Electronic Properties of Semiconductor Interfaces

Part of the Springer Handbooks book series (SPRINGERHAND)

Abstract

In this chapter, we discuss electronic properties of semiconductor interfaces. Semiconductor devices contain metal–semiconductor, insulator–semiconductor, insulator–metal, and/or semiconductor–semiconductor interfaces. The electronic properties of these interfaces determine characteristics of the device. The band structure lineup at all these interfaces is determined by one unifying concept, the continuum of interface-induced gap states (IFIGS ). These intrinsic interface states are the wave function tails of electron states that overlap the fundamental band gap of a semiconductor at the interface; in other words, they are caused by the quantum mechanical tunneling effect. IFIGS theory quantitatively explains the experimental barrier heights of well-characterized metal–semiconductor or Schottky contacts as well as the valence-band offsets of semiconductor–semiconductor interfaces or semiconductor heterostructures. Insulators are viewed as semiconductors with wide band gaps.

In his pioneering article entitled Semiconductor Theory of the Blocking Layer, Schottky [8.1] finally explained the rectifying properties of metal–semiconductor contacts, which had first been described by Braun [8.2], as being due to a depletion of the majority carriers on the semiconductor side of the interface. This new depletion-layer concept immediately triggered a search for a physical explanation of the barrier heights observed in metal–semiconductor interfaces, or Schottky contacts as they are also called in order to honor Schottky’s many basic contributions to this field.

The early Schottky–Mott rule  [8.3, 8.4] proposed that n-type (p-type) barrier heights were equal to the difference between the work function of the metal and the electron affinity (ionization energy) of the semiconductor. A plot of the experimental barrier heights of various metal–selenium rectifiers versus the work functions of the corresponding metals did indeed reveal a linear correlation, but the slope parameter was much smaller than unity [8.4]. To resolve the failure of the very simple and therefore attractive Schottky–Mott rule, Bardeen [8.5] proposed that electronic interface states in the semiconductor band gap play an essential role in the charge balance at metal–semiconductor interfaces.

Heine [8.6] considered the quantum-mechanical tunneling effect at metal–semiconductor interfaces and noted that for energies in the semiconductor band gap, the volume states of the metal have tails in the semiconductor. Tejedor and Flores [8.7] applied this same idea to semiconductor heterostructures where, for energies in the band-edge discontinuities, the volume states of one semiconductor tunnel into the other. The continua of interface-induced gap states (IFIGS) , as these evanescent states were later called, are an intrinsic property of semiconductors and they are the fundamental physical mechanism that determines the band-structure lineup at both metal–semiconductor contacts and semiconductor heterostructures: in other words, at all semiconductor interfaces. Insulator interfaces are also included in this, since insulators may be described as wide-gap semiconductors. Figure 8.1 shows schematic band diagrams of an n-type Schottky contact and a semiconductor heterostructure.
Fig. 8.1a,b

Schematic energy-band diagrams of (a) metal–semiconductor contacts and (b) semiconductor heterostructures. WF, Fermi level; ΦBn, barrier height; Wv and Wc, valence-band maximum and conduction-band minimum, respectively; ΔWv and ΔWc, valence- and conduction-band offset, respectively; i and b, values at the interface and in the bulk, respectively; r and l, right and left side, respectively

The IFIGS continua derive from both the valence- and the conduction-band states of the semiconductor. The energy at which their predominant character changes from valence-band-like to conduction-band-like is called their branch point . The position of the Fermi level relative to this branch point then determines the sign and the amount of the net charge in the IFIGS. Hence, the IFIGS give rise to intrinsic interface dipoles. Both the barrier heights of Schottky contacts and the band offsets of heterostructures thus divide up into a zero-charge-transfer term and an electric-dipole contribution.

From a more chemical point of view, these interface dipoles may be attributed to the partial ionic character of the covalent bonds between atoms right at the interface. Generalizing Pauling’s [8.8] electronegativity concept, the difference in the electronegativities of the atoms involved in the interfacial bonds also describes the charge transfer at semiconductor interfaces. Combining the physical IFIGS and the chemical electronegativity concept, the electric-dipole contributions of Schottky barrier heights as well as those of heterostructure band offsets vary proportional to the difference in the electronegativities of the metal and the semiconductor and of the two semiconductors, respectively.

The theoreticians appreciated Heine’s IFIGS concept at once. The initial reluctance of most experimentalists was motivated by the observation that the predictions of the IFIGS theory only marked upper limits for the barrier heights observed with real Schottky contacts [8.9]. Schmitsdorf et al. [8.10] finally resolved this dilemma. They found a linear decrease in the effective barrier heights with increasing ideality factors for their Ag/n-Si(111) diodes. Such behavior has been observed for all of the Schottky contacts investigated so far. Schmitsdorf et al. attributed this correlation to patches of decreased barrier heights and lateral dimensions smaller than the depletion layer width [8.10, 8.11]. Consequently, they extrapolated their plots of effective barrier height versus ideality factor to the ideality factor determined by the image-force or Schottky effect [8.12] alone; in this way, they obtained the barrier heights of the laterally homogeneous contacts. The barrier heights of laterally uniform contacts can also be determined from capacitance–voltage measurements (C ∕ V) and by applying ballistic-electron-emission microscopy (BEEM ) and internal photoemission yield spectroscopy (IPEYS ). The I ∕ V, C ∕ V, BEEM, and IPEYS data agree within the margins of experimental error.

Mönch [8.13] found that the barrier heights of laterally homogeneous Schottky contacts as well as the experimentally observed valence-band offsets of semiconductor heterostructures agree excellently with the predictions of the IFIGS-and-electronegativity theory.

8.1 Experimental Database

8.1.1 Barrier Heights of Laterally Homogeneous Schottky Contacts

I ∕ V Characteristics

The current transport in real Schottky contacts occurs via thermionic emission over the barrier provided the doping level of the semiconductor is not too high [8.14]. For doping levels larger than approximately 1018 cm−3, the depletion layer becomes so narrow that tunnel or field emission through the depletion layer prevails. The current–voltage characteristics then become ohmic rather than rectifying.

