Chapter 3 Metal-semiconductor interfaces and ballistic electron emission microscopy

This chapter is divided into two parts. In the first part the metal-semiconductor (M-S) inter- faces are discussed - viz. the formation of the Schottky barrier (SB), followed by the models to determine the Schottky barrier height (SBH), possible barrier lowering mechanisms like image force lowering, lowering due tunneling and due to electrostatic screening. In the sec- ond part we discuss the basic concepts of hot electron transport, as used in ballistic electron emission microscopy (BEEM). The various modes of operation in BEEM are presented. This is then followed by discussions of the various possible scattering mechanisms for hot elec- trons. The most commonly used model to determine the local Schottky barrier height, called the Bell-Kaiser model is discussed.

3.1    Transport at metal-semiconductor interfaces

We discuss the different transport mechanisms that occur at biased  and  non biased Schottky  interfaces between a metal and a semiconductor. First we discuss common transport models such as thermionic emission and  tunneling across such interfaces [1], [2].  We explain  the  relevance of incorporating tunneling mechanisms to ex- plain the observed current-voltage (I-V) characteristics in our devices  (as presented in Chapter 5). Further, we also discuss in details  the hot electron  transport at simi- lar Schottky  interfaces using the technique of ballistic electron  emission microscopy (BEEM). We explain  the  different contributions of hot  electron  scattering in met- als, semiconductors and  their  interfaces to hot electron  transport. We also discuss the factors  that  influence the hot electron  attenuation length  in metals.   We finally discuss the Bell-Kaiser  model  that  is commonly used  to extract  the local Schottky barrier height  at metal-semiconductor (M-S) interfaces [3].

3.1.1   Schottky barrier formation

When  an n-type semiconductor is brought in contact  with  a metal,  electrons will flow from  the semiconductor to the metal  if the Fermi  level of the semiconductor (SC) is higher than  that of the metal.  Such flow of electrons causes  the Fermi levels of the metal  and  the semiconductor to align.  The electrons moving from the semi- conductor to the metal  leave depleted donors in a region  close to the interface that create  an electric field in the semiconductor. This field causes  band  bending in the semiconductor close to the interface,  leading to the formation of a Schottky  barrier as shown in Fig.  3.1.  Such a barrier is a rectifying barrier for electronic  transport across  the metal  semiconductor interface.  

Figure 3.1: Energy  band  diagram of formation of a metal-semiconductor (n-type)  (M-S) con- tact.  (a) before contact,  (b) after the contact; the formation of a Schottky  junction  for the case where φm > φS . The M-S interface shown in (b) is at equilibrium.

Figure 3.2:  The metal  and  the  semiconductor are  shown in contact  at the  top.   (a) shows the charge  density, (b) electric field and  (c) electrostatic potential in the semiconductor as a function of the distance from the interface (x) into the semiconductor.

In Fig.  3.1 (a) the conduction band,  va- lence band  and Fermi level of the semiconductor are given by EC , EV  and EF S . φm is the work  function of the metal,  which  corresponds to the energy difference between  the vacuum level and the Fermi level of the metal.  χ is the electron  affinity of the semiconductor, which  is measured from  the bottom of the conduction band  to the vacuum level.  The obtained Schottky  barrier allows  electrons to flow from the semiconductor to the metal, but blocks it in the opposite direction, which  makes it a rectifying junction. In this thesis,  the two most important parameters that are to be considered are the depletion layer width (W ) and the Schottky  barrier height  (φB ).

3.1.2   Depletion layer

As mentioned above, when  a metal is brought in contact with a n-type semiconduc- tor, electrons flow from the semiconductor to the metal.  This leaves a region, close to 

the interface,  depleted of mobile  electrons. This region  is called the depletion layer. The depletion width (W ) in a Schottky  junction  can be determined analytically us- ing Poisson’s  equation. The depletion layer  width depends on the semiconductor permittivity ( s ), donor concentration (ND ), built-in  potential (Vbi ) and applied bias (V) [1], [2], following the equation:

(3.1)

Because  of the static  charge  in the depletion layer  an electric  field is present.  The strength of this field depends on the charge carrier density (ND ), the depletion width (W ), the permittivity of the semiconductor ( s ) and  the distance from the interface (x). This dependence is given by [1], [2]:

