Influence of Nanostructure Geometry on Electronic Properties

Recently, new quantum features have been studied in the area of nanostructured layers. It emerges that properties of nanostructures depend not only on their size but also on their geometry. Particularly, a nanograting (NG) on the surface of the thin layer imposes additional boundary conditions on electron wave function and forbids some quantum states. Density of quantum states reduces. Unlike conventional quantum well, state density per volume, is reduced in the case of NG layer. This leads to changes in electronic properties. Electrons, rejected from forbidden quantum states, have to occupy states with higher energy. In the case of semiconductor layers, electrons rejected from the valence band have to occupy empty quantum states in the conduction band. Such increase in conduction band electron concentration can be termed as geometry-induced doping or G-doping. G-doping is equivalent to donor doping from the point of view of the increase in electron concentration. However, there are no ionized impurities. This preserves charge carrier scattering to the intrinsic semiconductor level and increases carrier mobility with respect to the donor-doped layer. As rejected electrons occupy quantum states with the higher energy, the chemical potential of NG layer increases and becomes NG size dependent. We regard a system composed of NG layer and an additional layer on the top of the NG forming periodic series of p-n junctions. In such system, charge depletion region develops inside the top of NG and its effective height reduces, becoming a rather strong function of temperature T. Consequently, T-dependence of chemical potential magnifies and Seebeck coefficient S increases. Calculations show one order of magnitude increase in the thermoelectric figure of merit ZT relative to bulk material. In the case of metal layers, electrons rejected from forbidden quantum states below Fermi energy, occupy quantum states above Fermi energy. Fermi energy moves up on energy scale and work function (WF) reduces. WF reduction was observed in thin amorphous Au films grown on thermally oxidized Si substrates. WF was measured using Kelvin Probe and PEEM microscope. WF reduction depended on film internal structure and NG sizes. Maximum reduction in WF of 0.56 eV has been obtained.


Introduction
Developments in nanotechnology allow fabrication of densely packed, periodic structures [1][2][3][4]. Recently, ultra-short period nanopore arrays and nanogratings have been obtained by block copolymer lithography [1,2]. Another method for nanograting (NG) fabrication is a multi-beam interference lithography. Using these techniques, gratings with 10-nm [3] and even sub-10 nm pitch [4] were fabricated. At the same time, it has been shown that NG dramatically improves thermoelectric [5] and electron emission properties [6] when grating pitch becomes comparable with electron de Broglie wavelength. This is owing to special boundary conditions imposed by NG on electron wave function. Supplementary boundary conditions forbid some quantum states, and density of quantum states (DOS) reduces (in all bands). Electrons rejected from NG-forbidden quantum states have to occupy empty states with a higher energy. Fermi energy increases and electronic properties of NG layer change.
Qualitatively, DOS reduction can be described by the de Broglie wave reflection. Let us regard a solid/vacuum interface with NG on the surface (figure 1). Let us consider an electron with energy F E E < (where F E is Fermi energy), moving towards the border of the solid as a planar de Broglie wave. It will reflect back from the interface as the electron does not have enough energy to leave the solid. There will be two reflected waves. One reflected from the top of the indent and another from the bottom of the indent. If the indentation depth where λ is electron de Broglie wavelength, two reflected waves will interfere destructively resulting in no reflected wave. Consequently, an electron of certain energy cannot reflect back from the surface. On the other hand, the electron cannot leave solid to vacuum as it does not have enough energy to overcome the potential barrier ( F E E < ). It cannot reflect under arbitrary angle as it is in the Fermi gas in which all quantum states are already occupied. Therefore, all possible final quantum states for that particular electron are forbidden. As all the final states are forbidden, the initial state with wave vector a k 2 / π = is also forbidden. Obviously, all states with wave vector component where p is integer are also forbidden. As a result, DOS near the interface is reduced with respect to the plain solid.
Qualitative description of DOS reduction is demanding. The calculation requires solving of the time-independent Schrödinger equation in the NG geometry. Mathematically, there is no difference between DOS reduction and the electromagnetic (TM) mode depression [7,8]. Helmholtz equation and Dirichlet boundary conditions are used in both cases. Unfortunately, there is no exact analytical solution for NG geometry. The approximate analytical expression known as Weyl's formula [9,10] allows calculation of DOS using a ratio of layer surface and volume. Perturbation method [11] has been used to find an approximate analytical expression for G in NG geometry. DOS for electromagnetic modes in resembling geometries has been numerically calculated in the literature related to the Casimir effect [12].
Influence of periodic structures on electronic properties has been studied in related geometries such as periodic curved surfaces [13,14], nanotubes [15], cylindrical surfaces with non-constant diameter [16] and strain-driven nanostructures [17].

