consider a negative nitrogen ion situated at r (t)=Ωˆ(ϕ)[r R(t)] , (10)
ϕ
a distance z from a dielectric surface, the relative static −
R
permittivityofwhichwedenotebyεbr.Sincetheionisneg- with an appropriate rotation matrix Ωˆ(ϕ) [8].
atively charged its overall energy is lowered by the value The expression V specifies the Coulomb potential of
M
oftheimagepotentialatzR,forwhichweusetheclassical the screened molecular nuclei [26]
expression (see however below)
Z e2 1 1
Vi(zR)≈−εεbrbr+−114πe2ε04z1R . (6) VM(r,t)=− 4πεe0ffεr(z)(cid:20)|r−R1(t)| + |r−R2(t)|(cid:21)(1,1)
Inordertoobtaintheeffectiveionizationenergyε (z ),
0 R with Z 4 [8] and
eff
we virtually move the negative ion from the position z ≈
R
to a point infinitely far away from the surface, ionize the (cid:37)
R (t)=R (12)
electron bound in the 0 state to the vacuum level, and re- 1/2 ± 2
locate the resulting neutral molecule back to the original
distance z . The net energy change during this process denoting the positions of the two nuclei, which are given
R
determines the effective ionization energy intermsofthecenterofmasscoordinateRandtheinter-
nuclear vector (cid:37). The static dielectric constant ε (z) is
r
ε (z )= ∆ε= V (z )+( ε∞)+ 0 (7a) approximated by
0 R − − − i R − 0
(cid:2) N2−→∞ N2−→N2 N2→zR(cid:3) εb , z 0
=ε∞0 +Vi(zR(cid:124)). (cid:123)(cid:122) (cid:125) (cid:124) (cid:123)(cid:122) (cid:125) (cid:124)(cid:123)(cid:122)(cid:125) (7b) εr(z)=(cid:40)1r , z >≤0 . (13)
4 Johannes Marbach et al.: Resonant charge transfer at dielectric surfaces
The term V labels the electron’s self image interac- and σ given by the infinite series
SM 0
tion potential [26]
t t1
σ (t,t(cid:48))=1 dt dt ∆ (t ,t )
VSM(z)=Θ(z) Θ(zc−z)V0+Θ(z−zc)Vi(z) . (14) 0 −(cid:90)t(cid:48) 1(cid:90)t(cid:48) 2 0 1 2
Here we have used(cid:2) the usual truncation of the(cid:3)classical + tdt t1dt t2dt t3dt ∆ (t ,t )∆ (t ,t ) (22)
1 2 3 4 0 1 2 0 3 4
image potential close to the surface [27] with (cid:90)t(cid:48) (cid:90)t(cid:48) (cid:90)t(cid:48) (cid:90)t(cid:48)
... .
! −
V (z )=V . (15)
i c 0
The quantity ∆ emerges from the self-energy and reads
0
The wave functions of the metal electron Ψ and the
k
1
otrneeatofthtehemmagonleectuiclarquealenctturmonnΨu0mabreertmak,enwhfricohmis[8i]n.tWroe- ∆0(t1,t2)= (cid:126)2 Vk∗(t1)Vk(t2)
k (23)
duced within the LCAO representation of the molecular (cid:88)
wave functions [8], as an external parameter. e−(cid:126)iεk(t1−t2)e−(cid:126)i (cid:82)tt12dt¯ε0(t¯) .
×
By employing now the Dyson equation for GR , with
0k
2.2 Quantum kinetics theselfenergy(16b),theoccupancyofthemolecularlevel
(18b) can be written as
TheHamiltonian(5)describesastandardproblemofmany
particlephysics.ItcanbetreatedexactlyusingtheKeldysh t t
n (t)= dt dt ∆ (t ,t )σ (t,t )σ∗(t,t ). (24)
technique[28].Theself-energysolelyconsistsofafirstor- 0 1 2 0 1 2 0 1 0 2
der retarded and advanced term (cid:90)t0 (cid:90)t0
(cid:101)
ΣkR0/A(t1,t2)= (cid:126)iVk(t1)δ(t1−t2) (16a) Hdeerreth∆e0kissueqmu.aIlntoge∆n0erwailtthhaisnsaudbdtilteitoynwalofualcdtodrennyk(ftu0r)tuhner-
∗ simpli(cid:101)fication of Eq. (24). However, since for a dielectric
=− Σ0Rk/A(t1,t2) . (16b) surface the active energy band is the valence band, which
(cid:104) (cid:105) is completely filled, the terms ∆0 and ∆0 are essentially
The Keldysh component of the self-energy vanishes, the same. Note that even for metallic surfaces with an
the Dyson equation for the Keldysh Green’s function can incompletely filled conduction band thes(cid:101)e terms will still
thus be solved to give be almost equal, as long as the molecular resonance stays
inside the occupied portion of the band.