For thermionic emission over the barrier, the current–voltage characteristics of n-type Schottky contacts may be written as [8.13]
$$I_{\text{te}}=I_{\mathrm{s}}\exp\left(\frac{e_{0}V_{\mathrm{c}}}{nk_{\mathrm{B}}T}\right)\left(1-\exp\left(\frac{-e_{0}V_{\mathrm{c}}}{k_{\mathrm{B}}T}\right)\right)\;,$$
(8.1)
where
$$I_{\text{s}}=AA_{\mathrm{R}}^{\ast}T^{2}\exp\left(\frac{-\Phi_{\text{Bn}}^{\text{eff}}}{k_{\mathrm{B}}T}\right)$$
(8.2)
is the saturation current and A is the diode area, A R * is the effective Richardson constant of the semiconductor, and kB, T, and e0 are Boltzmann’s constant, the temperature, and the electronic charge, respectively. The effective Richardson constant is defined as
$$A_{\mathrm{R}}^{\ast}=\frac{4\uppi e_{0}k_{\mathrm{B}}m_{\mathrm{n}}^{\ast}}{h^{3}}=A_{\mathrm{R}}\frac{m_{\mathrm{n}}^{\ast}}{m_{0}}\;,$$
(8.3)
where \(A_{\mathrm{R}}={\mathrm{120}}\,{\mathrm{A{\,}cm^{-2}{\,}K^{-2}}}\) is the Richardson constant for thermionic emission of nearly free electrons into vacuum, h is Planck’s constant, and m0 and m n are the vacuum and the effective conduction-band mass of electrons, respectively. The externally applied bias Va divides up into a voltage drop Vc across the depletion layer of the Schottky contact and an IR drop at the series resistance Rs of the diode, so that \(V_{\mathrm{c}}=V_{\mathrm{a}}-IR_{\mathrm{s}}\). For ideal (intimate, abrupt, defect-free, and, above all, laterally homogeneous) Schottky contacts, the effective zero-bias barrier height Φ Bn eff equals the difference Φ Bn hom  − δΦ if 0 between the homogeneous barrier height and the zero-bias image-force lowering [8.13]
$$\delta\Phi_{\mathrm{if}}^{0}=e_{0}\left[{\frac{2e_{0}^{2}N_{\mathrm{d}}}{(4\uppi)^{2}\varepsilon_{\infty}^{2}\varepsilon_{\mathrm{b}}\varepsilon_{0}^{3}}\left(e_{0}\left|V_{\mathrm{i}}^{0}\right|-k_{\mathrm{B}}T\right)}\right]^{1/4}\;,$$
(8.4)
where Nd is the donor density, \(e_{0}|V_{\mathrm{i}}^{0}|\) is the zero-bias band bending, ε and εb are the optical and the bulk dielectric constant, respectively, and ε0 is the permittivity of vacuum. The ideality factor n describes the voltage dependence of the barrier height and is defined by
$$1-\frac{1}{n}=\frac{\partial\Phi_{\mathrm{Bn}}^{\mathrm{eff}}}{\partial e_{0}V_{\mathrm{c}}}\;.$$
(8.5)
For real diodes , the ideality factors n are generally found to be larger than the ideality factor
$$n_{\mathrm{if}}=\left({1-\frac{\delta\Phi_{\mathrm{if}}^{0}}{4e_{0}|V_{\mathrm{i}}^{0}|}}\right)^{-1}\;,$$
(8.6)
which is determined by the image-force effect only.
The effective barrier heights and the ideality factors of real Schottky diodes fabricated under experimentally identical conditions vary from one specimen to the next. However, the variations of both quantities are correlated, and the Φ Bn eff values become smaller as the ideality factors increase. As an example, Fig. 8.2 displays Φ Bn eff versus n data for Ag/n-Si(111) contacts with ( 1 × 1)i-unreconstructed and ( 7 × 7)i-reconstructed interfaces [8.10]. The dashed and dash-dotted lines are the linear least-squares fits to the data points. The linear dependence of the effective barrier height on the ideality factor may be written as
$$\Phi_{\mathrm{Bn}}^{\mathrm{eff}}=\Phi_{\mathrm{Bn}}^{\mathrm{nif}}-\varphi_{\mathrm{p}}(n-n_{\mathrm{if}})\;,$$
(8.7)
where Φ Bn nif is the barrier height at the ideality factor nif. Several conclusions may be drawn from this relation. First, the Φ Bn eff  − n correlation shows that more than one physical mechanism determines the barrier heights of real Schottky contacts. Second, the extrapolation of Φ Bn eff versus n curves to nif removes all mechanisms that cause a larger bias dependence of the barrier height than the image-force effect itself from consideration. Third, the extrapolated barrier heights Φ Bn nif are equal to the zero-bias barrier height Φ Bn hom  − δΦ if 0 of the laterally homogeneous contact.
Fig. 8.2

Effective barrier heights versus ideality factors determined from I ∕ V characteristics of Ag/n-Si(111)-( 7 × 7)i and -( 1 × 1)i contacts at room temperature. The dashed and dash-dotted lines are the linear least-squares fits to the data. (After [8.10])

The laterally homogeneous barrier heights obtained from Φ Bn eff versus n curves to nif are not necessarily characteristic of the corresponding ideal contacts. This is illustrated by the two data sets displayed in Fig. 8.2, which differ in the interface structures of the respective diodes. Quite generally, structural rearrangements such as the ( 7 × 7)i reconstruction are connected with a redistribution of the valence charge. The bonds in perfectly ordered bulk silicon, the example considered here, are purely covalent, and so reconstructions are accompanied by electric \(\mathrm{Si^{+{\Updelta}\textit{q}}-Si^{-{\Updelta}\textit{q}}}\) dipoles. The Si(111)-( 7 × 7 )  reconstruction is characterized by a stacking fault in one half of its unit mesh [8.15]. Schmitsdorf et al. [8.10] quantitatively explained the experimentally observed reduction in the laterally homogeneous barrier height of the ( 7 × 7)i with regard to the ( 1 × 1)i diodes by the electric dipole associated with the stacking fault of the Si(111)-( 7 × 7 )  reconstruction.

Patches of reduced barrier height with lateral dimensions smaller than the depletion layer width that are embedded in large areas of laterally homogeneous barrier height is the only known model that explains a lowering of effective barrier heights with increasing ideality factors. In their phenomenological studies of such patchy Schottky contacts , Freeouf et al. [8.11] found that the potential distribution exhibits a saddle point in front of such nanometer-size patches of reduced barrier height. Figure 8.3 explains this behavior. The saddle-point barrier height strongly depends on the voltage drop Vc across the depletion layer. Freeouf et al. simulated the current transport in such patchy Schottky contacts and found a reduction in the effective barrier height and a correlated increase in the ideality factor as they reduced the lateral dimensions of the patches. However, they overlooked the fact that the barrier heights of the laterally homogeneous contacts may be obtained from Φ Bn eff versus n plots, by extrapolating to nif.
Fig. 8.3

Calculated potential distribution underneath and around a patch of reduced interface potential embedded in a region of larger interface band-bending . The lateral dimension and the interface potential reduction of the patch are set to two-tenths of the depletion layer width zdep and one-half of the interface potential of the surrounding region. (After [8.13])

It has to be mentioned that the effective barrier heights and the ideality factors of real Schottky contacts generally vary as a function of temperature. The so-called Richardson plots ln ( Is ∕ T2 ) -versus-1 ∕ T will neither result in correct values of effective barrier heights nor will the effective Richardson constants obtained equal the theoretical values because such graphs inherently assume the barrier heights to be independent of temperature [8.16, 8.17, 8.18] in contrast to what is experimentally observed with real Schottky contacts.

C ∕ V Characteristics

Both the space charge and the width of the depletion layers at metal–semiconductor contacts vary as a function of the externally applied voltage. The space-charge theory gives the variation in the depletion layer capacitance per unit area as [8.13]
$$C_{\mathrm{dep}}=\left(\frac{e_{0}^{2}\varepsilon_{\mathrm{b}}\varepsilon_{0}N_{\mathrm{d}}}{2}\left[e_{0}\left(\left|V_{\mathrm{i}}^{0}\right|-V_{\mathrm{c}}\right)-k_{\mathrm{B}}T\right]\right)^{1/2}.$$
(8.8)
The current through a Schottky diode biased in the reverse direction is small, so the IR drop due to the series resistance of the diode may be neglected. Consequently, the extrapolated intercepts on the abscissa of 1 ∕ C dep 2 versus Va plots give the band-bending \(e_{0}|V_{\mathrm{i}}^{0}|\) at the interface, and together with the energy distance \(W_{\mathrm{n}}=W_{\mathrm{F}}-W_{\text{cb}}\) from the Fermi level to the conduction band minimum in the bulk, one obtains the flat-band barrier height \({\Phi}_{\text{Bn}}^{\text{fb}}{\equiv}{\Phi}_{\text{Bn}}^{\text{hom}}=e_{0}|V_{\mathrm{i}}^{0}|+W_{\mathrm{n}}\), which equals the laterally homogeneous barrier height of the contact.
As an example, Fig. 8.4 displays the flat-band barrier heights of the same Ag/n-Si(111) diodes that are discussed in Fig. 8.2. The dashed- and dash-dotted lines are the Gaussian least-squares fits to the data from the diodes with ( 1 × 1)i and ( 7 × 7)i interface structures, respectively. Within the margins of experimental error, the peak C ∕ V values agree with the laterally homogeneous barrier heights obtained from the extrapolations of the I ∕ V data shown in Fig. 8.2. These data clearly demonstrate that barrier heights characteristic of laterally homogeneous Schottky contacts can be only obtained from I ∕ V or C ∕ V data from many diodes fabricated under identical conditions rather than from a single diode. However, the effective barrier heights and the ideality factors vary as a function of the diode temperature. Hence, the effective barrier heights and ideality factors evaluated from the I ∕ V characteristics for one and the same diode recorded at different temperatures are also suitable for determining the corresponding laterally homogeneous barrier height [8.13].
Fig. 8.4