Semiconductor nanograting layers
In the case of semiconductor materials, electrons rejected from the valence band (VB) occupy empty quantum states in the conduction band (CB). Electron concentration n in the CB increases which can be termed as geometry-induced electron doping or G-doping. G-doping is equivalent to donor doping from the point of view of increase in n and Fermi energy. However, there are no ionized impurities. This maintains charge carrier scattering to intrinsic semiconductor level and increases carrier mobility with respect to the donor-doped layer of same electron concentration. G-doping is temperature independent as it originates from layer geometry and no ionized impurities are involved.
Other methods of doping without impurities are well known modulation doping and recently introduced polarization doping [18]. Both are 2D in their nature. However, a 3D approach of modulation doping was introduced in [19] to improve thermoelelectric characteristics of nanocomposites [20].
Electron confinement to the NG layer is needed to obtain G-doping. The layer can be made free standing but in practice it is usually sandwiched between wide band gap layers. NG layer thickness is fundamentally limited by the requirement of having quantum properties. However, thin layers have low optical absorption. Layer thickness also limits lateral charge and heat transport. Figure 2 shows a cross section of a single NG layer. The grating has depth a and period w 2 . To make a comparison, we choose reference layer as plain layer with thickness H such that it has the same cross section area. Nanograting imposes additional boundary conditions on electron wave function and forbids some quantum states. The DOS in energy ) (E ρ reduces [21] with respect to the reference well ( Where ) ( 0 E ρ is the DOS (number of quantum states within the unit energy region and in the unit volume) in a reference well and 1 ) , is the geometry factor. With increasing NG depth DOS reduces.
Nanograting reduces DOS in all bands. Electrons rejected from the NG-forbidden quantum states have to occupy empty quantum states with higher E. In a semiconductor, electrons rejected from the VB have to occupy empty (and not forbidden by NG) energy levels in CB (figure 3). Electrons reject from low energy levels and occupy high energy ones. During this process, Fermi energy increases from To simplify the presentation, in figure 3, we presume that T=0 (T is absolute temperature) and energy levels are equidistant on the energy scale (geometry-induced energy level shift is also ignored).  Figure 3. Energy diagrams of a) reference semiconductor quantum well and b) NG layer. Green lines depict occupied energy levels. Red lines depict NG-forbidden energy levels.
We will employ (1) to calculate the DOS and investigate G -dependence of n and F E . The density of the NG-forbidden quantum states is To determine the number of rejected electrons r n (per unit volume), (2) should be integrated over the electron confinement energy region. con con Where, ∫ = con 0 con . r n Here, we assume that electron confinement takes place only in narrow energy intervals inside which G is energy-independent. In most cases, thin layer is grown on a semiconductor substrate and confinement energy regions are band offsets. There are distinctions and similarities between the NG forbidden quantum state and a hole. The state is forbidden by the boundary conditions and cannot be occupied. However, it is not forbidden in an irreversible way. If the boundary conditions change (e.g., owing to charge depletion), then it can recombine with the electron (like the hole recombines with the electron). As the NG forbidden state is confined to the boundary conditions (macroscopic geometry), it is not localized in the lattice and cannot move like a hole.