(3.2)

The electric field is the largest  at the interface,  i.e. for x = 0

(3.3)

The presence of an internal electric  field  across  the  M-S interface results in a po- tential  difference between the metal  and  the semiconductor bulk called the contact potential (V ) which  is given as:

(3.4)

3.1.3   Schottky barrier height

From  Fig.  3.1 (b) it is seen  that  the  Schottky  barrier height  depends on the  work function of the metal (φm ) and the electron  affinity of the semiconductor (χ) as:

φB = (φm − χ)                                        (3.5)

This relation is called the Schottky-Mott relation. This model of determining a Schot- tky  barrier is based  on a few assumptions: (a) The surface  dipole  contribution to φm and  χ do not change  when  the metal  and  semiconductor are brought together. (b) There  are no localized states  present on the surface  of the semiconductor, and it forms  a perfect  contact  with  the metal.   In more  complex  approximations deter- mining the Schottky  barrier height, the influence of image potential, tunneling, and electrostatic screening should be taken  into  account.  These  three  mechanisms are discussed as follows:

Figure 3.3: (a) Left: field caused by an electron  close to the metal-semiconductor interface and surface  charges.   Right:  field caused by two opposite charges on either  side of the interface. (b) Representation of a Schottky  barrier showing the  image  force effect which  lowers  and pulls  the SBH maximum inside  the semiconductor, indicated by the shaded blue region.

1. Image force lowering.

2. Lowering due to tunneling.

3. Lowering due to electrostatic screening.

Schottky barrier lowering by image charge potential

When  an electron  approaches the metal-semiconductor interface,  it attracts surface charges of opposite sign in the metal.  These surface charges in the metal film exactly balance  the field generated by the electron  in the semiconductor, so that it does not penetrate into the metal  as shown in the left side of Fig. 3.3 (a). The field produced by these  surface  charges and  the electron  in the semiconductor is the same  as the field generated by an electron  in the semiconductor and another particle  of opposite charge  in the metal  as shown in the right  side of Fig.  3.3 (a).  This other  particle  is called the image charge.  This image charge in the metal film creates an image charge

Figure 3.4: Field and thermionic-field emission under forward bias. EF M and EF S represent the Fermi levels of the metal and the semiconductor respectively, V is the applied voltage  and Em is the energy where the contribution of thermionic-field emission has its maximum.

potential close to the barrier. This field is the highest at the barrier, because  there the electron  is very close to its image  charge.  At a large distance from the interface,  the electron  hardly feels the attraction of its image  charge  anymore and  the attractive force goes to zero. The potential energy caused by this image charge as a function of distance from the interface is schematically depicted in Fig. 3.3 (b). When the image potential energy is added to the original potential in the depletion layer, we find the barrier shape  that accounts for the image force. This resulting barrier height  is lower by an amount ∆EI given by [1], [2]:

(3.6)

In addition to lowering of the Schottky  barrier, the image charge potential also pulls the potential maximum into the semiconductor as shown in Fig. 3.3 (b) over a dis- tance of ∆z given as follows [1], [2]:

(3.7)

Because  the  maximum potential lies inside  the  semiconductor, the  electrons first travel  a short  distance through the semiconductor before  they  reach  the top of the barrier.

Schottky barrier lowering by tunneling

The  second  mechanism that  can  cause  lowering of the  effective  Schottky  barrier height  is tunneling, either direct or thermally assisted. Under forward bias, in heav- ily doped semiconductors at low temperatures, electrons can tunnel directly from the Fermi level of the semiconductor, through the Schottky  barrier, to the metal.  For reverse bias, tunneling from the metal  to the semiconductor can happen under the same circumstances. The current that arises from these electrons is called field emis- sion as shown in Fig.  3.4. When  electrons have  a certain  thermal energy, they  can also tunnel through the barrier with  thermal assistance. Since the barrier is thinner at higher energies, electrons with  higher energies have  higher tunneling probabil- ity.  On the other  hand, the number of electrons with  higher energies are few.  This implies  that  the electrons with  a certain  amount of energy have  maximum contri- bution to thermionic-field emission (denoted by Em ), as shown in Fig.  3.4.  When the  Schottky  barrier is approximated as a triangular potential barrier, the  tunnel- ing probability (P) for an electron  having an energy ∆E less than  the height  of the barrier is given by following equation [2]:

(3.8)

Here ∆E is the energy of the electron  below the top of the barrier and Vbi is the built- in potential. E00  is a parameter which plays an important role in tunneling theory.  It is the diffusion potential of a Schottky  barrier such that the transmission probability for an electron  whose  energy coincides with  the bottom of the conduction band  at the edge of the depletion region  is equal  to e−1  [2], and is given by:

(3.9)

Here ~ is the Planck constant, Nd  the donor concentration and   s the permittivity of the semiconductor.

