Theory of Quasi Zero-Dimensional Localization
4.4. Localization with a holding field
trough, a simple Drude-like model would prove useful, if discrepancies between the model and the experimental data could demarcate effects due to Coulomb interactions. In any case, before elaborating, it is always useful to examine how much can be explained by simple means, but no simpler.
4.4. Localization with a holding field
Assume that an electron resides above a flat helium film and a holding field E⊥ is present. Then, the potential will contain a term−eE⊥z, wherezis the distance of the electron from the helium film surface, and a term−Λs/d, where dis the thickness of the helium film andΛshas been defined in eq. (2.1).
If one imagines that the film surface is deformed such that troughs develop, whose depression is measured by∆d(x, y)60as defined in eq. (4.18), the influ-ence of the substrate in the area of a trough will be given by−Λs/ z+d(x, y)
, withd(x, y)defined in eq. (4.10) and there will also exist an additional potential termeE⊥∆d(x, y)< 0. This additional potential term reflects the fact that a depression is an area of lower potential energy when a holding field ‘pushes’
electrons towards the bottom plate. The total potential of this setting would then be written down as
V(x, y, z) = −Λ
z −eE⊥z− Λs
z+d(x, y)+eE⊥∆d(x, y). (4.21) Observe that for a flat substrate it is∆d(x, y) =0andd(x, y) =dand eq. (4.21) is reduced in a form similar to the potential of eq. (2.2).
At the limit of thick films (zd) a Taylor expansion of z+d(x, y)−1
up to first order terms leads to
V(x, y, z) = −Λ
From eq. (4.22) it is seen that the potential is separable into a component V(z)for transverse and a componentV(x, y)for lateral electron motion. Film thicknessd(x, y)can be substituted by either a constant value inΛs/d2, or the whole contributionΛs/d2can be dropped out completely, because it is much smaller than the already small perturbation coefficienteE⊥. Then, Schrödinger equation can be solved separately forV(z)andV(x, y), yielding the common Stark-shifted surface states of TDES on liquid helium in thez-direction, as it was intuitively claimed in the beginning of this chapter.
Electron dynamics in the(x, y)plane will be determined from theV(x, y) part of eq. (4.22). Substitution of∆d(x, y)as approximated in eq. (4.18) into V(x, y)and the additional approximation
d(x)−1≈ 1
which can be expressed as
d(x)−1≈8R b2 −32R
b4 x2, (4.24)
after making use ofK1/R −1≈−b2/8R2, accompanied by algebraic manipu-lations that will not be reproduced, leads ultimately to the harmonic oscillator potential
On a ridge,∆d(x, y) =0and the film thickness equals the constant value dtop. Consequently, the potential on the ridges will be just the constant−Λs/dtop. A redefinition of the potential by addingΛs/dtopon both ridges and troughs leads to zero potential on a ridge andV(x, y)as in eq. (4.25) in a trough.
From eq. (4.25) it is evident that a parabolic potential well of depth Λs/(dtop+δ0) −mω2Eb2/8is formed. Except for the dimension of the trough, which is reflected in the diameterb, the depth of the parabolic well is determined by the thicknessdtopof the helium film on the ridges and the cyclic frequency ωE. In turn,ωEis the sum of a termωs that depends on the nature of the substrate and a termωethat depends on the applied holding field7.
It is seen from eqs. (4.25) and (4.26) that an increasing holding fieldE⊥ deepens the potential well and therefore enhances electron localization in the troughs. For the same holding field and electron density, stronger dielectric substrates (largerΛs) increase the depth of the potential well, because electrons interact stronger with the substrate.
Substitution of numerical values for the physical quantities in eq. (4.26) reveals thatωsis several orders of magnitude larger thanωe. As a result, elec-tron localization in the troughs of helium films that wet periodically structured substrates is determined almost exclusively from the interaction of electrons with the dielectric substrate.
The hamiltonian for an electron in a trough can be written down as H^ = p^2
This is the hamiltonian of a two-dimensional harmonic oscillator shifted by a constant energy. The energy shift will be carried over in the energy spectrum,
7If it is assumed that electron density corresponds to the saturated electron density, the radius of curvatureRof the charged surface, which is present in bothωsandωe, will be a function of the holding field as well.
4.4. LOCALIZATION WITH A HOLDING FIELD 95
which will be given by
Eν= (2ν+1)hωE+ Λs
dtop+2δ0
−mω2Eb2
8 , ν=0, 1, . . . (4.28) From the form of eigenenergies it is seen that the discrete density of states, taking into account the two possible orientations of electron spin, is simply D(Eν) =2(2ν+1). The density of states in the continuous limit of the Thomas-Fermi approximation is easily found to beD(E)dE= (1/hωE)dE, where again electron spin states were taken into account.
If the number of electrons localized in the troughs is denoted withNloc, and the potential in the troughs spans the energy range from−V0(bottom of the potential well) up to0, one can write, in accordance with eq. (4.19),
Nloc(µ, β) = 1 It is important to note that the lower integration limit (−V0) does not coin-cide with the bottom of the potential well,Λs/(dtop+2δ0) −mω2Eb2/8, but it is higher by the amounthωE, which is the energy due to the zero-point motion of the quantum harmonic oscillator with two degrees of freedom. For very weak dielectric substrates and holding fields, the cyclic frequencyωEmight not be high enough to makeV0> 0, whereby the potential well is so shallow that no electrons would be localized in the troughs anyway. In this case,Nloc=0and eq. (4.29) is no longer valid.