Electronic properties of multiple homojunction nanograting layer
To obtain electron confinement in multiple NG layers transitional barrier layers are required. We regard cases of wide band gap material forming homojunction. Confinement regions originate from barrier layer donor doping.
Here, we investigate G-doping in multiple NG layers. Such layers are quasi-3D and have improved optoelectronic and thermoelectric characteristics with respect to a single layer. Multiple NG layers consist of a replicated structure including main and barrier layers forming a series of hetero or homo junctions. Main layer is thicker than a barrier layer and plays a leading role in carrier transport and optical absorption. Barrier layer forms electron confinement energy regions. It also contributes to carrier transport and optical absorption. Both main and barrier layers have NG geometry. One of the objectives of this work is to calculate n and in multiple NG layers and find out how they depend on layer thicknesses and material properties. First, we introduce G-doping in a single NG layer. Next, we calculate n and in a homojunction multilayer structure. Subsequently, we calculate the same for a heterojunction multilayer structure. Finally, the possibility of realization of such structures and their advantages for optoelectronic and thermo-electric devices are discussed. An analysis was made within the limits of parabolic band, wide quantum well and degenerate electron gas approximations.
We begin from calculating n in a homojunction NG layer, as it is relatively straightforward. Homojunction is formed between intrinsic main layer and donor-doped barrier layer. Let us consider a case when all layers have a plain geometry (NG is not formed yet). Figure 4a shows an energy diagram of such structure (typically obtained by homoepitaxal growth) [22]. Charge depletion regions form inside the both i-type main layer and n-type barrier layer. Band edges curve, shaping confinement regions in the main layer VB and in the barrier layer CB. Electrons with energies are confined to main layers and ones with energies are confined to barrier layers. Here, we assume that both main and barrier layers are relatively thick so that electron wave functions do not overlap and we can ignore the mini band formation. Green lines depict occupied energy levels. Red lines depict NG-forbidden energy levels.

Influence of Nanograting
Next, we fabricate NG on the surface of all layers simultaneously (fabrication of such structure will be discussed later in this work). Nanograting forbids some energy levels (red lines) and induces G-doping depending on geometry factor value. We introduce two geometry factors m , the bang edges remain intact in barrier layer. With the downward shift in main layer, VB confinement region will shrink and fewer electrons will reject to the CB. This will continue until some equilibrium value of rejected electrons is achieved. In other words, band edge downward shift provides negative feedback to the G-doping process. Next, let us regard the case Here, barrier layer band edges also shift down. Owing to the downward shift in barrier layer, confinement energy regions widen. This increases the number of rejected electrons in main layer. In other words, barrier layer G-doping provides positive feedback to main layer G-doping. Finally, we have a combination of negative feedback originating from main layer and positive feedback origination from barrier layer. In the case of overall negative feedback G-doping of two layers will saturate and equilibrium value of con E ∆ will be attained (figure 4b). Last depends on m G , b G and semiconductor material parameters.

Electron confinement energy region
To find con E ∆ we have to calculate electron concentrations in both main and barrier layers first. Applying (2) to the main layer we find the number of rejected electrons is the initial (to NG) DOS in the main layer VB. Let us regard the case when the rejected electron number is much more than the initial electron number in CB 0 n n r >> . Then, the CB electron concentration will be merely r n . Further, we use the following well known expression for valence band DOS to calculate r n [23] Here, V N is the effective VB density of states. Inserting (5) in (4) and integrating, choosing zero of energy scale as Further, we find Fermi levels in both layers using corresponding electron concentrations. According to the well known expression for degenerate semiconductor [24] Fermi level in main layer can be found from [ ] where, C N is the effective CB density of states. CB electron concentration of barrier layer does not change during NG formation, as there are no electrons transferred from the VB to the CB. However, Fermi level in the barrier layer moves up on an energy scale owing to NG-induced reduction in C N Here, d n is barrier layer initial electron concentration (obtained by donor doping). Energy diagram (figure 4) shows that . Subtracting (7) from (8) and inserting in the last expression we find Inserting (6) in (9) and depicting we get following nonlinear equation for η Equation (10) was solved numerically to find con E ∆ using two geometry factor and d n , C N , V N values for Si and GaAs materials.

Electron concentration and Fermi level
Numerical solutions of (10) were inserted in (6) to find r n . Next, we found the Fermi level of the main layer by inserting values of r n in (7). Formula (8) was used to find a Fermi level position in barrier layer. Table 1 shows the results for geometry factors m main layer G-doping levels of 10 17 -10 18 cm -3 were obtained. We chose this range of G-dropping levels, as they are frequently used in applications. Higher and lower G-doping levels can be obtained as well. Table 1 shows that one order higher donor doping d n (10 20 cm -3 ) is needed in Si with respect to GaAs (10 19 cm -3 ) to obtain a G-doping level 2.8 10 18 cm -3 . It is convenient to give an interpretation of this result using terms of negative and positive feedback. Negative feedback in main layer is weaker for Si material as it has higher 0 n value with respect to GaAs. Owing to the high value of 0 n , band edges shift down less rapidly with increasing r n , following well known logarithmic dependence  (7) and (8) are valid). For lower doping levels, other expressions should be used instead of (7) and (8).