From Eqn.  3.8 it can be deduced that  an E00 value  of 0 leads  to a tunnel proba- bility of zero and  a higher value  leads  to a higher tunnel probability. Also, a lower

∆E value,  which  means  a higher electron  energy, leads  to a higher tunnel proba- bility.  When  ∆E becomes  zero,  i.e.  the electron  has an energy equal  to that  of the Schottky  barrier, the tunnel probability goes to 1, which  we would expect, since the electron  has enough energy to overcome the barrier. Because the tunneling electrons can cross the barrier at an energy lower  than  the maximum of the Schottky  barrier, direct  and  thermally assisted tunneling lower  the effective  Schottky  barrier height as shown in Fig. 2.8. The amount of Schottky  barrier height  lowering due  to these

Figure 3.5: Schottky  barrier lowering due to electrostatic screening. φ and φef f represent the original and effective Schottky  barrier height  and ∆EES is the Schottky  barrier lowering due

to electrostatic screening. Adapted from [4].

effects is given by [12]:

(3.10)

Schottky barrier lowering due to electrostatic  screening

The third mechanism that can cause Schottky barrier lowering is electrostatic screen- ing. In ideal Schottky  theory,  the potential distribution in the metal is assumed to be constant. However, this condition may be violated in the metal close to the interface with  the semiconductor, when  the magnitude of the free charge  carriers  induced at the  surface  of the  metal  becomes  large  [5].  This is the  case at an interface with  a large permittivity semiconductor, such as Nb:SrTiO3 . This large permittivity causes a voltage  drop on the  metal  side  of the  junction  due  to the  conservation of elec- tric displacement, as shown in Fig 3.5 [6]. The barrier potential corresponds to the energy that  is needed to excite an electron  from  the bulk  of the metal  to the semi- conductor. From  Fig.  3.5 we can see that  this barrier is lowered by the amount of the voltage  drop in the metal.  For zero applied bias, this value  can be calculated as:

(3.11)

Figure 3.6: Transport processes in a forward-biased Schottky  junction. (a) Thermionic emis- sion over the barrier, (b) Thermally assisted tunneling through the barrier and (c) Direct tun- neling from the bottom of the conduction band.

In contrast to barrier lowering by image  force and  tunneling, the barrier lowering due to electrostatic screening is proportional to the square root of the semiconductor permittivity. Thus for higher values  of the relative  permittivity the Schottky  barrier is reduced by a larger  amount.

3.2    Electronic  transport across a Schottky barrier

For macroscopic characterization of Schottky  junctions, current-voltage measure- ments  (I-V measurements) are  most  commonly used.    In  such  measurements, a varying voltage  is applied across  the  interface and  the  current through the  inter- face is measured as shown in Fig.  3.6. In this circuit,  electrons cross the barrier at the interface between the semiconductor and  the metal.  Following are the various mechanisms by which  electronic  transport across the barrier can take place (Fig 3.6) [1], [2].

1. Thermionic emission over the top of the barrier.

2. Thermally assisted tunneling through the barrier.

3. Direct tunneling through the barrier.

3.2.1   Thermionic  emission

The electrical  transport across  an ideal  Schottky  barrier is described by thermionic emission [1]. By subtracting the current which  flows from the metal to the semicon- ductor JM →S   from the current flowing  from the semiconductor to the metal  JS→M the following expression for the total current I is obtained [1], [2]:

(3.12)

where q is the charge  of the electron,  kB  the Boltzman constant, T the temperature, A* the Richardson constant, φB the barrier height and n the ideality factor (unity  for purely thermionic emission dominated transport), and A∗ is given by:

(3.13)

where me ∗ is the effective mass  of the electron  in the semiconductor. The value  of A∗, the Richardson constant, used in this thesis for Nb doped SrTiO3  semiconductor is 156 Acm−2 K−2 [8]. When temperature is kept constant during a measurement, the only  variables are the  ideality factor  and  Schottky  barrier height  for zero  applied voltage,  so these  parameters can be determined by fitting  the  experimental data. However in practice, resistances appear in the  semiconductor and  the  rest  of the circuit  as well,  and  contribute to the  series  resistance causing the  current-voltage characteristics to deviate from thermionic emission theory at high voltages. Because of the series resistances in the circuit, the applied voltage  does not drop completely at the Schottky  barrier, but  also drops partially in the rest of the circuit  and  in the semiconductor. To reckon for the voltage  loss due to these resistances we can adjust Eqn. 3.12. The voltage  drop over the interface is then  given by V  minus I R instead of V , where R is the  total  resistance of all elements in the  circuit.   This results in following equation for current:

(3.14)

3.2.2   Electron transport by tunneling

Thermally assisted  tunneling and direct tunneling

Since electrons have a higher thermal energy at higher temperatures, thermal emis- sion (represented by (a) in Fig. 3.6) is more dominant in that case. For lower temper- atures, electrons lose their thermal energy and direct  tunneling becomes  dominant. In between these  two  regimes lies the thermally assisted tunneling regime.   While at very  low temperatures electrons tunnel directly through the barrier (direct  tun- neling,  represented by (c) in Fig.  3.6), at intermediate temperatures electrons first get thermally excited and then tunnel at a higher energy corresponding to a thinner part  of the barrier (thermally assisted tunneling, represented by (b) in Fig.  3.6). In the direct tunneling regime  the current is given by [7]:

(3.15)

and

(3.17)

where, E00 is a tunneling parameter (also called as characteristic energy). E00 (T

= 0 K) is 1, (Eqn. 3.9.) hence E0 (T = 0 K) equals  E00 . It also implies  that for this case

Eqn. 3.16 approaches to Eqn. 3.15 at very low temperatures.

3.3    Ballistic electron emission microscopy (BEEM)

Introduction

The technique of ballistic  electron  emission microscopy (BEEM) was developed by Kaiser and Bell in late 1980’s [3]. BEEM is a non destructive technique and is based on a scanning tunneling microscope (STM) [10]. It is a modified form of STM, with an additional contact  at the bottom of the semiconducting substrate, which  can col- lect the electrons traveling through the metal overlayer and across a Schottky  inter- face. Here, the STM tip is used  to inject a distribution of hot electrons into the metal overlayer to be investigated. The hot electrons travel  through the metal  overlayer and  get scattered. A fraction  of these  electrons are able to cross the Schottky  inter- face when  they have the necessary energy and momentum to do so.

BEEM has been used  for studying hot electron  transport in thin films and multi- layers using  conventional semiconductors like Si, GaAs [11], [12], [13], [14]. Energy and spatial dependence of carrier  transport, at the nanoscale and across buried lay- ers  and  interfaces using  current perpendicular to the  plane  of the  device  can  be

Figure 3.7: Schematic  of a BEEM setup  with  its circuit  diagram. A STM tip injects hot elec- trons  in a metal  overlayer via tunneling through the vacuum barrier. The electrons travel  to the  M-S interface.   The white  arrows represent the  spatial distribution of injected  hot  elec- trons.   The electrons with  proper momentum and  enough energy reach  the semiconductor. Due to the momentum criteria,  electrons outside the acceptance cone (in purple) are reflected back from the M-S interface.

investigated using  BEEM. The basic schematic of BEEM with  its circuit  diagram is shown in Fig.  3.6. A negative bias, VT , is applied to the tip to inject electrons into the metal film, as tunnel current, IT . The electrons travel through the film, across the interface and  are collected  in the semiconductor as a BEEM current, IB . The BEEM current constitutes a fraction  of electrons which  have  the  proper energy and  mo- mentum to overcome the Schottky  barrier height. Such an energy and  momentum filter is represented by an acceptance cone at the Schottky  interface as shown in Fig.