Histograms of flat-band barrier heights determined from C ∕ V characteristics of Ag/n-Si(111)-( 7 × 7)i and -( 1 × 1)i contacts at room temperature. The data were obtained with the same diodes discussed in Fig. 8.2. The dashed and dash-dotted lines are the Gaussian least-squares fits to the data. (After [8.10])

Ballistic-Electron-Emission Microscopy

In BEEM  [8.19], a tip injects almost monoenergetic electrons into the metal film of a Schottky diode. These tunnel-injected electrons reach the semiconductor as ballistic electrons provided that they lose no energy on their way through the metal. Hence, the collector current Icoll is expected to set in when the ballistic electrons surpass the metal–semiconductor barrier; in other words, if the voltage Vtip applied between tip and metal film exceeds the local potential barrier Φ Bn loc  ( z )  ∕ e0. Bell and Kaiser [8.20] derived the square law
$$I_{\mathrm{coll}}(z)=R^{\ast}I_{\mathrm{tip}}\left(e_{0}V_{\mathrm{tip}}-\Phi_{\mathrm{Bn}}^{\mathrm{loc}}(z)\right)^{2}$$
(8.9)
for the BEEM Icoll ∕ Vtip characteristics, where Itip is the injected tunnel current. BEEM measures local barrier heights; most specifically, the saddle-point barrier heights in front of nanometer-sized patches rather than their lower barrier heights right at the interface.
BEEM is the experimental tool for measuring spatial variations in the barrier height on the nanometer-scale. The local barrier heights are determined by fitting relation (8.9) to measured Icoll/Vtip characteristics recorded at successive tip positions along lateral line scans. Figure 8.5 displays histograms of the local BEEM barrier heights of two Pd/n-6H-SiC(0001) diodes [8.21]. The diodes differ in their ideality factors, 1.06 and 1.49, which are close to and much larger, respectively, than the value nif = 1.01 determined solely by the image-force effect. Obviously, the nanometer-scale BEEM histograms of the two diodes are identical, although their macroscopic ideality factors and therefore their patchinesses differ. Two important conclusions were drawn from these findings. First, these data suggest the existence of two different types of patches: intrinsic and extrinsic ones. The intrinsic patches might be correlated with the random distributions of the ionized donors and acceptors which cause nanometer-scale lateral fluctuations in the interface potential. A few gross interface defects of extrinsic origin, which escape BEEM observations, are then responsible for the variations in the ideality factors. Second, Gaussian least-squares fits to the histograms of the local BEEM barrier heights yield peak barrier heights of 1.27 ± 0.03 eV. Within the margins of experimental error, this value agrees with the laterally homogeneous value of 1.24 ± 0.09 eV, which was obtained by extrapolation of the linear least-squares fit to a Φ Bn eff versus n plot to nif. The nanometer-scale BEEM histograms and the macroscopic I ∕ V characteristics thus provide identical barrier heights of laterally homogeneous Schottky contacts.
Fig. 8.5

Histograms of local BEEM barrier heights of two Pd/n-6H-SiC(0001) diodes with ideality factors of 1.06 (gray solid bars) and 1.49 (empty bars). The data were obtained by fitting the square law (8.9) to 800 BEEM Icoll ∕ Vtip spectra each. (After Im et al. [8.21])

Internal Photoemission Yield Spectroscopy

Metal–semiconductor contacts show a photoelectric response to optical radiation with photon energies smaller than the width of the bulk band gap. This effect is caused by photoexcitation of electrons from the metal over the interfacial barrier into the conduction band of the semiconductor. Experimentally, the internal photoemission yield, which is defined as the ratio of the photoinjected electron flux across the barrier into the semiconductor to the flux of the electrons excited in the metal, is measured as a function of the energy of the incident photons. Consequently, this technique is called internal photoemission yield spectroscopy (IPEYS ). Cohen et al. [8.22] derived that the internal photoemission yield varies as a function of the photon energy ℏω as
$$Y(\hbar\omega)\propto\frac{1}{\hbar\omega}\left(\hbar\omega-\Phi_{\mathrm{Bn}}^{\mathrm{IPEYS}}\right)^{2}\;.$$
(8.10)
Patches only cover a small portion of the metal–semiconductor interface, so the threshold energy Φ Bn IPEYS will equal the barrier height Φ Bn hom of the laterally homogeneous part of the contact minus the zero-bias image-force lowering δΦ if 0 .
In Fig. 8.6 , experimental \((Y(\hbar{\omega})\times\hbar{\omega})^{1/2}\) data for a Pt/p-Si(001) diode [8.23] are plotted versus the energy of the exciting photons. The dashed line is the linear least-squares fit to the data. The deviation of the experimental \((Y(\hbar{\omega})\times\hbar{\omega})^{1/2}\) data towards larger values slightly below and above the threshold is caused by the shape of the Fermi–Dirac distribution function at finite temperatures and by the existence of patches with barrier heights smaller and larger than Φ Bn hom .
Fig. 8.6

Spectral dependence of the internal photoemission yield of a Pt/p-Si(001) diode versus the photon energy of the exciting light. The dashed line is the linear least-squares fit to the data for photon energies larger than 0.3 eV. (After Turan et al. [8.23])

8.1.2 Band Offsets of Semiconductor Heterostructures

Semiconductors generally grow layer-by-layer, at least initially. Hence, core-level photoemission spectroscopy (PES ) is a very reliable tool and the one most widely used to determine the band-structure lineup at semiconductor heterostructures. The valence-band offset may be obtained from the energy positions of core-level lines in x-ray photoelectron spectra recorded with bulk samples of the semiconductors in contact and with the interface itself [8.24]. Since the escape depths of the photoelectrons are on the order of just 2 nm, one of the two semiconductors must be sufficiently thin. This condition is easily met when heterostructures are grown by molecular beam epitaxy (MBE ) and photoemission spectra (PES) are recorded during growth interrupts. The valence-band discontinuity is then given by (Fig. 8.7)
$$\begin{aligned}\displaystyle\Updelta W_{\mathrm{v}}&\displaystyle=W_{\mathrm{vir}}-W_{\mathrm{vil}}=W_{\mathrm{i}}(n_{\mathrm{r}}l_{\mathrm{r}})-W_{\mathrm{i}}(n_{\mathrm{l}}l_{\mathrm{l}})\\ \displaystyle&\displaystyle\quad+[W_{\mathrm{vbr}}-W_{\mathrm{b}}(n_{\mathrm{r}}l_{\mathrm{r}})]-[W_{\mathrm{vbl}}-W_{\mathrm{b}}(n_{\mathrm{l}}l_{\mathrm{l}})]\;,\end{aligned}$$
(8.11)
where nrlr and nlll denote the core levels of the semiconductors on the right (r) and the left (l) side of the interface, respectively. The subscripts i and b characterize interface and bulk properties, respectively. The energy difference Wi ( nrlr )  − Wi ( nlll )  between the core levels of the two semiconductors at the interface is determined from energy distribution curves of photoelectrons recorded during MBE growth of the heterostructure. The energy positions Wvbr − Wb ( nrlr )  and Wvbl − Wb ( nlll )  of the core levels relative to the valence-band maxima in the bulk of the two semiconductors are evaluated separately.
Fig. 8.7

Schematic energy band diagram at semiconductor heterostructures. Wvb and Wvi are the valence-band maxima and Wb ( nl )  and Wi ( nl )  are the core levels in the bulk and at the interface, respectively. The subscripts l and r denote the semiconductors on the right and the on the left side of the interface. ΔWv is the valence-band offset. The thin dashed lines account for possible band-bending from space-charge layers

Another widely used technique for determining band offsets in heterostructures is internal photoemission yield spectroscopy. The procedure for evaluating the IPEYS signals is the same as described in Sect. 8.1.1.