G-doping for high electron mobility applications
Realization of G-doping is attractive from the point of view of high mobility applications. Let us use a multi junction solar cell [25] as an example to estimate G-doping improvement of its characteristics. In this device, window, emitter and tunnel junction layer doping levels are roughly 10 18 cm -3 . At this level of donor doping, ionized impurities reduce electron mobility by a factor of 4 in GaAs [26] (at T=300 K) and by the factor of 10 in Si [27]. Most types of solar cells use transparent conductive oxides with doping levels of 10 20 -10 21 cm -3 . At this donor doping level ionized impurities reduce electron mobility by the factor of 30-50 in GaAs. Thus, using G-doping in these layers can dramatically improve characteristics of solar cells.
Multiple NG layers can be realized by epitaxial growth on the top of lead grating, previously formed on the substrate. Such structure has been modelled [28] and fabricated [29] on GaAs substrate using epitaxial growth technique. Layers, grown using this technique, have diverse geometry compared to single NG layer. They have gratings from both sides and interfaces resemble sine instead of square. With increasing number of layers sine amplitude reduces and finally one gets plain layers. Geometry factor and G-doping level reduce with increasing of layer number. It is challenging to obtain high G values or considerable DOS reduction in both NG and alike geometries. However, as table 1 shows, G-doping required values are quite close to unity (especially for the main layer). The obtaining of such values in multiple NG systems seems to be straightforward.
The above analysis was made using the assumption of G energy independence (3). Usually, electron confinement energy regions are small (tens or hundreds of meV) and this assumption is valid. However, the NG layer can be made free standing or grown on very wide band gap substrate. In these cases, confinement energy intervals are much wider and care should be taken when assuming G energy independence.

Thermoelectric properties of nanograting layers
Thermoelectric materials are characterized in terms of the dimensionless figure of merit ZT. Here, T is the temperature and Z is given by κ is an electron gas thermal conductivity, and l κ is lattice thermal conductivity. The difficulty in increasing ZT is that materials having high S usually have lowσ . When σ is increased, it leads to an increase in e κ , following Wiedemann-Franz law, and ZT does not improve much. Another approach is to eliminate the lattice thermal conductivity by introducing vacuum nanogap between the hot and cold electrodes [30][31][32] and using electron tunnelling. Cooling in such designs was observed in [33] and theoretically studied in [34]. However, vacuum nanogap devices appear extremely difficult to fabricate. In this work, we present a solution that allows large enhancement of S without changes in e , κ σ , and l κ . It is based on having a series of p-n junctions on the top of the NG. It is another objective of this work is to calculate ZT of NG with series of p-n junctions and compare it with Z0T of reference nanograting (RNG), in which ) , and bulk material. The reduced figure of merit T Z ZT 0 / is calculated and µ dependences are presented for such traditional thermoelectric materials as Si and Ge. Analysis was made within the parabolic bands approximation and the abrupt junction's approximation. Since only heavy G-doping was considered, we neglected the hole contribution in transport.