3.8. The energy schematic of the BEEM is shown in Fig. 3.9.

Hot electrons and their scattering mechanisms

When  the injected  electrons have  an energy a few tenths of an electron  volt above the Fermi  level of the system they  are referred to as "hot" electrons. By applying a bias of a few eV to the STM tip with  respect  to the Fermi level of the metal  layer we inject a distribution of electrons into the metal  overlayer. As kB T is 25 meV at room temperature, a similar  analysis yields an equivalent temperature of ≈ 12000 K for 1 eV [14]. Such an analogy leads to the term "hot" electrons when  the energy of the injected  electron  is few eV above  the Fermi level of the metallic  film. Scattering

Figure 3.8: Energy  schematics of the  BEEM technique.  It shows  the  hot  electron  distribu- tion  injected  into  the metal  overlayer.  Subsequently, a fraction  of them  get collected  in the conduction band  of the semiconductor.

mechanisms for hot  electrons and  for electrons at the  Fermi  level  are very  differ- ent.  Hot  electrons injected  at an energy eVT  can scatter  into all unoccupied states between eVT  and  EF , according to Fermi’s golden rule.  Such an electron-electron (e-e) scattering for hot electrons results in inelastic  scattering (loss of energy)  and  is a dominant scattering mechanism. In contrast, at the Fermi  level elastic  or quasi- elastic scatterings are the dominant scattering mechanism. When  the hot electrons reach the interface without being scattered inelastically or elastically,  they are called "ballistic" electrons.

The hot electron  transport in BEEM can be divided in different steps:

1. Injection of the hot electrons from the tip into the metal base.

2. Transport of hot electrons through the metal base.

3. Transmission of hot electrons across the metal-semiconductor interface.

4. Collection  at the conduction band  of the semiconductor.

Charge carriers  are  injected  from  the  tip  by tunneling into  unoccupied states of the  thin  metal  base.   This results in momentum and  energy distribution of the injected carriers  at the metal surface.  After injection, the hot electrons travel through the  metal  film and  are  scattered by cold  electrons (lying  close  to Fermi  level)  by inelastic  scattering.  However, an  energy independent elastic  [15] or quasi-elastic scattering by with  either  defects,  grain boundaries, phonons, magnons etc. can also occur. When the electrons reach the interface and satisfy the energy and momentum criteria  at the interface,  they  can be transmitted through and  enter  the conduction band  of the semiconductor and constitute the BEEM current. Due to the local nature of injecting  electrons and the requirement of lateral  momentum conservation at the Schottky  interface,  this technique results in a very high spatial resolution [16] .

3.4    BEEM Theory

In order  to extract  the Schottky  barrier height  from spectroscopy measurements, a theoretical model  is needed to fit the  data.   The first theoretical description deal- ing with the transport of hot charge  carriers  through a metal-semiconductor system in a BEEM setup  was  proposed by L.D. Bell and  W.J. Kaiser  [3].  For all the work presented in this thesis,  electrons are the charge  carriers  responsible for the BEEM current, due to the use of n-type semiconducting substrates (Nb doped SrTiO3 ).

3.4.1   Tunnel  injection  of non-equilibrium charge carriers

The  applied potential between the  tip  and  the  metal  base,  called  the  tip  voltage VT , will determine the energy of the injected  electrons. Tunneling across the poten- tial barrier between the tip and  the metal  will always result  in a distribution of the energy and momentum of the electrons. In common BEEM theory [3], the tunnel in- jection of non-equilibrium electrons from the tip into the base is assumed to behave according to the planar tunneling theory [17]. Although it has been shown that it is not always valid to use planar tunneling theory,  the voltage  spectroscopy measure- ments  with  BEEM are found to agree well with  planar tunneling based  theory [18]. At tip voltages close to the threshold this  results in a sharply peaked distribution of the injected  electrons perpendicular to the M-S interface.   Therefore the injected electrons will have little momentum parallel to the metal base (kk     k⊥).