Realistic dielectric substrates (e.g silicon, withs=11) are always associated with sufficiently deep parabolic wells, where the Hartree-Fock continuum ap-proximation is well justified. As an example, one can calculate that the potential well of a trough with diameterb=800nm of a suspended helium film that wets a silicon substrate and is formed0.5mm above bulk liquid helium, can contain a total of 4525 quantahωE.
Holding fields for which eq. (4.29) can be applied, being thus capable of causing electron localization in the troughs, should be stronger than the critical fieldE(c)⊥ . The critical field is determined from the conditionV0=0, which leads to a criticalωEcalculated from eq. (4.29). Substitution of the criticalωEin the relevant expression of eq. (4.26) yields the critical holding field
E(c)⊥ =16h2R
Except for the limiting cases of a weak dielectric substrate (Λs≈0) or extremely high electron densities that would makeRsmall enough, a potential well can be formed for any (positive) holding field.
The integration in the expression forNloc, eq. (4.29), can be carried analyti-cally, furnishing the final expression for the number of localized electrons in the presence of holding field,
For progressively deeper potential wells (V0−→∞), the number of localized electrons is independent of temperature and it behaves asymptotically asNloc∼ ωE. At the limit of high (β−→0) and low (β−→∞) temperatures indeterminate forms arise that can be easily dealt with an application of L’ Hôpital rule8, finding that for high temperatures it isNloc∼0, whereas for low temperatures it is Nloc∼ωE.
As electrons on the ridges are free (the potential there is almost zero), both their hamiltonian and their energy spectrum should be those of an electron on the surface of bulk helium.
The density of states of an electron, moving with two degrees of freedom in the Thomas-Fermi approximation, isD(E)dE= (m/πh2)dE, if the two possi-ble spin states are taken into account. The numberNfr(µ, β)of free electrons, according to eq. (4.19), will be then
Nfr(µ, β) =mb2 At the limit of low temperatures (β−→∞) the number of free electrons tends asymptotically to zero asNfr∼1/β. The limit of high temperatures (β−→
0) has the asymptotic formeβµ/β, which is an indeterminate form0/0, sinceµ tends to−∞at high temperatures. The indeterminacy is removed, if one recalls that the dependence of chemical potential on temperature is logarithmic at the high-temperature limit, so thatµ∼lnβ. This implies thatNfr∼eβlnβ/β, which tends to infinity at the limit of high temperatures, as one would intuitively expect.
A trough extends in an areaπb2/4, and in equilibrium the total number of electrons at this area will bensπb2/4, wherensis the saturated electron density.
Of these electrons,Nlocare localized andNfrfree. In equilibrium, according to eq. (4.20), it will be
Nloc(µ, β) +Nfr(µ, β) =πb2
4 ns. (4.34)
8The asymptotic expression derived from L’ Hôpital rule contains combinations ofβµwhose asymptotic properties, as known from statistical mechanics, areβµ−→−∞forβ−→0, and βµ−→0forβ−→∞. The chemical potential forT=0is the Fermi energy.
4.4. LOCALIZATION WITH A HOLDING FIELD 97 nloc/nsof localized electrons, as a function of temperature Tat the limitΛs→0(no localized at best. Scale of the y-axis is logarithmic.
The only unknown parameter in eq. (4.34) is the chemical potential, whose valueµ0in equilibrium can then be determined. Inspection of the equilibrium equation (4.34) suggests that an analytic expression forµ0is not possible, and nu-merical methods, like Newton’s method, should be employed instead. Provided thatµ0has been determined, the partial fraction of free electrons in equilibrium for a given temperature and holding electric field is
nfr
while the partial fraction of localized electrons at the same conditions of temper-ature and holding electric field will be
nloc
The numerical solution of eq. (4.34) for realistic dielectric substrates (e.g silicon, or silicon oxide) indicates complete electron localization in the troughs, due to the very strong interaction of electrons with the substrate.
At the limitΛs−→0, which corresponds to weak dielectric substrates, or no substrate at all, it turns out that a significantly smaller fraction of electrons is localized in the troughs. Fig. 4.4 depicts the behavior ofnloc/nswith respect to temperature and magnitude of holding field, as derived from the numerical solution of eq. (4.34) forΛs−→0. Stronger fields (higher electron densities) are able to localize a larger fraction of electrons. Across the various holding fields, lower temperatures are generally associated with more localized electrons.
A striking feature of fig. 4.4 is that extremely strong holding fields are re-quired for an appreciable fraction of localized electrons in the case of weak dielectric substrates. Holding field magnitudes in the order of106V/m imply
that voltages in the kV range must be applied between the bottom and the top plate. In practice, this means that in the case where no dielectric substrate is present, electrons on deformed films should be in principle free as they cannot be localized by means of a holding field alone.
On the other hand, dielectric substrates like silicon interact so strongly with TDES on deformed films that most, if not all electrons, should be already localized in the troughs even for the weakest holding fields.