Charge and heat transport in the NG layer with p-n junctions
Here, we propose a system composed of NG layer and an additional layer on the top of the ridges forming periodic series of p+-n+ junctions ( Figure 5). In such system, charge depletion region and consequently the geometry factor G becomes temperature-dependent, All parameters of NG layer including µ become stronger functions of T than it will be in an NG layer without the junctions. Seebeck coefficient and thermoelectric figure of merit increase.
Analysis was made using the Boltzmann transport equations. First, we calculate µ ∇ for the system of NG layer and junctions and express it as Here, 0 µ ∇ is the chemical potential gradient in an RNG (NG layer without p-n junctions) and J µ ∇ is introduced by the junctions. Next, we insert µ ∇ in Boltzmann transport equations and calculate S for the system of NG layer and junctions, expressing it as J 0 , where 0 S is Seebeck coefficient of RNG and J S is introduced by the junctions. Cross-section of the system of NG and periodic junction is shown in figure 5. We assume that there is a temperature gradient T ∇ in the Y-dimension. Consequently, depletion depth depends on the Ycoordinate, and geometry factor gradient G ∇ appears in the Y-direction. The presence of G ∇ and T ∇ modifies the electron distribution function and causes electron motion from the hot side to the cold side. This motion is compensated by thermoelectric voltage. Let us write Boltzmann transport equations [35]  ( ) where 0 f is the electron distribution function, ) τ(E is the electron lifetime, y v is the electron velocity in the Y-direction, and α = 0, 1, 2. Let us find how NG affects where are the coefficients of bulk material having the value of µ obtained by the conventional doping (for instance). Further, we have to find µ for the system of NG layer and junctions and insert it in (12) together with (14). The chemical potential of degenerated semiconductor ( can be written as [37] (note that frequently Fermi energy is used instead of chemical potential in semiconductor literature) where n is the electron concentration and NG N is the effective conduction band density of states of NG layer. In the case of heavy G-doping ( i n n >> , where i n is intrinsic concentration), we can use (3) Further, we rewrite (16) as defines reference material as n+-type semiconductor with electron concentration of ) 1 ( , or RNG layer having constant Next, we calculate the gradient of (10) taking into account . Inserting (18) and (14) in Boltzmann equations, we find charge and heat currents in the system of NG layer + Junctions Electrical and thermal conductivity in the system of NG layer and junctions remain unaffected (with respect to RNG layer or bulk material having the same µ value) by a series of junctions, and S change according to (21). To calculate θ and then S, we have to find T G ∂ ∂ / first (19). Such performance of transport coefficients has following physical interpretation. The geometry factor G is a rather strong function of temperature. Consequently, T-dependence of chemical potential magnifies and Seebeck coefficient, having approximate value is G independent since, the reduction of state density leads to a proportional increase of lifetime, according to Fermi's golden rule (resulting in (14)). In a degenerate electron gas (n+-type semiconductor) Wiedemann-Franz law works well. Thus, the thermal conductivity e κ follows σ and is also G independent.

Seebeck coefficient of NG layer with p-n junctions
It is reasonable to calculate the Seebeck coefficient of the NG layer with junctions relative to RNG layer, in which geometry factor is constant 0 . This will allow comparison of dimensionless figure of merit ZT with Z0T, using similarity of electric (23) and heat conductivities (22). In (17) (16). Inserting this in (19) and further inserting the obtained result in (21), we obtain for the Seebeck coefficient of the NG layer with junctions 3D Seebeck coefficient of reference RNL does not differ from the Seebeck coefficient of the bulk material, and within the parabolic band approximation it is equal to [38] Here, r refers to scattering parameter and it is assumed that electron lifetime is the reduced chemical potential, and *) F(µ are Fermi integrals. In the case of no impurities (G-doping) and low energies, acoustic phonons are responsible for electron scattering and 0 = r . It should be noted here that NG layer has G times more width [21] with respect to the conventional layer. As 3D ) (E ρ in wide quantum wells tends to ) (E ρ of the bulk material of the same width, using (28) for RNG is a good approximation. However, in the case of thin layers oscillatory behaviour of transport coefficients should be considered [39].
To find the ratio 0 / Z Z , we use relations (22) and (23)  was used to convert X-axis so that in the same figure is calculated according to (26). We present µ dependences as they allow the understanding of possible µ ranges, within which real devices can operate without changing the sign of Ge, we choose CON n so that, for value G=10, µ was close to the optimum value 2 / 1 / B OPT + = r T k µ [38]. It should be noted here that NG layer exhibits quantum properties at G times more widths with respect to the conventional quantum well. Figure 6 shows large enhancement of figure of merit in p+-n+ junction NG layer both for Si and Ge materials. The dependences are quite identical for materials having rather different band gaps (1.12 eV for Si and 0.66 eV for Ge). This shows that despite the band gap value entering expressions for depletion depth (Reference 5) and ) dependencies can be obtained by matching such parameters as β and γ .