3.4.2   Transport across the metal base

Due to scattering the spatial and energetic distribution of the electrons will broaden when  traversing the metal  base.  When  assuming a free electron  like behavior the attenuation of the electrons can be described by a single parameter called the atten- uation length, λ(E), which  in principle is energy dependent.  The attenuation can

Figure 3.9: Four different scattering mechanisms in a forward biased  BEEM, where the solid spheres represent electrons and hollow  spheres represent holes. (1) is a purely ballistic trans- port (red), (2) is inelastic scattering of hot electrons in the metal overlayer (pink) can also lead to secondary electrons, (3) elastic scattering in the metal  over layer and (4) impact  ionization where an electron-hole pair is created (green).

then be described by an exponentially decaying function depending on the injection angle θ away  from the surface  and metal film thickness d:

(3.18)

Since the electrons are injected  with  almost  zero parallel momentum kk = 0 we can assume cos(θ) ≈ 1, simplifying the equation.

3.4.3   Scattering mechanisms

All of the different scattering processes which  are relevant in this thesis occur in the metal base and interface.  In Fig. 3.9, the most prominent scattering mechanisms are depicted, which  are:

Ballistic  charge carriers   Ballistic transport is the unscattered propagation of elec- trons  through the  metal  base.   These  electrons do  not  lose  energy or  undergo a change  in momentum and  form  an  important contribution to the  BEEM current.

If the electrons travel  ballistically through the metal  base they  might  have  enough energy, depending on VT , to surmount the Schottky  barrier at the M-S interface.

Inelastic  scattering    If the electrons are scattered inelastically their  energy will be reduced. The processes dominating this form of scattering, at the energies relevant for this  thesis  is electron-electron (e-e) scattering [3] and  will  typically result  in a reduction of half the electron  energy. At low tip voltage  this effectively  means  that any  inelastically scattered electron  will  not  have  enough energy to surmount the Schottky  barrier. However at higher tip voltages, at least twice that of the Schottky barrier, the  collision  might  result  in a secondary electron  with  enough energy to surmount the Schottky  barrier, while the primary electron still has energy above the Schottky  barrier, thereby increasing the BEEM current. Although phonon scattering can also result  in energy loss, they  are not  taken  into  account since the change  in energy is negligibly small  (in the order  of kB T ) in comparison with  e-e scattering. Although plasmon excitations also cause  inelastic  scattering they  are not  relevant since the electron  energies relevant for this thesis are too low for plasmon excitations to occur.

Elastic  scattering    This  form  of scattering will  change  the  momentum but  con- serves  the  total,  kinetic,  energy of the  electrons.  Therefore any  elastic  scattering will result  in a broadening of the distribution of angular momentum. Since trans- mission across  the M-S interface depends sensitively on the momentum, as shown in section 3.4.4, elastic scattering will also have an effect on the BEEM current. Grain boundaries, defects and any inhomogeneities in general are the main elastic scatter- ing sites.

Impact ionization   When an electron  with high enough energy enters  the semicon- ductor, it could  transfer a part  of this energy to an electron  in the valence  band.   If enough energy is transferred, it could  excite the  electron  to the  conduction band creating an electron-hole pair.  This electron  could then contribute to the BEEM cur- rent.  However, for this to occur  the impacting electron  should have  an excess en- ergy  nearly  more  than  twice  the  semiconductor band  gap.   Since all experiments performed in this thesis are below this limit, impact  ionization is absent.

Transport across the metal base

Scattering causes broadening of the spatial and energetic distribution of the injected electrons while  traveling through the metal  overlayer. Scattering in the metal  over- layer  can be quantified by hot  electron  attenuation length  When  assuming a free electron  like behavior the attenuation of the electrons can be described by a single parameter called  the attenuation length, λ(E), which  is usually an energy depen- dent  parameter (Eqn. 3.18).

3.4.4   Transmission across the M-S interface

The transmission across  the  barrier is dependent on the  energy and  the  momen- tum  of the incoming electrons. Assuming the electrons satisfy  the 2-d free electron model,  their energy would be given as:

(3.19)

where m is the  rest  mass  of the  electron  and  k⊥ and  kk are the  momentum of the electron  perpendicular and parallel to the M-S interface,  respectively. Here, kk is assumed to have both Kx and ky  components. The energy of the electron  just at the maximum of the Schottky  barrier height  can now be expressed as:

(3.20)

where m∗ is the effective mass of the electron  inside  the semiconductor and  the subscript of kS   denotes the  momentum in the  semiconductor.  If we now  assume conservation of parallel momentum we can obtain  an analytical expression for the maximum allowed parallel moment kkS .  This argument would only be fully con- vincing  for a fully  epitaxial system without defects,  any  deviations from  such  a system would break  the symmetry and  therefore conservation of parallel momen- tum  could  be lost to a certain  degree. Despite this, experimental evidence for non- epitaxial Au/Si systems showing momentum conservation has been observed [19]. We can equate Eqns. 3.20 and 3.21 which  will give us an expression for Ek:

(3.21)

The maximum amount of parallel energy Ekmax would be obtained if the elec tron would have exactly zero perpendicular energy left after crossing the Schottky barrier i.e. E⊥S = 0. This shows that due to the effective mass there is a restriction on the amount of parallel momentum:

(3.22)

If the electron has more parallel momentum than Ekmax it cannot be transmitted across the M-S interface and will bounce back into the metal base. This effect is much  like the total refraction of light at an interface of two media  having different refractive indices.  Since we have found the maximum parallel momentum that elec- trons can have and we know that the maximum total energy is the Fermi energy, the minimum perpendicular energy Emin is equal  to:

(3.23)

We can now express this momentum requirement in the form of an acceptance cone at the M-S interface:

(3.24)

3.4.5   BEEM transport models

In order  to extract  the Schottky  barrier height  from  spectroscopy measurements a theoretical model  is used.  The first theoretical description dealing with the transport of hot-charge carriers  through a metal-semiconductor system in a BEEM setup  was proposed by L.D. Bell and  W.J. Kaiser [3]. The tunnel current between tip and  top metal based  on planar tunneling theory can be written as:

(3.25)

T (E⊥) is the tunnel probability for an electron  to tunnel through the vacuum barrier over  the  transverse and  parallel (to the  interface) energies, E⊥  and  Ek.   A is the constant related to effective tunneling area, f (E) is the Fermi distribution function, and VT  is the applied tip voltage.

According to the widely used  Bell-Kaiser (BK) model  [3], BEEM transmission is the fraction  of the ballistically transmitted tunnel current:

(3.26)

where R is an attenuation factor  due  to scattering in the  metal  base  and  the  M-S interface.  According to the BK model,  R is considered to be energy independent but it can also be weakly dependent on energy (other  parameters are:  

and      )

For VT  just above φB , close to threshold, above Eqns. 3.26 and 3.27 predict:

(3.27)

Figure 3.10: Typical spectra for direct (left) and the reverse (right) BEEM spectroscopy.

Such  a quadratic onset  considers classical  transmission across  the  M-S interface with parabolic conduction band minimum in the semiconductor. Considering quan- tum  mechanical transmission across the M-S interface another model  was given by Ludeke-Prietsch (LP model)  according to which  IB ∝IT (VT  − φB )2.5  [12].  It was found that near the threshold regime,  no significant difference between the BK and LP models can  be resolved beyond experimental error.   For the  Schottky  barrier extraction in our experimental measurement we have  considered the BK model  in- stead of the LP model  and we have seen a better match of SBH with the macroscopic I − V measurements.

3.4.6   BEEM spectroscopy

Direct and Reverse BEEM spectroscopy

In direct  BEEM spectroscopy, which  is one of the most  commonly used  modes in BEEM, a negative bias, VT , is applied to the tip with  respect  to the metal  forming a Schottky  barrier contact  with  a n-type semiconductor. When  the sample-tip bias is below  the Schottky  barrier height, no BEEM transmission is observed. However, IB  increases after a certain  onset  that corresponds to the local SBH. A typical  direct BEEM spectra thus  can be obtained by recording the BEEM current with  respect  to the sample-tip bias at a fixed tip position (shown in fig.  3.8 ). Usually, to improve

Figure 3.11: Energy  schematic of the reverse BEEM spectroscopy.

the signal to noise ratio, several  BEEM spectra are recorded to obtain  a single aver- aged spectrum. The final BEEM spectrum provides valuable information on energy dependence of hot electron  transport in the metal  film as well as the M-S interface. The onset  of the  BEEM spectra determines the  local Schottky  barrier height  with high  accuracy ( of ±0.02 eV) whereas the spectral shape  carries  information about scattering in the metal film, across the M-S interface and in the semiconductor.