8.2 IFIGS-and-Electronegativity Theory

Because of the quantum-mechanical tunneling effect, the wave functions of bulk electrons decay exponentially into vacuum at surfaces or, more generally speaking, at solid–vacuum interfaces. A similar behavior occurs at interfaces between two solids [8.6, 8.7]. In energy regions of Schottky contacts and semiconductor heterostructures where occupied band states overlap a band gap, the wave functions of these electrons will tail across the interface. The only difference to solid–vacuum interfaces is that the wave function tails oscillate at solid–solid interfaces. Figure 8.8 schematically explains the tailing effects at surfaces and semiconductor interfaces. For the band-structure lineup at semiconductor interfaces, only the tailing states within the gap between the top valence and the lowest conduction band are of any real importance since the energy position of the Fermi level determines their charging state. These wave function tails or interface-induced gap states (IFIGS) derive from the continuum of the virtual gap states (ViGS ) of the complex semiconductor band structure. Hence, the IFIGS are an intrinsic property of the semiconductor.
Fig. 8.8a,b

Wave functions at clean surfaces (a) and at metal–semiconductor and semiconductor–semiconductor interfaces (b) (schematically)

The IFIGS are made up of valence-band and conduction-band states of the semiconductor. Their net charge depends on the energy position of the Fermi level relative to their branch point, where their character changes from predominantly donor- or valence-band-like to mostly acceptor- or conduction-band-like. The band-structure lineup at semiconductor interfaces is thus described by a zero-charge-transfer term and an electric dipole contribution.

In a more chemical approach, the charge transfer at semiconductor interfaces may be related to the partly ionic character of the covalent bonds at interfaces. Pauling [8.8] described the ionicity of single bonds in diatomic molecules by the difference between the electronegativities of the atoms involved. The binding energies of core-level electrons are known to depend on the chemical environment of the atoms or, in other words, on the ionicity of their chemical bonds. Figure 8.9 displays experimentally observed chemical shifts for Si(2p) and Ge(3d) core levels induced by metal adatoms on silicon and germanium surfaces as a function of the difference Xm − Xs between the Pauling atomic electronegativity of the metal and that of the semiconductor atoms. The covalent bonds between metal and substrate atoms still persist at metal–semiconductor interfaces, as ab-initio calculations [8.25] have demonstrated for the example of Al/GaAs(110) contacts. The pronounced linear correlation of the data displayed in Fig. 8.9 thus justifies the application of Pauling’s electronegativity concept to semiconductor interfaces.
Fig. 8.9

Chemical shifts of Si(2p) and Ge(3d) core levels induced by metal adatoms on silicon and germanium surfaces, respectively, as a function of the difference Xm − Xs in the metal and the semiconductor electronegativities in Pauling units . (After [8.13])

The combination of the physical IFIGS and the chemical electronegativity concept yields the barrier heights of ideal p-type Schottky contacts and the valence-band offsets of ideal semiconductor heterostructures as
$$\Phi_{\mathrm{Bp}}=\Phi_{\mathrm{bp}}^{\mathrm{p}}-S_{\mathrm{X}}(X_{\mathrm{m}}-{X}_{\mathrm{s}})$$
(8.12)
and
$$\Updelta W_{\mathrm{v}}=\Phi_{\mathrm{bpr}}^{\mathrm{p}}-\Phi_{\mathrm{bpl}}^{\mathrm{p}}+D_{\mathrm{X}}({X}_{\mathrm{sr}}-{X}_{\mathrm{sl}})\;,$$
(8.13)
respectively, where \({\Phi}_{\text{bp}}^{\mathrm{p}}=W_{\text{bp}}-W_{\mathrm{v}}({\Gamma})\) is the energy distance from the valence-band maximum to the branch point of the IFIGS or the p-type branch-point energy. It has the physical meaning of a zero-charge-transfer barrier height. The slope parameters SX and DX are explained at the end of this section.
The IFIGS derive from the ViGS of the complex band structure of the semiconductor. Their branch point is an average property of the semiconductor. Tersoff [8.26] calculated the branch-point energies Φ bp p of Si, Ge, and 13 of the III–V and II–VI compound semiconductors. He used a linearized augmented plane-wave method and the local density approximation. Such extensive computations may be avoided. Mönch [8.27] applied Baldereschi’s concept [8.28] of mean-value k-points to calculate the branch-point energies of zincblende-structure compound semiconductors. He first demonstrated that the quasi-particle band gaps of diamond, silicon, germanium, 3C-SiC, GaAs, and CdS at the mean-value k-point equal their average or dielectric band gaps [8.29]
$$W_{\mathrm{dg}}=\frac{\hbar\omega_{\mathrm{p}}}{\sqrt{\varepsilon_{\infty}-1}}\;,$$
(8.14)
where ℏωp is the plasmon energy of the bulk valence electrons. Mönch then used Tersoff’s Φ bp p values, calculated the energy dispersion of the topmost valence band in the empirical tight-binding approximation (ETB ), and plotted the resulting branch-point energies at the mean-value k-point versus the widths of the dielectric band gaps Wdg. The linear least-squares fit to the data of the zincblende-structure compound semiconductors [8.30]
$$\Phi_{\mathrm{bp}}^{\mathrm{p}}={0.449}W_{\mathrm{dg}}-[W_{\mathrm{v}}(\Gamma)-W_{\mathrm{v}}(\underline{k}_{\mathrm{mv}})]_{\text{ETB}}$$
(8.15)
indicates that the branch points of these semiconductors lie 5% below the middle of the energy gap at the mean-value k-point kmv. Table 8.1 displays the calculated p-type branch-point energies of the Group IV elemental semiconductors, of SiC, of III–V and II–VI compound semiconductors and of some I–III–VI2 chalcopyrites.
Table 8.1

Optical dielectric constants , widths of the dielectric band gap , and branch-point energies of diamond-, zincblende, and chalcopyrite-structure semiconductors. (After [8.26, 8.29, 8.30])

Semiconductor

ε

Wdg (eV)

Φ bp p (eV)

C

5.70

14.40

1.77

Si

11.90

5.04

0.36 a

Ge

16.20

4.02

0.18 a

3C-SiC

6.38

9.84

1.44

3C-AlN

4.84

11.92

2.97

AlP

7.54

6.45

1.13

AlAs

8.16

5.81

0.92

AlSb

10.24

4.51

0.53

3C-GaN

5.80

10.80

2.37

GaP

9.11

5.81

0.83

GaAs

10.90

4.97

0.52

GaSb

14.44

3.8

0.16

3C-InN

6.48

1.51

InP

9.61

5.04

0.86

InAs

12.25

4.20

0.50

InSb

15.68

3.33

0.22

ZnS

5.14

8.12

2.05

ZnSe

5.70

7.06

1.48

ZnTe

7.28

5.55

1.00

CdS

5.27

7.06

1.93

CdSe

6.10

6.16

1.53

CdTe

7.21

5.11

1.12

CuGaS2

6.15

7.46

1.43 b

CuInS2

6.3*

7.02

1.47 b

CuAlSe2

6.3*

6.85

1.25 b

CuGaSe2

7.3*

6.29

0.93 b

CuInSe2

9.00

5.34

0.75 b

CuGaTe2

8.0*

5.39

0.61 b

CuInTe2

9.20

4.78

0.55 b

AgGaSe2

6.80

5.96

1.09 b

AgInSe2

7.20

5.60

1.11 b

\({}^{*}\,{\varepsilon}_{{\infty}}=n^{2}\), a [8.26], b [8.29]