Metal nanograting layers
In the case of metal layers, electrons rejected from forbidden quantum states below Fermi energy, occupy quantum states above Fermi energy. Fermi energy moves up on energy scale and work function reduces.
Low work function (WF) electrodes [40] are essential for cold emission and room temperature operation of thermionic [41,42] and thermotunnel [43][44][45]  some compounds show φ e =2-3 eV. This is one order of magnitude higher than required. WF values of about 1 eV were obtained in sophisticated systems like Mo-Cs and Ag-O-Cs. However, these types of electrodes have a short lifetime even in good vacuum conditions. To overcome difficulties, quantum mechanical tunnelling was utilized. Tunnelling through vacuum nanogap allows sufficiently large currents from the electrodes, having relatively high φ e values. It was found that image force reduces the potential barrier and increases tunnelling current, giving a cooling power of 100 W/cm 2 for 1 ≈ φ e eV [43] in mixed thermotunnel and thermionic regime. Electrons were filtered by collector coating, to increase the cooling coefficient [44] and confor mal electrode growth technology was developed [45]. However, the vacuum nanogap device appears extremely difficult to fabricate as it requires an electrode spacing of 5-10 nm. Earlier, WF reduction in Au, Nb and Cr metal NG layers was observed experimentally [46]. If an φ e <1eV electrode could be obtained, poor thermionic regime can be realized at increased electrode spacing. Fabrication of wide vacuum gap, using conformal electrode technology, is much more straightforward. Low WF energy converters are essential for heat to electricity direct conversion in the temperature range 400-1000 K in which waste heat is available from the combustion sources. Low WF coolers are essential for integrated circuit cooling and other applications where low weight and ecological purity are major requirements.
Here, we calculate φ e in the metal NG layer, grown on semiconductor and metal substrates, and find out how it depends on NG dimensions and electron confinement. In the beginning we calculate φ e values in the metal NG layer forming a Schottky barrier or ohmic contact with a semiconductor substrate. Next, we calculate φ e in the metal NG layer grown on a metal substrate. Finally, the possibility of realization of such electrodes using conventional thermionic and other materials is discussed. An analysis was made within the limits of parabolic band, wide quantum well and degenerate electron gas approximations.

Work function of metal NG layer grown on a semiconductor substrate
To maintain the uniform vacuum nanogap over the whole area, electrodes should have plane geometry and smooth surface. The simplest solution is to use semiconductor substrate as an electrode base [47] and grow a thin metal NG layer on it.
The Energy diagram of the metal film grown on n+ type semiconductor substrate is shown in Figure 7a. We begin from the case when the difference between metal initial work function 0 φ e and is satisfied. Then, electrons having energies within the region (27) are confined to the metal film. Within ) 0 ( con E ∆ , quantum states for electrons are filled in the metal film and forbidden in the semiconductor substrate. Next, we fabricate ridges on the metal film surface (Fig. 7b) The φ e value was calculated using electron number conservation in the NG layer conduction band.
The number of electrons rejected from the forbidden quantum states was equal to the number of electrons accommodated above the initial Fermi level ACC n . The rejected electron number, according to Eq. (3) was In Eq. (28), we take into account that electrons were rejected from the energy interval . They were accommodated in the interval . If r n was low, the last interval fit within the con E ∆ (Fig. 7b) where the density of states is reduced Eq. (1). The accommodate electron number was equal to the number of empty quantum states between (0) Using condition ACC r n n = and putting in , which is true within the limit of parabolic band approximation, we found after integration and simplification that Here, we use 3D density of states changed sign as shown in figure 7c. Here, r n was calculated using Eq. (28) again.
However, in this case, electrons were accommodated in quantum states from two different energy intervals. The primary interval was above (0) F E and within the energy where the DOS was reduced, and the secondary interval was above (0) F E and out of con . The two intervals differ by the DOS. The DOS was equal to Eq. (1) in the primary interval and ) ( 0 E ρ in the secondary one. Consequently, ACC n was

