sample tip VB
CB
EF
t s
0 d
BB
z
Figure 3.8: Tip induced band bending at zero bias. If no bias is applied, the difference between semiconductor and tip work functions (in our case = s t= 0:8 eV) lead to downwards bending of conduction band (CB) and valence band (VB) at the sample surface. Part ofdrops within the sample due to charge accumulation, the rest drops linearly across the vacuum gap.
level is unpinned. That volume is often called space charge region, since the charge needed to screen the tip potential is spread out over it, determined by the materials limited capacity to accumulate charge.
Within this space charge region, the presence of the charges creates a locally varying electrical potential. At every point in the sample, this potential shifts the semiconductor bands accordingly. At the sample surface, the energy bands shift up or down towards the STM tip, depending on differences in tip and sample work functions and on the potential difference across the vacuum gap, which has led to the term tip-induced band bending. The bands stay parallel to each other since the width of the band gap is an intrinsic property of the material. Figure 3.8 illustrates this situation.
Since band bending shifts the sample LDOS and is affected by the applied bias, we have a situation where the measurement process and the measurement parameters affect the configuration of the sample. So a reasonably accurate pic-ture of the band bending will be crucial to our understanding of measurement results.
For the following we assume that equilibrium has been reached, meaning the Fermi levels of tip and sample have aligned, either by tunneling electrons or by shorting tip and sample, i.e. applying a bias of 0 V. If we apply an additional bias voltageU to the sample, we can already calculate the surface potential in a very straightforward way. The difference between tip work function and sample work function plus the applied sample bias equals the total potential drop:
= t (s eU) (3.17)
In our case the semiconductor’s work function (p-InAs: 5:3 eV) is larger than the tip’s work function (W: 4:9 eV). Therefore at zero bias, there will be an
accu-sample tip VB
CB EF
0 d
BB
a b c
zero bias:
weak downwards BB
negative bias:
inversion
positive bias:
accumulation Figure 3.9: Tip induced band bending and states contributing to the tunnel current at zero, negative and positive bias. (a)At zero bias, there is downwards band bending and no juxtaposition of unoccupied and occupied states and therefore no tunneling. (b) At negative bias the downwards band bending is increased. Tunneling happens between valence band and unoccupied tip states; if conduction band states are occupied, also from those. If the conduction band edge gets pushed below the bulk Fermi energy a triangular potential well is created, in which quantum dot states can exist. (c)At positive bias, the bands are pushed upwards at the surface. Tunneling happens from tip states to conduction band states. Again, a triangular well is formed, in which quantum dot states can exist and participate in tunneling.
mulation of negative charges in the sample and positive charge at the tip surface.
As there are now charges on both sides of the vacuum gap, there is an electric field from the tip to the sample.
The above mentioned is the total energy shift between the metal and the region of the semiconductor with flat bands. This potential difference drops partly in the vacuum and partly in the semiconductor. If we call the amount of band bending at the semiconductor surfacebb and the tip to sample distanced, we get
e = d
@U
@z
z=0+ bb
e
We use a z-scale where the sample surface isz = 0and z increases into the vacuum and towards the tip and is negative inside the sample. Above, @U@z is the potential gradient in the vacuum. z=+0
To get a deeper understanding of the spatial structure of the space charge region, we need to calculate the actual charge density which accumulates. A typical procedure of finding the local charge density and electrostatic potential is to first determine the samples’ Fermi energy, then calculate the electrostatic potential at the sample surface bb, then use the fact that charge density and curvature of the potential are proportional and integrate the potential gradient
going into the bulk of the sample until zero potential†is reached. This is basically a one-dimensional approach, but can be generalized to a two-dimensional or even three-dimensional treatment.
The Fermi energyEF needed in the first step is temperature dependent, since the width of the Fermi distribution affects the occupation statistics. The correct value can be found by taking the temperature dependent Fermi function and the density-of-states functions of valence band, conduction band, acceptors and donors, and looking for a value ofEF that provides charge neutrality. (see [89]
sections 4.4-4.7).
The surface potential is found from the continuity principle for the electrical displacement at the sample surface. At the boundary between sample and vac-uum (z = 0), the electric field in the semiconductor Esc and in the vacuum Evac
and the electric potentialU must satisfy Evac = scEsc
Evac= sc
@U
@z
0
The local potential and the charge density always satisfy Poisson’s equation:
U(~r) = (~r)
sc0 (3.18)
So we can find the local potential by integrating the electric field, itself the in-tegral of the charge density. But additionally, the local potential also determines the occupation statistics of the bands and dopants and therefore the charge den-sity according to:
total charge density: (~r) = e NA(~r) ND+(~r) + n(~r) p(~r)
(3.19) electron density in
conduction band: n(~r; T ) = Z 1
EC(~r)DC() f() d (3.20) hole density in
valence band: p(~r; T ) =
Z EV(~r)
1 DV() [1 f()] d (3.21)
Fermi function: f(x) = 1
1 + e(x EF)=kBT
with NA: density of charged acceptors EC: conduction band minimum ND+: density of charged donors EV: valence band maximum
†or something close enough
-0.5 0 0.5 1 1.5 2 -0.6
-0.4 -0.2 0 0.2
Ubias (V)
bb (eV)
Ufb
depopulation of valence band states
ionization of acceptors
Figure 3.10: Calculated surface band bending versus bias voltage. The curve shows the amount of band bending at the sample surface. Flat bands (bb = 0 eV) occur at about 0:8 V. Below this voltage, negative charge is accumulated throug ionization of acceptors. Above this point, positive charge is accumulated through depopulation of valence band states. The different slopes of bb vs. Ubias are caused by the different densities of states of dopants and conduction band.
If there are no quantum dot states in conduction or valence band, one can use the density of states of a three-dimensional electron system, which is
DC=V() = gSgV(2mC=V)3=2 42~3
pj EC=Vj (3.22)
where mC: electron effective mass gS: spin degeneracy mV: hole effective mass gV: valley degeneracy
This determines the band bending at the semiconductor surfacebb and the local energetic position of the band edges, which in turn determines the occupa-tion ratio and therefore the charge density. This can be optimized until a self-consistent solution is found. Figure 3.10 shows a result of such a calculation for our sample system. More examples of calculated charge densities and band bend-ing profiles will appear in Chapter 5 and Chapter 6. Above the flat band voltage, there is upwards band bending and conduction band states become depopulated.
Below flat band voltage, there is downwards band bending and acceptors are pushed below the Fermi level and become negatively charged. Due to the higher density of states in the valence band compared to the acceptor states, less band bending is needed to accumulate holes in the valence band than electrons on ac-ceptors. This is reflected in the different slopes of bb vs. Ubias, ca. 0:2 eV=V aboveUfband about0:4 eV=Vfor lower voltages.