A simple phenomenological model of Schottky contacts with a continuum of interface states and a constant density of states Dis across the semiconductor band gap yields the slope parameter [8.31, 8.32]
$$S_{\mathrm{X}}=\frac{A_{\mathrm{X}}}{\left(1+\frac{e_{0}^{2}}{\varepsilon_{\mathrm{i}}\varepsilon_{0}}D_{\mathrm{is}}\delta_{\mathrm{is}}\right)}\;,$$
(8.16)
where εi is an interface dielectric constant. The parameter AX depends on the electronegativity scale chosen and amounts to 0.86 eV ∕ Miedema-unit and 1.79 eV ∕ Pauling-unit. For Dis → 0, relation (8.16) yields SX → AX or, in other words, if no IFIGS were present at the metal–semiconductor interfaces one would obtain the Schottky–Mott rule. The extension δis of the interface states may be approximated by their charge decay length 1 ∕ 2qis. Mönch [8.32] used theoretical D gs mi and q gs mi data for metal-induced gap states (MIGS ), as the IFIGS in Schottky contacts are traditionally called, and plotted the \((e_{0}^{2}/{\varepsilon}_{0})D_{\text{gs}}^{\text{mi}}/(2q_{\text{gs}}^{\text{mi}})\) values versus the optical susceptibility ε − 1. The linear least-squares fit to the data points yielded [8.32]
$$\frac{A_{\mathrm{X}}}{S_{\mathrm{X}}}-1=0.1(\varepsilon_{\infty}-1)^{2}\;,$$
(8.17)
where the reasonable assumption εi ≈ 3 was made.

To a first approximation, the slope parameter DX of heterostructure band offsets may be equated with the slope parameter SX of Schottky contacts, since the IFIGS determine the intrinsic electric-dipole contributions to both the valence-band offsets and the barrier heights.

In early Gedanken experiments, Schottky contacts and semiconductor heterostructures were assembled by gradually approaching the respective two solids in vacuum [8.3, 8.33]. As long as a vacuum gap exists between the two surfaces facing each other, the vacuum level Wvac is the appropriate reference and, for example, the work function Wvac − WF of the metal and the electron affinity Wvac − Wc or the ionization energy Wvac − Wv of the semiconductor may be used to describe the charge distribution and the energy-band alignment in such a metal–vacuum–semiconductor array. As the vacuum gap is closed, that is, an intimate Schottky contact is made, chemical bonds between metal and semiconductor atoms will have formed at the now intimate metal–semiconductor interface. Hence, the work function of the metal as well as the electron affinity or the ionization energy of the semiconductor are inappropriate reference properties, but instead the electronegativities of the metal and the semiconductor may be used to describe the electric dipole layer at Schottky contacts [8.34]. The same arguments also hold for semiconductor heterostructures.

8.3 Comparison of Experiment and Theory

8.3.1 Barrier Heights of Schottky Contacts

Experimental barrier heights of intimate, abrupt, clean, and (above all) laterally homogeneous Schottky contacts on n-Si, n-GaAs, n-GaN, and SiO2, as an example for insulators, are plotted in Figs. 8.10a,b and 8.11a,b versus the difference in the Miedema electronegativities of the metals and the semiconductors. Miedema’s electronegativities [8.35] are preferred since they were derived from properties of metal alloys and intermetallic compounds, while Pauling [8.8] considered covalent bonds in small molecules. The p- and n-type branch-point energies, \({\Phi}_{\text{bp}}^{\mathrm{p}}=W_{\text{bp}}-W_{\mathrm{v}}({\Gamma})\) and \({\Phi}_{\text{bp}}^{\mathrm{n}}=W_{\mathrm{c}}-W_{\text{bp}}\), respectively, add up to the fundamental band-gap energy \(W_{\mathrm{g}}=W_{\mathrm{c}}-W_{\mathrm{v}}({\Gamma})\). Hence, the barrier heights of n-type Schottky contacts are
$$\Phi_{\mathrm{Bn}}^{\mathrm{hom}}=\Phi_{\mathrm{bp}}^{\mathrm{n}}+S_{X}({X}_{\mathrm{m}}-{X}_{\mathrm{s}})\;.$$
(8.18)
The electronegativity of a compound is taken as the geometric mean of the electronegativities of its constituent atoms.
Fig. 8.10a,b

Barrier heights of laterally homogeneous n-type silicon (a) and GaAs Schottky contacts (b) versus the difference in the Miedema electronegativities of the metals and the semiconductors. The □ and ○, △, ◇, and ▽ symbols differentiate the data from I ∕ V, BEEM, IPEYS, and PES measurements, respectively. The dashed and the dash-dotted lines are the linear least-squares fits to the respective experimental data. The solid IFIGS lines are drawn with \(S_{\mathrm{X}}={\mathrm{0.101}}\,{\mathrm{e{\mskip-2.0mu}V/Miedema\text{-}unit}}\) and Φ bp p  = 0.36 eV for silicon (a) and with \(S_{\mathrm{X}}={\mathrm{0.08}}\,{\mathrm{e{\mskip-2.0mu}V/Miedema\text{-}unit}}\) and Φ bp p  = 0.5 eV for GaAs (b). (After [8.13])

Fig. 8.11a,b

Barrier heights of laterally homogeneous n-GaN(0001) (a) and SiO2 Schottky contacts (b) versus the difference in the Miedema electronegativities of the metals and the semiconductors. (a) The solid IFIGS line is drawn with \(S_{\mathrm{X}}={\mathrm{0.29}}\,{\mathrm{e{\mskip-2.0mu}V/Miedema\text{-}unit}}\) and Φ bp p  = 2.37 eV. (a,b) The □, ○, △, and ▽ symbols differentiate the data from I ∕ V, C ∕ V, IPEYS, and PES measurements. The dashed lines are the linear least-squares fits to the respective data. (After [8.13, 8.36])

First, the experimental data plotted in Figs. 8.10a,b and 8.11a,b clearly confirm that the different experimental techniques, I ∕ V, C ∕ V, BEEM, IPEYS, and PES, yield barrier heights of laterally homogeneous Schottky contacts which agree within the margins of experimental error.

Second, the experimental Si, GaAs, and GaN data are quantitatively explained by the branch-point energies (8.15) and the slope parameters (8.17) of the IFIGS-and-electronegativity theory. As was already mentioned in Sect. 8.1.1 , the stacking fault, which is part of the interfacial Si(111)-( 7 × 7)i reconstruction [8.15], causes an extrinsic electric dipole in addition to the intrinsic IFIGS electric dipole. The latter one is present irrespective of whether the interface structure is reconstructed or ( 1 × 1)i-unreconstructed. The extrinsic stacking fault-induced electric dipole quantitatively explains the experimentally observed barrier-height lowering of 76 ± 2 meV [8.10].

