While interesting, your predictions are incorrect.
The cause of your mistake is rooted in the very first paragraph of the question - you misinterpret the meaning of built-in voltage. Allow me to write a step-by-step answer - you may already know most of the theory, but there are others who don't.
Doped Silicon
Without loss of generality let me talk about Silicon.
Doping is a process of adding non-silicon atoms into (otherwise pure) bulk of silicon. The dopants (=atoms which were added) are neutral, therefore the material stays neutral too. However, dopants have very interesting property - each dopant atom contribute one free charge carrier. This charge carriers can be used (and are used) as the main current carriers in semiconductor devices.
"Donor" dopants add negatively charged free carriers - electrons. The resulting material is called \$n\$-type silicon.
"Acceptor" dopants add positively charged free carriers - holes. The resulting material is called \$p\$-type silicon.
Depletion region
When bringing two oppositely doped pieces of Silicon into a contact, the severe difference in free carriers concentrations give rise to diffusion currents (there are much more holes at the \$p\$ side and much more electrons at the \$n\$ side). While free carriers diffuse across the boundary they "leave behind" static dopant ions. The carriers themselves neutralize each-other (recall that they have negative polarities), but the ions stay in their places and give rise to local electric fields:
The diffusion described above continues until the magnitude of the electric field, induced by the "exposed" dopant ions, is just enough to balance the tendency of free carriers to diffuse.
The region containing exposed ions is called Depletion Region. There is an electric field inside this region which prevents from free carriers to diffuse further.
Built-in voltage
Since there is an electric field in depletion region, there is a potential difference associated with this field. This potential difference is called "Built-in voltage" (usually denoted by \$V_{bi}\$; it is this voltage that you call \$V_{eq}\$).
Now is the most confusing fact about built-in voltage: it can't be observed externally. This means that if you take a voltmeter and try to measure the voltage between \$p\$ and \$n\$ sides of the diode when it is in thermal equilibrium (i.e. no external bias) - you'll read 0V.
Wrong interpretation
The first paragraph of your question suggests that you think of built-in voltage as representing the potential difference between the two sides of a diode. This is not the case: the potential difference is 0V in thermal equilibrium, and the built-in voltage only compensates for a diffusion tendency.
However, it is true that when an external bias is applied, the magnitude of the voltage across depletion region is reduced:
$$V_{depletion}=V_{bi}-V_{applied}$$
If \$V_A>0\$ (forward bias): the voltage across depletion region reduces and the width of the region reduces. In this case the diode will conduct a forward-bias current which has exponential dependence on \$V_A\$.
If \$V_A<0\$ (reverse bias): the voltage across depletion region increases and the width of the region increases. The diode will not conduct any appreciable current in reverse bias mode.
Now I suggest you'll try to reconsider your models, taking into account that there is no potential differences between between parts of semiconductors devices in thermal equilibrium.
The noise that is created is truly random as it is generated by recombination of electrons with the atoms on the other side of the junction. The mean power density (RMS) can be calculated using the following formula:
$$i_n = \sqrt{2 I q \Delta B}$$
where \$q\$ is the electron charge, \$I\$ is the current that runs through the diode and \$\Delta B\$ is the bandwidth of the detector that measures the noise.
Apart from that, it is a stochastic process and therefore random. Read more about it here:
http://en.wikipedia.org/wiki/Noise_(electronic)#Shot_noise
Best Answer
The H11L1 chip uses four diodes in series as reference for its very simple voltage regulator:
It does not even try to supply a constant current, so its characteristics vary depending on supply voltage and temperature:
(source)
If you need a voltage that actually can be called "constant", you need a better reference.