GKαβ(t,t(cid:48))=−i 1−2nγ(t0) To simplify Eq. (24), we first note that when t is kept
γ (17) constant the derivative of Eq. (22) with respect to t(cid:48) can
(cid:88)(cid:2) (cid:3)
GR (t,t )GA (t ,t(cid:48)), be shown to read
× αγ 0 γβ 0
∂σ (t,t(cid:48)) t
where α and β can take the values 0 or k and γ runs 0 = dt¯σ (t,t¯)∆ (t¯,t(cid:48)). (25)
over all states {0,k} of the system. Note that t0 is short- ∂t(cid:48) (cid:90)t(cid:48) 0 0
hand for . Equation (17) can be used to calculate the
−∞ Using this expression we can now rewrite (24) into
occupation of the molecular level
1 t dσ (t,t )
n (t)= 1 iGK(t,t) (18a) n (t)=2 dt σ∗(t,t ) 0 2 (26a)
0 2 − 00 0 (cid:60) 2 0 2 dt
(cid:20)(cid:90)t0 2 (cid:21)
= (cid:2)nk(t0)|GR0k(t(cid:3),t0)|2 , (18b) =2 1−σ0(t,t0)σ0∗(t,t0)
(cid:88)k (cid:2)2 tdt σ (t,t )dσ(cid:3)0∗(t,t2) , (26b)
where we have used the relation [28,29] − (cid:60) 2 0 2 dt
(cid:20)(cid:90)t0 2 (cid:21)
GR (t,t )GA (t ,t)=1. (19) where we have used integration by parts in the last line.
αγ 0 γα 0
Combining (26a) and (26b), we obtain
γ
(cid:88)
EvaluationoftheGreen’sfunctionGR0kappearingin(18b) n0(t)=1−σ0(t,t0)σ0∗(t,t0). (27)
requires knowledge of the diagonal component GR, the
00 We have not yet used any approximation to σ . The
Dysonequationofwhichcanbesolvediterativelytoyield 0
result (27) is thus exact. Since V is small it is however
k
GR(t,t(cid:48))=GR(0)(t,t(cid:48))σ (t,t(cid:48)), (20) sufficient to approximately re-sum the infinite series (22).
00 00 0 Using the exponential re-summation technique described
with in [8] and retaining only the lowest order term we obtain
GR00(0)(t,t(cid:48))=−iΘ(t−t(cid:48))e−(cid:126)i (cid:82)tt(cid:48) dt1ε0(t1) , (21) σ0(t,t(cid:48))≈e−(cid:82)tt(cid:48) dt1(cid:82)tt(cid:48)1dt2∆0(t1,t2) , (28)
Johannes Marbach et al.: Resonant charge transfer at dielectric surfaces 5
which after insertion into (27) yields Here we have rescaled r , k and t according to
ϕ
n0(t) 1 e−(cid:82)tt0dt1(cid:82)tt0dt2∆0(t1,t2) . (29) r 1r , k κk, t 1 t, (36)
≈ − ϕ → κ ϕ → → κv
0
Evaluating (29) at t = t we see that the initial condi-
0
tthioenmne0t(at0ll)ic=w0aviesffuulnficltleiodn.T[8a],ktinhgetkhseulmimiintL(23→)t∞urnwsitinhtino whaitnhdκno=ta2tZiaeofBfn.fFourrthermore,theexpressionsrϕ± areshort-
an integral. For a numerical treatment it is convenient to
(cid:37)
einxcthhaenegxeptohneenktaonfd(2t19/)2tihnetnegtraaktieosntshienf(o2r9m). The integral rϕ± = rϕ± 2ez(cid:48) . (37)
(cid:12) (cid:12)
t t L3 Inserting (34) into Eq. (3(cid:12)(cid:12)1) yields (cid:12)(cid:12)
dt dt ∆ (t ,t )= dk Λ (k,t)2
(cid:90)t0 1(cid:90)t0 2 0 1 2 2(cid:126)2(2π)3 (cid:90) | 0 |(30) Λ(k,t)= CmκTvk∗0zVk (cid:90)t0tdt1 e−κkzzR(t)e(cid:126)κiv0εkt1 (38)
with e(cid:126)κiv0 (cid:82)(cid:101)0t1dt2ε0(t2) ,
t ×
Λ0(k,t)= dt1 Vk(t1)e(cid:126)iεkt1e(cid:126)i (cid:82)0t1dt2ε0(t2) , (31) which enables us to calculate the involved space and time
(cid:90)t0 integrals independently within a numerical simulation.
where the L3 factor in (30) cancels with the L−3/2 factor
in V .
k
3 Decay of the negative ion
2.3 Matrix element separation
3.1 Surface-induced decay
In order to make a numerical calculation of the ground
The surface-induced decay reaction can be described by
state level’s occupancy feasible, we need to introduce ap-
the Hamiltonian
proximations to the matrix element.