Third, the IFIGS line in Fig. 8.11a,ba was drawn using the branch-point energy calculated for cubic 3C-GaN since relation (8.15) was derived for zincblende-structure compounds only. However, the Schottky contacts were prepared on hexagonal wurtzite-structure 2H-GaN. The good agreement between the experimental data and the IFIGS line indicates that the p-type branch-point energy seems to be rather insensitive to the specific bulk lattice structure of the semiconductor. This conclusion is further justified by the barrier heights of laterally homogeneous Schottky contacts on cubic 3C-SiC and its hexagonal polytypes 4H and 6H [8.13] as well as by the experimentally observed band-edge discontinuities of semiconductor heterostructures, which will be discussed in Sect. 8.3.2.

8.3.2 Band Offsets of Semiconductor Heterostructures

In the bulk and at interfaces of sp3-coordinated semiconductors, the chemical bonds are covalent. The simplest semiconductor–semiconductor interfaces are lattice-matched heterostructures. However, if the bond lengths of the two semiconductors differ, then the interface will respond with tetragonal lattice distortions. Such pseudomorphic interfaces are under tensile or compressive stress. If the strain energy becomes too large, then it is energetically more favorable to release the stress by the formation of misfit dislocations. Such metamorphic interfaces are almost relaxed.

In contrast to isovalent heterostructures, the chemical bonds at heterovalent interfaces require special attention, since interfacial donor- and acceptor-type bonds may cause interfacial electric dipoles [8.37]. No such extrinsic electric dipoles will exist normal to nonpolar (110) interfaces. However, polar (001) interfaces behave quite differently. Acceptor bonds or donor bonds normal to the interface would exist at abrupt heterostructures. But, for reasons of charge neutrality, they have to be compensated by a corresponding density of donor bonds and acceptor bonds, respectively. This may be achieved by an intermixing at the interface which, on the other hand, causes extrinsic electric dipoles. Their components normal to the interface will add an extrinsic electric-dipole contribution to the valence-band offset.

The valence-band offsets at nonpolar heterostructures of compound semiconductors should equal the difference in the branch-point energies of the two semiconductors in contact provided the intrinsic IFIGS electric-dipole contribution can be neglected, see relation (8.13). This assumption is justified for heterostructures among the elemental Group IV semiconductors Ge and Si, SiC, and the III–V and II–VI compounds – with the nitrides and the oxides being excepted – because the elements that constitute these semiconductors are all placed in the middle of the Periodic Table so that their electronegativities only differ slightly. Figure 8.12a,ba displays respective experimental results for diamond- and zincblende-structure semiconductors as a function of the difference in the calculated branch-point energies given in Table 8.1. The dashed line clearly demonstrates that the experimental data are excellently explained by the theoretical branch-point energies or, in other words, by the IFIGS theory.
Fig. 8.12a,b

Valence-band offsets at nonpolar (110)-oriented (a) and metamorphic semiconductor heterostructures (b) versus the difference between the p-type branch-point energies of the semiconductors in contact. (After [8.13])

As an example of lattice-matched and isovalent heterostructures, Fig. 8.13 shows valence-band offsets of Al1xGa x As/GaAs heterostructures as a function of the alloy composition x. The IFIGS branch-point energies of the alloys were calculated assuming virtual cations [8.30] and were found to vary linearly as a function of composition between the values of AlAs and GaAs. More refined first-principles calculations yielded identical results [8.38, 8.39, 8.40]. Figure 8.13 reveals that the theoretical IFIGS valence-band offsets excellently fit the experimental data.
Fig. 8.13

Valence-band offsets of lattice-matched and isovalent heterostructures as a function of alloy composition x. (After [8.13])

Figure 8.12a,b b displays valence-band offsets of metamorphic heterostructures versus the difference in the branch-point energies of the two semiconductors. The dashed line indicates that the experimental results are again excellently described by the theoretical IFIGS data. This is true not only for heterostructures between cubic (C) zincblende- and hexagonal (H) wurtzite-structure compounds but also for wurtzite-structure Group III nitrides grown on both cubic 3C- and hexagonal 6H-SiC substrates. These observations suggest the following conclusions. First, all of the heterostructures considered in Fig. 8.12a,bb are only slightly (if at all) strained, although their lattice parameters differ by up to 19.8%. First-principles calculations [8.41] revealed that misfit dislocations change the valence-band offsets of metamorphic heterostructures by less than 0.1 eV only. Second, the calculations of the IFIGS branch-point energies assumed zincblende-structure semiconductors. These values, on the other hand, reproduce the experimental valence-band offsets irrespective of whether the semiconductors have zincblende, wurtzite, or, as in the case of 6H-SiC, another hexagonal-polytype structure. These findings again support the conclusion drawn from the GaN Schottky barrier heights in the previous section that the IFIGS branch-point energies are rather insensitive to the specific semiconductor bulk lattice structure.

Quite generally, the contribution of the IFIGS electric-dipole term SX ( Xsl − Xsr )  to the valence-band offset ΔWv of heterostructures may not be neglected. A simple rearrangement of terms in relation (8.13) gives
$$\Updelta W_{\mathrm{v}}-S_{\mathrm{X}}(X_{\text{sr}}-X_{\text{sl}})=\Phi_{\mathrm{bpr}}^{\mathrm{p}}-\Phi_{\mathrm{bpl}}^{\mathrm{p}}\;.$$
(8.19)
For heterostructres of one and the same semiconductor, the valence-band offsets less the IFIGS-related electric-dipole contributions thus vary as the p-type branch-point energies of the respective other semiconductors. Figure 8.14 displays experimental valence-band offsets of SiO2 heterostructures minus the respective electric-dipole contributions as a function of the calculated branch-point energies of the respective other semiconductors [8.26, 8.30] (Table 8.1); the SX value of SiO2 was adopted from Fig. 8.11a,bb. The linear least-squares fit has a slope parameter of 0.94 ± 0.07 which value is close to 1 as is to be expected from (8.19), and its intercept on the ordinate gives the empirical p-type branch-point energy of SiO2 as 3.73 ± 0.07 eV.
Fig. 8.14

Experimental valence-band offsets of SiO2 heterostructures minus the respective IFIGS electric-dipole contributions as a function of the calculated p-type branch-point energies of the respective other semiconductors. The dashed line is the linear least-squares fit to the data points. (After [8.42])

Figures 8.15 and  8.16a,b display experimental \({\Updelta}W_{v}-S_{\mathrm{X}}(X_{\text{semi}}-X_{\text{Si,Ge,GaN}})\) data of Si, Ge, and GaN heterostructures as a function of the branch-point energies of the other semiconductors and insulators involved. For the semiconductors, the calculated values [8.26, 8.30] were taken again while for the insulators empirical values [8.42, 8.43, 8.44] were considered. Again, the slope parameters of the linear least-squares fits (0.97 ± 0.03, 0.90 ± 0.04, and 0.87 ± 0.04) are close to 1, the value expected from (8.19). The empirical p-type branch-point energies result as 0.30 ± 0.09 eV for Si, 2.34 ± 0.4 eV for GaN, and 0.08 ± 0.10 eV for Ge. Within the margins of experimental error, these empirical values excellently agree with the calculated ones (Table 8.1).
Fig. 8.15

Experimental valence-band offsets of Si heterostructures minus the respective IFIGS electric-dipole contributions (□) and the IFIGS electric-dipole contributions (○) as a function of calculated and empirical p-type branch-point energies of the respective other semiconductors and insulators, respectively. The dashed and dash-dotted lines are the linear least-squares fits to the data points. (After [8.43])

Fig. 8.16a,b

Experimental valence-band offsets of GaN (a) and Ge heterostructures (b) minus the respective IFIGS electric-dipole contributions as a function of the calculated and empirical p-type branch-point energies of the respective other semiconductors and insulators, respectively. The dashed lines are the linear least-squares fits to the data points. (After [8.43])

Figure 8.15 also displays some of the IFIGS electric-dipole contributions SX ( Xsemi − XSi )  in Si heterostructures. The example of SiO2/Si heterostructures demonstrates that this intrinsic dipole term measures 1.3 eV and this large value thus amounts to approximately 30% of the total valence-band offset.