In Reverse  BEEM spectroscopy, a positive bias is applied to the  STM tip  with respect  to the metal layer and a distribution of electrons is extracted from the metal overlayer (grown on a n-type SC) to the STM tip.  Reverse  BEEM is based  on the collection  of only  secondary electrons in the  conduction band  of the semiconduc- tor. These secondary electrons are produced by electron-electron scattering which is similar  to Auger  like scattering.

In the case of R-BEEM [20], hot holes  injected  by the tip (corresponding to hot electron  extraction from  the  metal  overlayer) lose energy by creating a secondary electron-hole (e-h) pair.   The energy of the  injected  holes  is transferred to the  ex- cited  electrons up  to a maximum of EF,m + eVT .  If the  secondary electrons have enough energy and  momentum to surmount the  barrier, they  can be collected  as collector current with the same sign as direct BEEM. Considering free electrons and zero temperature, R-BEEM transmission can be written as:

(3.28)

where EF,m  is the base Fermi energy, P (E, E⊥)  is the probability of creation of ex- cited electrons from the injected  hot holes.  The excited  electrons are then  collected above φB with proper momentum. Near threshold, the above expression of R-BEEM transmission can be simplified as:

(3.29)

3.4.7   Hot electron attenuation  length

With increasing metal  base thickness the BEEM current is attenuated. The total at- tenuation length, λ, is related to the inelastic  attenuation length, λI , and  the elastic attenuation length, λe , as described by Matthiessen’s rule:

(3.30)

λe   corresponds to scattering due  to different factors  viz.  defects,  grain  bound- aries, phonons, magnons, polarons etc. From equation 3.18 it is clear that the trans- mission is exponentially dependent on the thickness of the metal  base.  Therefore, by varying the  metal  base  layer  thickness and  measuring the  BEEM transmission at a particular energy, a plot can be obtained of the transmission versus metal  base thickness and energy. The slope of the plot (semi-log)  gives the electron  attenuation length  at a particular energy. The energy dependence of the attenuation length  can now be obtained by repeating this process  at different energies. Although there  are possibilities of extracting the two different attenuation lengths λi and λe  from λ, this is generally not straight forward.

3.4.8   BEEM sample requirements

In BEEM, the measured signals  are often very low (tenths  of pA) and  it is thus  im- portant to reduce noise in the system to measure such tiny currents. The most  im- portant source  of noise in the BEEM signal  arises from the feedback resistors of the operational amplifier (op-amp) circuits,  which  amplify the BEEM current, and from the Schottky  interface.  The feedback resistors of the op-amp circuits  (BEEM current is monitored by a two-stage-op-amp) add noise to the BEEM channel. Although this is hard  to avoid,  the other  noise source  related to the sample can be reduced.  The

Figure 3.12: As the electrons pass through the metal over layer they are inelastically scattered resulting in an exponential decay of the transmission.

voltage  fluctuation across  a resistor at finite temperature is known as Johnson noise

and is given by

(3.31)

where ∆ V is the root mean  square of the voltage  fluctuations, kB  is the Boltzmann constant, T the measurement temperature, B the measurement bandwidth and R the value  of the resistor.

The current passing through a Schottky diode,  described by thermionic emission theory is given  by Eqn.  3.14. In BEEM we are interested in the zero bias resistance of the diode  which  is given as

(3.33)

Thus, by increasing the zero bias resistance R0  of the diode,  its contribution to noise can be decreased. To make sure the diode  is not dominating the noise, its resistance should be higher than the resistance of the op-amp. For common M-S interfaces such as Au/Si the SBH is ≈ 0.8 V, the junction  resistance is of the order  of 1 GΩ at room temperature for a diode  area with  a diameter of 150 µm and thus  is high enough to make sure the sample is not dominating the noise. This is thus an important sample requirement for BEEM studies [11].

3.5    Conclusions

In this  chapter we  discussed the  different transport mechanisms across  a metal- semiconductor (M-S) interfaces along  with  the  different models that  are  essential to the determination of the Schottky  barrier height. We also discussed the different modes of ballistic  electron  emission microscopy that  are used  in this thesis  and  the commonly used  model  to interpret hot  electron  transmission in metal  layers  and across their interfaces with a semiconductor.

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