Numerical evaluation of the full matrix element (9) H(t)= ε (t)c† c +ε (t)c†c
suggeststhatV isclearlydominatingoverV andthat q q q 1 1 1
M SM q
in addition the matrix element may be neglected in the (cid:88) (39)
halfspacez 0.Makinguseofthesefindingswecansep- + V (t)c† c +V∗(t)c c† ,
≤ q q 1 q q 1
aratethematrixelement’stimedependencebyperforming (cid:88)q (cid:16) (cid:17)
the coordinate transformation
which is structurally identical to the resonant capture
r =Ωˆ†(ϕ)r +R(t), (32) Hamiltonian (5) with the associations k q and 0 1.
ϕ
↔ ↔
In (39) ε is the upper ionization level of the negative
r 1
ϕ(cid:48) nitrogen ion and ε denotes the energy of the free elec-
q
which is the inverse of(cid:124)(10(cid:123))(cid:122). U(cid:125)tilizing the relation tron states, which are centered around the moving mol-
ecule. Both of these energy levels shift downwards upon
(cid:37)
r R (t) = r R(t) Ωˆ†(ϕ)e (33a) approaching the surface, as can be shown using the anal-
| − 1/2 | | − ∓ 2 z| ysis presented in Sec. 2.1
(cid:37)
= Ωˆ†(ϕ)[r e ] (33b)
| ϕ∓ 2 z | ε (t)=ε∞+V (z (t)), (40a)
(cid:37) 1 1 i R
= r e (33c) ε (t)=ε∞+V (z (t)). (40b)
| ϕ∓ 2 z| q q i R
and approximating Θ(z(cid:48) +z (t)) by Θ(z(cid:48) +z ) as in [8] This fact implies that the difference ε (t) ε (t) is con-
ϕ R ϕ 0 1 − q
we then obtain stant in time.
The tunnelling matrix element V is given by
Vk(t)≈CTk∗ze−κkzzR(t)Vk , (34) q
where T is defined in [8], V (t)=α(ε∞,t) dr Ψ∗ (r (t))Ψ (r (t))
kz (cid:101) q qz qϕ ϕ 1 ϕ (41)
(cid:90)
C = mNk Zeffe2 ,and (35a) VS(z)+VSM(z) ,
2πN2πuκ 4πε0 with Ψ as de×fin(cid:2)ed in [8] and Ψ (cid:3)given by
1 1 1 q
Vk = (cid:112)drϕ Θ(zϕ(cid:48) −z0)e−κkzzϕ(cid:48) r+ + r− 1
(cid:90) (cid:20) ϕ ϕ(cid:21) Ψ (r)= eiq·r . (42)
(cid:101) e−rϕ+ +e−rϕ− (xϕ+imyϕ) (35b) q L32
×
e(cid:104)−i(kxx(cid:48)ϕ+kyyϕ(cid:48)(cid:105)) . culeS,iwnceehtahveeetolecetvraolnuaitseeinmiEttqe.d(4f1ro)mthethweamveofvuinngctimonoloe-f
×
6 Johannes Marbach et al.: Resonant charge transfer at dielectric surfaces
the free electron, Ψ (r), in terms of the molecular coordi- For the parallel orientation Φ takes the form
q ϕ
natesr andthewavevectors q whicharetheelectron’s
ϕ ϕ (cid:37)
wave vectors as seen from the molecule’s reference frame. Φ (z(cid:48),Q )=4i√2πsin q
(cid:107) (cid:107) (cid:107) x2
The plane wave ansatz (42) is valid in the present case,
QcosΦ (cid:16) (cid:17)
bhtwhieinetcrhdae.ufzCsoeerroeotnhtceeoeffpetetmhcuteiaitvtltleweyd,onu-uecscleielnnectgutersΨrocnqChϕloa(eurraglϕvoee)m.sinbaEnwqeau.vt(er4au1l)smecdoorlirenecsu[8pl]eonbbduest- ×z(cid:20)(cid:48)m(1|z+(cid:107)(cid:48)|Q23 2)(cid:107)54K|z(cid:107)(cid:48)|25zK(cid:48) 52(cid:16)1|z+(cid:107)(cid:48)|(cid:112)Q21+Q, 2(cid:17) (49)
counInts(f4o1r) twhee itnratrpopdiuncgedoftshleowteremlectαro(εn∞qsz,itn) twhheicihmaagce- − (cid:107)(1+Q2)34 32(cid:16)| (cid:107)|(cid:112) (cid:17)(cid:21)
The lateral wave vectors Q and Q are given by
potential close to the surface. When calculating the occu- ⊥ (cid:107)
pancies of the molecular levels, α must equal unity since
Q =(q ,q )=(Qcos(Φ ),Qsin(Φ )), (50a)
the decay of the molecular state does not depend on the ⊥ x y ⊥ ⊥
Q =(q ,q )=(Qcos(Φ ),Qsin(Φ )). (50b)
emittedelectronbeingabletoescape.Ontheotherhand, (cid:107) y x (cid:107) (cid:107)
whencalculatingthenumberofemittedelectrons,αneeds
Since the Hamiltonian (39) is structurally identical
to cut off the matrix element when the overall energy of
to(5),thequantum-kineticcalculationoutlinedinSec.2.2
the emitted electron is smaller or equal to zero.
holds for the current case as well. The only difference is
Theoverallenergyoftheelectronisgivenbythesumof
that the ion’s motion does not start at t=t = , but
theelectron’sperpendicularenergyε∞ andtheimagepo- 0 −∞
qz at some later time tg, at which the ion is generated by
tentialatthepositionofemission,whichmaybeapproxi-
electron capture. This can, however, easily be accounted
matedbythemolecule’scenterofmasscoordinate[8].We
for by replacing t with t in the previous calculation.