Material pairs for nanograting layer coated electrode
The thickness of a conventional metal quantum well is only 1-5 nm [48]. NG fabrication on such thin layer surface seems complicated. However, owing to reduced quantum state density, metal NG layer retains quantum properties at G times more thickness (as found in Ref. 21) with respect to a conventional metal quantum well. This makes NG fabrication straightforward. In some cases, the layer thickness can be increased even more than G times. As figure 7 shows, confined electrons have energies Consequently, electron scattering on phonons and lattice defects is limited. Such scattering requires the exchange of a small portion of energy with the environment, which is quantum mechanically restricted as all nearby quantum states are occupied. The mean free path of such electrons is very large and is limited only by structural defects, such as grain boundaries. In epitaxial films, such electrons allow phase coherence at large distances (in transverse direction) and metal NG thickness can be increased. In this work, we do not concentrate on thermal processes in the metal NG layer by following reasons. Transport coefficients in the NG have the same values as in a conventional NG layer (having the same value of F E ) [5]. In vacuum device, the temperature gradient is applied mainly to the vacuum gap and only significant thermal process inside the electrode is Joule heating [49]. Such heating inside the metal NG layer is negligible. We regard only n+ semiconductor substrates in which, the difference F C E E − is low and, heat absorption or release at the metal/semiconductor interface is insignificant.
Regular semiconductor materials can be used as base substrate. Thermionic and thermotunnel converters are high current low voltage devices. Consequently, the main limitation is electrical conductivity. Equations (30) and (32) show that the wide band gap material allows more reduction in φ e . Unfortunately, it is problematic to achieve low resistivity in such materials. The most promising seems GaN in which relatively low resistivity was obtained. Other possible substrate materials are GaAs and Si. Table 2 shows possible donor concentrations D N for GaN [50] , Si, GaAs [23] and corresponding . Table 3 shows the parameters of some metals frequently used as electrode materials (other interesting metals are included as well), collected from the literature:  scale. The last can only be increased by increasing band gap. However, wide band gap substrates have high resistivity and were not acceptable for thermotunnel and thermionic devices. Obviously 1-2.3 eV reduction in φ e was not enough for cold emission, but it was still interesting since the plain Mo and Ni electrodes, coated with an ultra thin layer of Cs atoms, show very low WF (1-1.5 eV). WF reduction by less than mono layer of Cs atoms is a surface effect. At the same time, quantum state depression is not a surface effect. Most probably, the two mechanisms of WF reduction will sum up and result in an φ e considerably less than 1 eV.
The same is true for Ag-Ba and W-Li electrodes.
Borides have 4 0 < φ e eV and for them 0 (Fig. 7). The most frequently used is Lanthanum hexaboride LaB6, which shows φ e =2-3.2 eV. Fermi energy for LaB6 is Band was used in all cases). These values were low enough for thermotunnel and thermionic devices operating at room temperatures.
Geometry factors, at which ideal ohmic contact was obtained, were calculated using Eq. (34). They were in the range for most material pairs. Such 0 G can easily be obtained in practice. Exceptions were Ni/Si ( 59 0 = G ) and Au/Si ( 16 0 = G ) pairs.

Experimental investigation of metal nanograting layers
Thin gold films, having NG on both sides, were prepared to observe the WF reduction ( Figure 8).
Gold was the material of choice because it does not form a natural oxide on the surface and allows exposure of the films to the atmosphere. Au film was vacuum deposited on a Si/SiO2 (dry thermal oxide) substrate in which NG was fabricated using both e-beam lithography and photolithography. Au was grown on a variable temperature NG substrate (T=80-300 K) using the quench deposition technique. Previously, the substrate was cooled to temperatures between -16 C and -22 C, and a thin film of 2-3 nm thickness, evaporated from a mixture of Au and Cr, was deposited on SiO2 to form an adhesive layer (not shown on figure 8 for simplicity). Following the deposition, the wafer was moved rapidly (maximum 5 s) to another location, where an Au film of thickness of 60 nm was deposited using the rapid thermal evaporation of Au wire (99.999% purity). In other cases NG in Au was made only on the surface of the film using etching techniques, while film base remained plain. Finally, large structures such as NG and plain circles and chess desk like areas containing both NG and plain squares were fabricated using photolithography. Measurements of the work function were made using Kelvin Probe (KP) method and Photo Electron Emission Microscope (PEMM). All measurements were comparative to exclude absolute inaccuracies: KP reading from the NG area was compared to the reading from the plane area of the film. PEEM images were collected using high-resolution photoelectron microscope [62]. The internal structure of the films was analyzed using X-ray diffraction.
For all samples measured, the indented regions showed a reduced work function (WF) compared to the plain regions. The magnitude of reduction depended on the structure of the film and the dimensions of NG. The internal structure of the film was varied by changing substrate temperature. Amorphous films show a much higher reduction in WF than films with polycrystalline structure. For amorphous films the reduction of WF was in the range of 0.2-0.5 eV. All polycrystalline films show a WF reduction less than 0.1 eV. The reduction in WF was more pronounced for samples that were deposited in a cleaner environment (by plasma cleaning the deposition chamber prior to deposition). Residual gas pressure and composition has considerable influence on the results. One order difference in WF reduction shows that the internal structure of the film has principal importance.
It was observed that the strength of the effect depends on the depth of the indents. Amorphous Au films with thickness of 60 nm, and an indent depth of 50, 20 and 10 nm, show WF reduction of 0.16 eV, 0.25 eV and 0.56 eV respectively.
A WF reduction of 0.1 eV was observed in NG films grown on plain substrate. Low value of WF reduction could be explained by the fact that Au films were deposited on room temperature substrate. Another explanation is the difference in geometry. One of the samples was prepared so that it contained five areas each having a different surface geometry. The first area was plain and the other four areas had various indent widths of 800, 600, 400, 200 nm, but the same depth and length. Comparative measurements made between 1-2, 1-3, 1-4, 1-5 show that WF difference increases with reduction of indent width.
Reduction of WF of 0.2 eV was also observed in the indented areas of SiO 2. The measurement was made on several samples and yielded almost identical results.
PEEM measurements show that NG areas in all microscope images appeared brighter than adjacent plain areas (two of them were placed as on chess desk). Brighter areas corresponded to more electron emission from NG areas. This was in quantitative agreement with KP measurements. Unfortunately