As was already mentioned, the p- and n-type branch-point energies, \({\Phi}_{\text{bp}}^{\mathrm{p}}=W_{\text{bp}}-W_{\mathrm{v}}({\Gamma})\) and \({\Phi}_{\text{bp}}^{\mathrm{n}}=W_{\mathrm{c}}-W_{\text{bp}}\), respectively, add up to the fundamental band-gap energy \(W_{\mathrm{g}}=W_{\mathrm{c}}-W_{\mathrm{v}}({\Gamma})\). Table 8.2 shows empirical Φ bp n and Φ bp p values that were obtained from experimental barrier heights of n-type Schottky contacts and valence-band offsets of heterostructures, respectively; see, for example, Figs. 8.10a,b, 8.11a,b, 8.148.16a,b, 8.19. Within the margins of experimental error, the empirical p- and n-type branch-point energies of these semiconductors and insulators indeed add up to the widths of their fundamental band gaps. These findings again strongly support the IFIGS-and-electronegativity concept of the band-structure alignment at semiconductor interfaces. The empirical Φ bp n values are of special importance since they were evaluated from experimental barrier heights with no other input than the metal and semiconductor electronegativities.
Table 8.2

Fundamental band gaps Wg and empirical branch-point energies Φ bp n and Φ bp p as obtained from barrier heights of Schottky contacts and valence-band offsets of heterostructures, respectively [8.13, 8.36, 8.42, 8.43, 8.44, 8.45, 8.46]

Semiconductor

Wg (eV)

Φ bp n (eV)

Φ bp p (eV)

Φ bp n  + Φ bp p (eV)

Si

1.12

0.80 ± 0.01

0.30 ± 0.09

1.10

Ge

0.66

0.59 ± 0.01

0.08 ± 0.10

0.67

GaN

3.39

1.12 ± 0.03

2.34 ± 0.09

3.46

GaAs

1.42

0.90 ± 0.01

0.68 ± 0.11

1.58

InP

1.34

0.50 ± 0.02

0.86 ± 0.11

1.36

SiO2

8.7–8.9

4.95 ± 0.19

3.73 ± 0.07

8.68

TiO2

3.3–3.5

1.13  (  ± 0 ) 

2.34 ± 0.36

3.47

HfO2

5.7–6.0

4.01 ± 0.19

2.27 ± 0.14

6.28

Al2O3

6.7–7.0

4.38 ± 0.34

2.98 ± 0.26

7.36

Ga2O3

4.9

1.34 ± 0.08

3.57 ± 0.50

4.91

Fig. 8.17

Contact resistance ratio of Al/Si3N4/n-GaAs Schottky contacts versus the thickness of the Si3N4 interlayer. (After [8.49])

Fig. 8.18

Band-structure lineup at metal-ultrathin insulator–semiconductor (MUTIS ) structures (schematically) with the assumption \(X_{\mathrm{m}}<X_{\mathrm{i}}> X_{\mathrm{s}}\). On the semiconductor side the extension of the band-bending regime is not drawn to scale. (After [8.50])

Fig. 8.19

Schottky barrier heights of intimate metal/n-Si (from Fig. 8.10a,ba) and metal/Si3N4/n-Si versus the difference Xm − XSi of the metal and the Si electronegativities. The full and the dashed lines are the predictions of the IFIGS theory and the linear-least squares fits to the experimental data, respectively. (After [8.50])

8.4 Modifications of Schottky Contacts

8.4.1 Nonalloyed Ohmic Contacts or MUTIS Schottky Contacts

Ideal metal–semiconductor or Schottky contacts are generally rectifying while genuine low-resistance or ohmic contacts are obtained with p-type Ge and GaSb and n-type InN, InAs, and InSb only because their IFIGS branch points are close to their valence-band maxima and above their conduction-band minima, respectively (Table 8.1). Recently, Connelly et al. [8.47, 8.48] studied a new path towards Ohmic contacts. They avoided alloying and demonstrated that the resistance of Si Schottky contacts is considerably lowered by ultrathin Si3N4 layers placed between the metal and silicon. As a typical example, Fig. 8.17 displays the contact resistance of Al/Si3N4/n-GaAs Schottky contacts normalized to the value of the intimate Al/n-GaAs contact in dependence on the insulator thickness [8.49]. The initial dramatic decrease of the contact resistance is accompanied by a gradual reduction of the Schottky barrier heights to a then constant value while the subsequent rise is easily attributed to the increasing tunnel resistance through the then thicker insulator interlayers.

Figure 8.18 schematically shows the energy-band diagram of ideal metal-ultrathin insulator–semiconductor or MUTIS structures resulting from the IFIGS-and-electronegativity concept. It is assumed that the electronegativity of the insulator is intermediate between the values of the metal and the semiconductor. In order to avoid that the IFIGS of the metal–insulator and the insulator–semiconductor interface overlap the thickness of the insulator has to be larger than four to five times the decay length at the IFIGS branch point of the insulator. Theoretical calculations found these dacay lengths to range between 0.09 and 0.28 nm [8.50]. The pronounced minimum value of the contact resistance ratio displayed in Fig. 8.17 is compatible with this estimate. Considering (8.12) and (8.13), one obtains
$$W_{\mathrm{F}}-W_{v}^{\mathrm{i}}=\Phi_{\text{bp}}^{\text{pi}}-S_{X}^{\mathrm{i}}(X_{\mathrm{m}}-X_{\mathrm{i}})$$
(8.20)
at the metal–insulator interface and
$$\begin{aligned}\displaystyle W_{\mathrm{F}}-W_{v}^{\mathrm{i}}&\displaystyle=\Updelta W_{v}+W_{\mathrm{g}}^{\mathrm{s}}-\Phi_{\text{Bn}}^{\mathrm{s}}\\ \displaystyle&\displaystyle=\Phi_{\text{bp}}^{\text{pi}}-\Phi_{\text{bp}}^{\text{ps}}+S_{\mathrm{X}}^{\mathrm{i}}(X_{\mathrm{s}}-X_{\mathrm{i}})+W_{\mathrm{g}}^{\mathrm{s}}-\Phi_{\text{Bn}}^{\mathrm{s}}\end{aligned}$$
(8.21)
at the insulator–semiconductor interface where the sub- and superscripts m, i, and s indicate metals, insulators, and semiconductors, respectively. W g s designates the width of the fundamental band gap of the semiconductor. A rearrangement of (8.20) and (8.21) then gives the effective n-type barrier heights of the MUTIS structures as
$$\begin{aligned}\displaystyle\Phi_{\text{Bn}}^{\mathrm{s}}&\displaystyle=W_{\mathrm{g}}^{\mathrm{s}}-\Phi_{\text{bp}}^{\text{ps}}+S_{\mathrm{X}}^{\mathrm{i}}(X_{\mathrm{m}}-X_{\mathrm{s}})\\ \displaystyle&\displaystyle=\Phi_{\text{bp}}^{\text{ns}}+S_{\mathrm{X}}^{\mathrm{i}}(X_{\mathrm{m}}-X_{\mathrm{s}})\end{aligned}$$
(8.22)
The effective n-type barrier height of an ideal n-type MUTIS structure is thus determined by the n-type branch-point energy Φ bp ns of the semiconductor and the slope parameter S X i of the insulator. The optical dielectric constant ε of Si3N4, and the same holds for other insulators such as Al2O3, Ge3N4, and MgO, is considerably smaller than the respective values of semiconductors as, for example, Ge, Si, and GaAs and, hence, (8.17) gives S X i  > S X s for the slope parameters. The insertion of the ultrathin insulators in MUTIS contacts thus alleviates the pinning of the Fermi level in comparison with the respective intimate MS contacts. Furthermore, (8.22) reveals that Ohmic rather than rectifying n-type MUTIS contacts will be obtained provided metals are chosen such that their electronegativities are sufficiently smaller than the values of the respective semiconductors.