0 g
thus define
UsinganadaptedversionofEq.(18b)wethusfindfor
the occupancy of a free electron state with wave vector q
1 , mol. level
α(ε∞,t)= . (43)
qz (cid:40)Θ ε∞qz +Vi(zR(t)) , q states nq(t)= 12 1−iGKqq(t,t) =|GRq1(t,tg)|2 . (51)
(cid:0) (cid:1)
The interaction potential VS is given through Employing th(cid:2)e Dyson equa(cid:3)tion for GR(t,t ) and the
q1 g
resultsofSec.2.2thetotalnumberofemittedelectronsis
V (z)=Θ( z)V , (44)
S − 0 then given by
and V is defined in Eq. (14). Employing the transfor-
SM n(t)= n (t) (52a)
mation (32) the matrix element takes the form q
q
(cid:88)
t t
V (t)=α(ε∞,t) dr Ψ∗ (r )Ψ (r ) = dt dt ∆ (t ,t )σ∗(t ,t )σ (t ,t ), (52b)
q qz ϕ qϕ ϕ 0 ϕ (45) 1 2 1 1 2 1 1 g 1 2 g
(cid:90) (cid:90)tg (cid:90)tg
× VS(zϕ(cid:48) +zR(t))+VSM(zϕ(cid:48) +zR(t)) . =1−σ1(t,tg)σ1∗(t,tg) (52c)
Focusing(cid:2)on the two principal molecule ori(cid:3)entations with
andusingtheexplicitformofthewavefunctionstogether
1
with the calculus of Fourier transform Eq. (45) can be ∆1(t1,t2)= (cid:126)2 Vq(t1)Vq∗(t2)
reduced to a single integral q (53)
(cid:88)
Vq(t)= α(εL∞qz32,t)(cid:90)−∞∞dzϕ(cid:48) e−iqzzϕ(cid:48) Φϕ(zϕ(cid:48) ,Qϕ) (46) and σ1 defined by an ex×pree(cid:126)sis(εio∞qn−sε∞1an)(atl1o−gto2)us to Eq. (22),
V (z(cid:48) +z (t))+V (z(cid:48) +z (t)) . but with ∆1 instead of ∆0.
× S ϕ R SM ϕ R Theequalityof (52b)and(52c),althoughnotobvious,
In the perpen(cid:2)dicular geometry Φ reads (cid:3) can easily be confirmed. To do so, we denote these two
ϕ
equations by ζ (t) and ζ (t), respectively, and calculate
1 2
Q their derivatives with respect to t
Φ (z(cid:48) ,Q )= (2i)√2π eimΦ
⊥ ⊥ ⊥ − (1+Q2)45 ⊥ dζ (t) t
1 = dt ∆ (t,t )σ∗(t,t )σ (t ,t )
5 dt 2 1 2 1 g 1 2 g
z+ 2 K5 z+ 1+Q2 (47) (cid:90)tg (54a)
× | | 2 | | t
(cid:20) (cid:16) (cid:112) (cid:17) + dt ∆ (t ,t)σ∗(t ,t )σ (t,t )
5 1 1 1 1 1 g 1 g
−|z−|2 K52 |z−| 1+Q2 , (cid:90)tg
(cid:16) (cid:112) (cid:17)(cid:21) = σ∗(t,t )dσ1(t,tg) σ (t,t )dσ1∗(t,tg) (54b)
with K denoting the modified Bessel function of the sec- − 1 g dt − 1 g dt
ond kind and dζ (t)
z± =z⊥(cid:48) ± (cid:37)2 . (48) = d2t . (54c)
Johannes Marbach et al.: Resonant charge transfer at dielectric surfaces 7
In addition, by inspection of Eq. (52) we see that ζ (t) Equations (57) and (58) hold when calculating the num-
1
andζ (t)bothvanishatt=t .Hence,ζ (t)andζ (t)are ber of emitted electrons n(t). They can, however, also be
2 g 1 2
solutionsofoneandthesameordinaryfirstorderdifferen- applied to calculate the occupancy of the molecular levels
tial equation with one and the same initial condition. By if one sets t equal to 0.