Conclusions
Geometry-induced electron doping (G-doping) was investigated in multiple NG layers. They were composed of main and barrier layers forming a series of isotype homo junctions. Barrier layers were used to form electron confinement energy regions. Both main and barrier layers were n-type. Barrier layers were donor-doped to obtain electron confinement and induce G-doping in main layers. Such parameters as electron concentration, Fermi level and confinement region width were calculated for this system. Main layer G-doping level of 10 17 -10 18 cm -3 was obtained at barrier layer donor doping of 10 18 -10 20 cm -3 and geometry factor values m G =1.02, 1 . 1 = b G . One order higher donor doping was required in Si with respect to GaAs to obtain the same G-doping level. For high electron mobility it was preferable to have high values of geometry factors. Electron mobility was higher in G-doped layers with respect to donor-doped layers of same doping level. G-doping opens prospects for new quasi-3D optoelectronic and thermoelectric systems. At the same time G-doping is temperature independent and can be used to extend working temperature ranges of cryogenic and power electronics.
Thermoelectric transport coefficients were investigated in the system of NG layers and periodic series of p-n junctions at the top of the ridges. An analysis was made on the basis of Boltzmann transport equations. It was shown that the Seebeck coefficient increases considerably. At the same time, electrical and thermal conductivities remain unaffected by the series of junctions. This allows large enhancement of thermoelectric figure of merit. Dependence of Seebeck coefficient on the geometry factor G and junction parameters was investigated and the analytical expression was obtained. Seebeck coefficient changes sign for some value of G. Dependences of S and ZT on chemical potential were presented for p-n junction NG layers. Calculations show one order of magnitude increase in thermoelectric figure of merit with respect to the bulk material.
Low work function was obtained in the metal NG layer having a reduced quantum state density. Metal LG layers, grown on semiconductor and metal substrates, were analyzed. Electron confinement to the NG layer was essential in both cases. When using semiconductor substrate, the wide band gap material allows more electron confinement and lower values of the resulting φ e . Dependence of φ e on the band gap was analyzed for a number of cases and the corresponding formulae derived. Thin Au films, having NG on both sides, were prepared using vacuum deposition, e-beam lithography and photolithography. WF reduction was measured using Kelvin Probe and verified using PEEM microscopy. Film internal structure was monitored using X-ray structural analysis. WF reduction dependence on film internal structure and NG dimensions were recorded. WF reduction of 0.56 eV was observed in Au NG films with amorphous internal structure.

Acknowledgements
Author thanks Ismat S. Shah and Juejun Hu for useful discussions. Apparatus received from EU TEMPUS project 530278-TEMPUS-1-2012-1-DE-TEMPUS-JPHES was used for preparation of this work.