Experimental barrier heights of Si3N4/n-Si and intimate n-Si Schottky contacts are plotted versus the electronegativity difference Xm − XSi in Fig. 8.19. The thickness of the Si3N4 layers measured approximately 1 nm and is thus sufficiently larger than the estimated decay length of the IFIGS at their branch point. Within the margins of experimental error, the experimental barrier heights of the metal/Si3N4/n-Si junctions clearly confirm prediction (8.22) of the IFIGS-and-electronegativity concept in that the MUTIS barrier heights are determined by the branch-point energy of the semiconductor and the slope parameter of the insulator. This observation most directly demonstrates the alleviation of the Fermi-level pinning in MUTIS structures in comparison with intimate MS interfaces.

8.4.2 Atomic Interlayers

The barrier heights of metal–semiconductor contacts may also be modified by atomic interlayers . Figure 8.20a,b displays barrier heights of p-type silicon and diamond Schottky contacts without and with hydrogen interlayers [8.48, 8.50, 8.51]. Irrespective of the metals used, the H-induced modifications amount to −1.4 eV for p-diamond(001) but to +0.3 eV for p-Si(111) and to +0.16 eV for p-Si(001) Schottky contacts. In contrast to the insertion of ultrathin insulators, the H-interlayers do not alter the slope parameters SX.
Fig. 8.20a,b

Barrier heights of p-Si (a) and p-diamond Schottky contacts (b) with and without H-interlayers. The full and the dashed lines are the predictions of the IFIGS theory and the linear least-squares fits to the experimental data, respectively. (After [8.50])

The metal-independent H-induced modifications of the Schottky barrier heights were explained by the partial ionic character of the covalent bonds between the H atoms and the interface atoms of the substrates [8.52]. The electronegativity of hydrogen ranges between the values of silicon and carbon, that is, \(X_{\text{Si}}<X_{\mathrm{H}}<X_{\mathrm{C}}\) [8.8]. Hence, the partial ionic character of covalent Si–H and C–H bonds differs in sign, and H−Δq-Siq but Hq-C−Δq bonds will exist at silicon and diamond interfaces, respectively. The opposite directions of the H-induced interface dipoles easily explain why the H-induced variations of the barrier height exhibit different signs with p-Si and p-diamond Schottky contacts.

These early conclusions were excellently confirmed by the recent observation that F-interlayers reduce the barrier heights of n-Ge Schottky contacts and do not change the slope parameter SX [8.53]. The electronegativity of F is larger than the one of Ge so that F−Δq-Geq dipoles are to be expected at F-doped Ge interfaces. These dipoles will reduce the barrier heights of metal/F:n-Ge Schottky contacts while, as with H:p-Si contacts, an increase would be observed with metal/F:p-Ge interfaces. The reduction of the barrier height observed with Fe/S:n-Ge contacts [8.54] is easily explained along the same lines.

The experimental data presented in Fig. 8.20a,b and [8.53] clearly demonstrate that atomic interlayers in Schottky contacts cause no depinning of the Fermi level and no alleviation of the Fermi-level pinning either.

8.5 Graphene Schottky Contacts

Graphene (Gr) is a single layer of sp2-bonded carbon atoms arranged in a hexagonal crystal lattice. The remaining 2p z orbitals are perpendicular to the graphene plane and form contiguous π and π* bands so that graphene may be described as a zero-overlap semimetal. As the conventional metal-semiconductor contacts discussed in the preceding chapters graphene-semiconductor contacts indeed exhibit rectifying properties. However, conventional and Gr Schottky contacts differ in that the bonds between the metal and the semiconductor atoms are covalent while the bonds between graphene and semiconductor substrates are of van der Waals type. Mönch [8.55] verified that the experimental barrier heights of intimate Gr contacts on conventional n-type semiconductors as well as on layered n-MoS2 agree with the values predicted by (8.17) and (8.18) but do not follow the Schottky–Mott rule. Hence, electric dipoles exist at Gr-semiconductor interfaces because graphene has a larger electronegativity than the semiconductors, with SiO2 being the only exemption. The respective electronic charge transfer from the semiconductors towards the more electronegative Gr layer was also confirmed by theoretical calculations (see Refs. cited in [8.55]). The barrier heights of graphene Schottky contacts are thus explained by the π and π* wave-function tails of the graphene across the interface in the energy range where the ππ* bands of the graphene overlap the band gap of the semiconductor. The IFIGS-and-electronegativity concept thus also explains the band structure lineup at graphene-semiconductor interfaces rather than the Schottky–Mott rule as assumed in almost all the articles reporting barrier heights of such contacts.

8.6 Final Remarks

The local density approximation to density functional theory (LDA -DFT ) is the most powerful and widely used tool in theoretical studies of the ground-state properties of solids. However, excitation energies such as the width of the energy gaps between the valence and conduction bands of semiconductors cannot be correctly obtained from such calculations. The fundamental band gaps of the elemental semiconductors C, Si, and Ge as well as of the III–V and II–VI compounds are notoriously underestimated by 25–50%. However, it became possible to compute quasi-particle energies and band gaps of semiconductors from first principles using the so-called GW approximation for the electron self-energy [8.56, 8.57]. The resulting band-gap energies agree to within 0.1–0.3 eV with experimental values.

For some specific metal–semiconductor contacts, the band-structure lineup was also studied by state-of-the-art ab-initio LDA-DFT calculations. The resulting LDA-DFT barrier heights were then subjected to a-posteriori corrections, which consider quasi-particle effects and, if necessary, spin–orbit interactions and semicore-orbital effects. However, comparison of the theoretical results with experimental data gives an inconsistent picture. The mean values of the barrier heights of Al- and Zn/p-ZnSe contacts, which were calculated for different interface configurations using ab-initio LDA-DF theory and a-posteriori spin–orbit and quasi-particle corrections [8.58, 8.59], agree with the experimental data to within the margins of experimental error. The same conclusion was reached for Al/Al x Ga1xAs Schottky contacts [8.60]. However, ab-initio LDA-DFT barrier heights of Al-, Ag-, and Au/p-GaN contacts [8.61, 8.62], as well as of Al– and Ti/3C-SiC(001) interfaces [8.63, 8.64], strongly deviate from the experimental results.

As already mentioned, ab-initio LDF-DFT valence-band offsets of heterostructures [8.39, 8.40, 8.41] reproduce the experimental results well. The same holds for mean values of LDF-DFT valence-band offsets computed for different interface configurations of GaN- and AlN/SiC-heterostructures [8.65, 8.66, 8.67, 8.68, 8.69].

The main difficulty, which the otherwise extremely successful ab-initio LDF-DFT calculations encounter when describing semiconductor interfaces, is not the precise exchange-correlation potential, which may be estimated in the GW approximation, but their remarkable sensitivity to the geometrical and compositional structure right at the interface. This aspect is more serious at metal–semiconductor interfaces than at heterostructures between two sp3-bonded semiconductors. The more conceptual IFIGS-and-electronegativity theory, on the other hand, quantitatively explains not only the barrier heights of ideal Schottky contacts but also the valence-band offsets of semiconductor heterostructures. Here again, the Schottky contacts are the more important case, since their zero-charge-transfer barrier heights equal the branch-point energies of the semiconductors, while the valence-band offsets are determined by the differences in the branch-point energies of the semiconductors in contact.

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© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.Faculty of PhysicsUniversity Duisburg-EssenDuisburgGermany

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