qz
virtue of the uniqueness of such a solution ζ (t) and ζ (t) In applying the analysis of Sec. 2.2 to the excited
1 2
are thus identical. molecular level, we find its occupancy to be
Thetimeintegrationofthemodulatedmatrixelement
n (t)=1 n(t)=σ (t,t )σ∗(t,t ). (59)
involved in (52c), requires evaluation of 1 − 1 g 1 g
Inordertoobtaintherateofthesurface-induceddecay
κ1v tdt1 Vq(t1)e(cid:126)κiv(ε∞q −ε∞1 )t1 panrodcetasskeΓSth,ewdeeurisveattihveeloofwEesqt.o(r5d9e)rwaiptphrroexsipmeacttiotontf,or σ1
(cid:90)tg (55)
= κv1L23 (cid:90)−∞∞dzϕ(cid:48) e−iqzzϕ(cid:48) Φϕ(zϕ(cid:48) ,Qϕ)I(t,tg,zϕ), dnd1t(t) =2(cid:60)(cid:20)σ1∗(t,tg)dσ1d(tt,tg)(cid:21)
t
where we have used the variable scaling (36) and intro- 2 dt [∆ (t,t )]σ (t,t )σ∗(t,t )
duced the quantity ≈− (cid:90)tg 2 (cid:60) 1 2 1 g 1 g
Γ (t,t )n (t)
t ≡ S g 1
I(t,tg,zϕ)=(cid:90)tgdt1 α(ε∞qz,t1)e(cid:126)κiv(ε∞q −ε∞1 )t1 (56) with
t
× VS(zϕ(cid:48) +zR(t1))+VSM(zϕ(cid:48) +zR(t1)) . ΓS(t,tg)=−2 dt2 (cid:60)[∆1(t,t2)] . (60)
(cid:2) (cid:3) (cid:90)tg
Although tedious, the time integration within Eq. (56)
Notethatinadditiontothetimeofobservationt,thede-
can be done analytically. For shortness, we merely give
cayrateΓ alsodependsonthetimeofgenerationt .The
the result S g
coupling of the surface-induced decay process with the
(t,tg,zϕ)= 2V0 Θ(t2 t1)Θ( tg)eiat1+2t2 pBreifoorren,ehgoawtievveeiro,nwgeetnuerrnatoiounr awtitlelnbteioonuttolintehde ninatSuerca.l3d.e3-.
I a − −
(cid:20) cay of the shape resonance N−(2Π ).
t t 2 g
2 1
sin a − +Θ(t)Θ(t t )
4 3
× 2 −
(cid:18) (cid:19)
3.2 Natural decay
eiat3+2t4 sin at4−t3
× 2 The N−(2Π ) shape resonance has a mean life time due
(cid:18) (cid:19)(cid:21) 2 g
κe2 (57) to natural decay of τ = 1.6 10−15s [30], which corre-
+ 16πε0(cid:20)Θ(t5−tg)Θ(−tg)eiaz˜ esparolniedrswtoorka [d9e]ctahyerlaintee sohfaΓpNe·o=f aτu−t1o-=dec0a.6y2p5r·o1c0e1s5se/ss.hIans
π
Ei ia(z˜ t ) + (1+isgn(a)) beenmodelledbymeansofaLorentziandistribution,also
5
× − − 2
known as Breit-Wigner distribution. In accordance with
Θ(cid:110)(t)(cid:0)Θ(t t6)e−(cid:1)iaz˜ (cid:111) this approach we define therefore the natural decay rate’s
− −
energy spectrum (cid:37) (ε∞,t) by
N q
Ei ia(z˜+t) Ei ia(z˜+t ) ,
6
× − ∞
(cid:110) (cid:0) (cid:1) (cid:0) (cid:1)(cid:111)(cid:21) ΓN(t)= dε∞q (cid:37)N(ε∞q ,t)
where we have introduced the following abbreviations (cid:90)0
t1 =max(tg,zϕ(cid:48) +z0−zc), (58a) = 4∆π3ετ (cid:90)0∞dε∞q (cid:90)0π2dϑq(cid:90)02πdϕq (61)
t2 =min(0,tqz,t), (58b) α ε∞q cos2(ϑq),t
t =max(t ,t ,0), (58c) .
3 g qz × (ε∞ ε )2+ ∆ε2
t =min(z z(cid:48) z ,t), (58d) (cid:0)q − 1 4(cid:1)
4 c− ϕ− 0 Here ∆ε is the line’s full width at half maximum, which
t¯ =z z(cid:48) z , (58e)
5 c− ϕ− 0 can be estimated using Heisenberg’s uncertainty relation
t =min( t¯, t ,0,t), (58f)
t5 =max(−t ,50,−t¯q,zt ), (58g) ∆ε(cid:38) h . (62)
6 g 5 qz τ
εb 1 κe2 1
t = r− z , (58h) In (61) we again introduced the image potential trap-
qz εb +116πε ε − 0 ping factor α(ε ,t). Using its explicit form (43) we can
r 0 qz qz
ε∞ ε∞ carry out the angle integration in (61) and obtain
a= q − 1 , (58i)
(cid:126)κv ∆ε ∞ α¯(ε∞,t)
z˜=zϕ(cid:48) +z0 . (58j) ΓN(t)= 2πτ dε∞q (ε∞ ε∞q)2+ ∆ε2 , (63)
(cid:90)εα q − 1 4
8 Johannes Marbach et al.: Resonant charge transfer at dielectric surfaces
with The generation of negative ions is realized by filling
the hole in the ground state level 0, that is, the lower ion-
α¯(ε∞q ,t)=211 acos εi(zR(t)) ,, mqostl.atleevsel . (64) iaznateiolenctlreovnelforfoNm−2t(h2eΠegx).ciTtehdeiloenvetlh1enwdheiccahysisbtyheemhiitgthinegr
π (cid:18) ε∞q (cid:19) ionization level of N−2(2Πg). Hence, the molecular frac-
(cid:113) tions n can be related to the single level occupan-
∗/−/g
Theintegrationboundaryεα appearingin(63)isgivenby cies n by means of
0/1
0 , mol. level n (t)=1 n (t)
εα =(cid:40)εi(zR(t)), q states . (65) ng∗(t)=1−−n01(t)(cid:27) ⇒ n−(t)=n0(t)+n1(t)−1. (71)
The rate equations for the complete electron emission Tocalculatethetotalnumberofemittedelectronsn(t)
processneedstotakeintoaccountnotonlythedecaypro- and their energy spectrum we first note that due to par-
cess,beitnaturalorsurface-induced,butalsothepreced- ticle conservation n(t) must equal the fraction of ground
ing generation of the negative ion. It is described next. state molecules n (t). Hence, we can employ Eq. (67c)
g
to calculate n(t). In analogy to (61) we write the decay
rate Γ (t) as
1
3.3 Consecutive decay approach
∞
Γ (t)= dε∞ (cid:37) (ε∞,t), (72)
BecauseofthelackofaneffectiveHamiltonianfortheac- 1 q 1 q
(cid:90)0
tive electrons describing simultaneously the formation of
the negative ion due to electron capture and the decay where we introduced the rate’s energy spectrum (cid:37)1(ε∞q ,t)
of the negative ion with subsequent electron emission we which needs to account for trapping of low energy elec-
cannotemploytheKeldyshtechniquetocalculatethesec- trons infront ofthe surfacedue tothe image potential. It
ondaryelectronemissioncoefficient.Asanalternative,we will be given below.
use rate equations with rates obtained from the Keldysh Afteremploying(72)in(67c)wedirectlyintegratethe
approach. resulting equation and obtain
The reaction chain consists of two consecutive pro-
∞ t
cesses.Consideringonlythemolecularstateswecanwrite n(t)= dε∞ dt (cid:37) (ε∞,t )n (t ). (73)
q 1 1 q 1 − 1
(cid:90)0 (cid:90)t0
N (3Σ+) Γ0(t) N−(2Π ) Γ1(t) N (1Σ+), (66)
2 u −−−−−→ 2 g −−−−−→ 2 g Inserting n− from Eq. (70b) we can use (73) to calculate
the overall number of emitted electrons as well as their
where Γ (t) and Γ (t) are the reaction rates of electron
0 1
energyspectrum.Fort= Eq.(73)definesthesecondary
captureandrelease,respectively.Thescheme(66)iseasily ∞
electron emission coefficient γ =n( ).
translated into a system of ordinary differential equations e ∞
Finally,weneedtobuildalinkbetweentheratesΓ
0/1
and the previously calculated quantities Γ . The rate
S/N
dn∗(t) = Γ (t)n (t), (67a) of electron capture Γ0 is defined by
0 ∗
dt −
d 1 n (t)
dn−(t) − 0 = Γ (t) 1 n (t) . (74)
= Γ (t)n (t)+Γ (t)n (t), (67b) 0 0
1 − 0 ∗ dt − −
dt − (cid:0) (cid:1)
dng(t) and can be calculated from (29). T(cid:0)he rate of(cid:1)negative ion
=Γ (t)n (t), (67c)
dt 1 − decay Γ contains contributions from the surface-induced
1
decay and the natural decay. For the latter the rate Γ
where n , n and n denote the respective fractions of N
∗ − g wasexplicitlygiveninSec.3.2.Fortheformerweneedto
metastablemolecules,negativeions,andgroundstatemol-
keep in mind that the rate Γ (t,t ) given in Sec. 3.1 also
ecules, which obey the relation S g
dependsonthetimeofnegativeioncreationt .Toobtain
g
the overall rate Γ (t), we therefore need to weight the
n∗(t)+n−(t)+ng(t) 1 t (68) S
≡ ∀ two-time rate Γ (t,t ) with the negative ion’s production
S g
and satisfy the initial conditions term along the trajectory, that is,
n (t )=1, n (t )=0, n (t )=0. (69) t
∗ 0 − 0 g 0 Γ (t)= dt Γ (t,t ) Γ (t )n (t ) . (75)
S g S g 0 g ∗ g
The system (67) can be solved straightforwardly to give (cid:90)t0
−dnd∗t(gtg)
n∗(t)=e−(cid:82)tt0dt1Γ0(t1) , (70a) Employing the explicit form of Γ (cid:124)(t,t )(cid:123)(cid:122)given(cid:125)by Eq. (60)
S g
n−(t)= tdt1 dn∗(t1) e(cid:82)tt1dt2Γ1(t2) , (70b) together with partial integration we obtain
− dt
(cid:90)t0 1 t
ng(t)=1−n∗(t)−n−(t). (70c) ΓS(t)=(cid:90)t0dt1 2(cid:60) ∆1(t,t1) 1−n∗(t1) . (76)
(cid:2) (cid:3)(cid:0) (cid:1)
Johannes Marbach et al.: Resonant charge transfer at dielectric surfaces 9
1 1
para.
para.
perp.
0.8 0.8 perp.
) 0.6
t) 0.6 (t−
( n 0.4
0
n
0.4
0.2
0.2
0
10 5 0 5 10
− −
0 t
10 8 6 4 2 0 2 4
− − − − − 0.02
t
para.
Fig. 2. Time evolution of the occupation of the molecu- 0.015 perp.
lar ground state level n in front of an Al O surface for
0 2 3 )
parallel (solid line) and perpendicular (dashed line) mol- (t 0.01
−
ecule orientation. The molecule’s kinetic energy was fixed n
to 50meV. The curves were calculated from Eq. (29).
0.005
0
The total rate of decay is given by
10 5 0 5 10
− −
t
Γ (t)=Γ (t)+Γ (t) (77)
1 S N
Fig. 3. Fraction of negative ions versus time for the
and the total rate’s energy distribution (cid:37)1(ε∞q ,t), which surface-induced decay process (upper panel) and the nat-
was introduced in (72), is related to the previous calcula- uraldecayreaction(lowerpanel)infrontofanAl O sur-
2 3
tion by means of (63) and (76), respectively. face at a molecular kinetic energy of 50meV. The solid
To be specific we note that for constructing (cid:37)1(ε∞q ,t) lines represent the parallel molecule orientation, whereas
theqsuminvolvedin∆ (seeEq.(53))hastobeconverted thedashedlinesdenotetheperpendicularorientation.The
1
into an integral by letting L . Afterwards one can results were calculated from Eq. (70b).
→ ∞
integrate the angle coordinates of the q vector to obtain
the energy spectrum of the rate for the surface-induced
decay. The final expression for (cid:37)1(ε∞q ,t) reads molecular vacancy is completely filled within the incom-
ingbranchofthetrajectoryforbothorientations.Theeffi-
(cid:37) (ε∞,t)=(cid:37) (ε∞,t)+(cid:37) (ε∞,t), (78)
1 q S q N q ciency is higher in parallel orientation. Complete filling of
thegroundstateholeisachievedinthatcaseataboutt=
with
1, corresponding to a molecule-surface distance of ap-
−
(cid:37) (ε∞,t)= 2me 32 L3 tdt π2dϑ 2πdϕ pthreoxoitmheartehlayn4d.,9c2oamBp.lIentetfihlelipngeripseonndlyicruelaacrhoerdieanttaatbioount,ton=
S q h2κ2 (cid:126)2 1 q q
(cid:18) (cid:19) (cid:90)t0 (cid:90)0 (cid:90)0 0 which corresponds to the molecule’s turning point z0 =
(79)
×(cid:60) Vq(t)Vq∗(t1)e(cid:126)i(ε∞q −ε∞1 )(t−t1) 4.42NaeBx.t,wefocusonthedecayofanegativeiongenerated
× (cid:110)ε∞q sin(ϑq) 1−n∗(t1) (cid:111) by electron capture. To estimate the relative strength of
thesurface-inducedandthenaturaldecaychannel,respec-
and (cid:37)N(ε∞q ,t) im(cid:112)plicitly defin(cid:0)ed in (61).(cid:1) tively, we treat them separately. Figure 3 shows the time
evolutionofthefractionofnegativeionsascalculatedfrom
Eq. (70b) for both processes in front of an Al O surface
2 3
4 Results at ε =50meV. As we see, within the incoming branch
kin
the portion of negative ions first increases due to the ef-
In the following we present numerical results for the two ficiency of the electron capture into the ground state and
principal molecule orientations, that is, for the molecular thefinitenessofthedecayrates.Atsomepointwithinthe
axis aligned perpendicular (ϕ = π) and parallel (ϕ = 0) incomingbranch,however,thedecayprocessstartstoout-
2
to the surface. balance the generating reaction, resulting in a decrease of
First, we consider the capturing of a metal electron the negative ion fraction. Since the surface-induced decay
intothemoleculargroundstatelevel,whichwasdescribed process is driven by the molecule-surface interaction (41),
inSec.2.Figure2showstheevolutionofthegroundstate which vanishes at large distances from the solid, the neg-
level’soccupationn (t)ascalculatedfromEq.(29)forthe ative ion share saturates to a finite value in this case (see
0
case of an Al O surface at a kinetic energy of 50meV. upper panel of Fig. 3). For the parallel orientation the
2 3
We clearly see that the process is very efficient, since the saturation level is much lower than for the perpendicu-
10 Johannes Marbach et al.: Resonant charge transfer at dielectric surfaces
1 0.05
para.
er 0.8 n 0.04 perp.
b ∗
m
nu 0.6 (cid:3)
n para. 1eV 0.03
o perp. (cid:2)
ati 0.4 )∞∞
up n(dεq0.02
c d
Oc 0.2 ng
0.01
0
10 8 6 4 2 0 2 4 0
− − − − − 0 2 4 6 8 10
t
ε∞q [eV]
Fig. 4. Time evolution of the fractions of metastables n
∗ Fig. 5. Spectrum of emitted electrons for an Al O sur-
andgroundstatemoleculesn inparallel(solidlines)and 2 3
g
face in parallel (solid lines) and perpendicular (dashed
perpendicular(dashedlines)orientationforanAl O sur-
2 3
lines) orientation at t = , as calculated from Eq. (73).
face at a molecular kinetic energy of 50meV. Only the ∞
The molecule’s kinetic energy was fixed to 50meV. Only
natural decay channel was considered. The curves were
the natural decay process was taken into account.
obtained by employing Eqs. (70a) and (70c), respectively.
larorientation.Thisiscausedbytheantisymmetryofthe means of the quantity α(εqz,t) (see Eq. (43)). Following
molecularwavefunctionsintheplanenormaltothemole- the low energy cut-off, the spectra show a strong peak
cule’saxis.Intheparallelorientationthisantisymmetryis at about 1.8 1.9eV, before they slowly fall off towards
−
broken by the surface, which results in a significantly en- zero for larger energies. Examining the curves we further
largedmatrixelement,andhenceinahigherrateofdecay. notethatthespectrumislargerintheparallelorientation
For the natural decay, on the other hand, the decay rate than in the perpendicular orientation. This can again be
is constant, and thus the negative ion fraction eventually attributed to the trapping effect. From Fig. 3 we know
drops to zero (see lower panel of Fig. 3). that in perpendicular geometry the negative ion is gener-
ated and destroyed closer to the surface than in parallel
Animportantpointtonoticeisthattheportionofneg-
orientation. The image potential and hence the trapping
ative ions is about two orders of magnitude higher in the
effect are thus particularly strong in that case which in
surface-induced decay as compared to the natural decay.
turn reduces the efficiency of electron emission.
The natural decay rate must thus be considerably larger
thanthesurface-induceddecayrate.Hence,thelattercan Finally, we calculate the secondary electron emission
be neglected in the present situation. It could, however, coefficientγe,whichistheareabeneathcurvesofthetype
become important for material combinations that allow shown in Fig. 5. We specifically consider Al2O3, MgO,
for projectile turning points closer to the surface. This SiO2, and diamond. The electronic parameters for these
would lead to an increase in the matrix element, which in substances are listed in Table 1. We assume the turning
turn results in a higher decay rate. point formula given in [8] to be valid for all of these ma-
Tocalculatethefractionsofmetastablesn andground terials. Hence, we can neglect the surface-induced decay
∗
statemoleculesn fromEqs.(70a)and(70c),respectively, process.
g
we focus therefore, in view of the above discussion, ex- Figure 6 depicts the variation of the secondary elec-
clusively on the natural decay and neglect the surface- tronemissioncoefficientγe withthemolecule’skineticen-
induced decay channel. The results are depicted in Fig. 4. ergy εkin for the aforementioned materials as calculated
Notethatwiththedecreaseofthefractionofmetastables fromEq.(73).LeavingMgOaside(foradiscussionseebe-
the ground state share almost immediately rises by the low), the γe-coefficients are on the order of 10−1 over the
same amount, which is due to the rather large decay rate whole range of kinetic energies. The rather large values
for natural decay. The fractions n∗ and ng can be related for γe can be attributed to the shape resonance N−2(2Πg)
to the occupancies of the molecular levels n and n by which is not only efficiently formed in front of the sur-
0 1
means of Eq. (71). facebutalsoquicklydecaystherebyreleasinganelectron.
We now turn to the spectrum of emitted electrons, Atomic projectiles, for instance, metastable argon, which
which can be calculated from Eq. (73). Figure 5 shows cannot form a metastable negative ion, will lead to much
the electronic spectrum at t = , again for the case of smaller γe values.
∞
Al O and ε = 50meV, while neglecting the surface- The secondary electron emission coefficient we find
2 3 kin
induceddecaychannel.Inbothmolecularorientationsthe for metastable nitrogen hitting a dielectric surface agrees
spectrum exhibits a strong cut-off for energies below ap- rather well with the γ -coefficient which is required for
e
proximately 1eV. This is a direct consequence of the im- a convergent, self-consistent kinetic simulation of DBDs
age potential, trapping low energy electrons close to the taking this particular surface collision process into ac-
surface, which was incorporated into our calculation by count[3].Inkeepingwithourinitialmotivation,theeffec-