I have read that resistance is proportional to resistivity and length of the material and inversely proportional to the area of the material.
Doesn't that make resistance a constant thing?
Then I read that under breakdown of material and in things like photodiodes, resistance changes/decreases. How is this possible when length, area and resistivity are constant?
At a constant temperature, all other things being constant, resistivity \$\rho\$, and thus resistance is constant or very nearly so for metals. This came from an empirical observation that current in metals was closely proportional to voltage, published by Georg Simon Ohm in "Die galvanische Kette Mathematisch Bearbeitet" (1827).
In general it's not constant. It may vary with voltage, with temperature, with self-heating, with pressure or other external factors, it may even be anisotropic, in which case \$\rho\$ is represented in tensor form.
For something like a diode or a varistor the current that flows when you apply a given voltage is not even remotely proportional to the voltage so it makes no sense to use \$\rho\$ or the R= \$\rho\$L/A equation.
Doesn't that make resistance a constant thing?
No.
Temperature coefficient of resistivity is one property that's used for RTD (resistor temperature device). One of the popular types used in industrial temperature control is the Pt100, where Pt is the chemical symbol for platinum and 100 is its resistance (100 Ω) at 0°C. At 200°C the resistance will increase to approximately 138Ω (38%) - so its resistance is definitely not a constant thing.
Figure 1. Constant? It's not even linear! Image source: Beamex.
It's all about electron mobility (or carrier mobility, in general) and electron density (or again, charge carrier density, in general).
In the resistance formula that you are familiar with, which is
$$ \mathrm{R = \rho \frac{l}{A}} $$
the resistivity, \$\rho\$ (= 1/\$\sigma\$ i.e. inverse of conductivity), is determined by the electron mobility and the carrier density.
For metals it's fairly constant (different for each metal, of course) as the carrier density is relatively the same along the entire length.
But for semiconductors it's not because the carrier quantity and/or density is not constant. So, even if you keep the mechanical dimensions constant (ignoring the effect of temperature), you don't have a constant resistivity due to the states of electrons and holes (carriers).
Doesn't that make resistance a constant thing?
Nope. In fact, many people's jobs are to use the non-constant resistance as a transducer from some other physical quantity, such as mechanical strain, thermal strain, temperature via the thermal coefficient of resistance, and so on.
Electronic bathroom scales work because you can change the resistance of a piece of metal by stretching it.
No. The simplest example is an incandescent lightbulb.
At room temperature, the resistance of the filament is very low, so there's a lot of current going through it at the set voltage, and it heats up quickly. But as its temperature rises, its resistance rises as well and the current decreases. That's why incandescent bulbs usually burn out right in the moment you turn them on.
Some materials are nearly incompressible, and the quantity of water that would be displaced by submerging a given mass of such a materials may be very nearly constant and predictable. Some other materials may be readily compressed or stretched to achieve more than a 2:1 variation in volume, and it would be impossible to know how much water a certain mass of such a material would displace without knowing the pressure depth of the water as well as any other stresses to which the material would be subjected. For the former materials, it would make sense not only to describe the density as the mass in kilograms required to displace one liter of water, but also to use that ratio to predict how much water would be displaced by a given mass, or what mass would be required to displace a given volume. The latter materials would have a certain mass and displace a certain volume of water, and would thus have a "density", but the ratio would be far less meaningful and useful with those materials than it had been with the incompressible ones.
The relation between voltage, current, and resistance in various materials might be seen as somewhat analogous to the relationship between mass, volume, and density in the above-described materials. Some materials have a resistance that is close enough to being constant that it can be used to make accurate predictions about how they will behave electrically, but other materials behave in ways that cannot be predicted by assuming a simple constant resistance.
I have read that resistance is proportional to resistivity and length of the material and inversely proportional to the area of the material.
This property (R = ρ.l/A) is only true for conductive materials with a uniform resistance distribution. Typical examples are metals, graphite, etc.
Some history: In the early 90's, I was obsessed with the idea of using conductive foam to make all sorts of "rubber resistors". It was particularly useful for educational purposes, for example to illustrate connecting resistors in series and parallel, voltage dividers, bridge circuits, etc.
CircuitLab experiments: Now I wondered if this could not be done with CircuitLab and hastily fabricated the story below. I have used as a basis an IEC resistor with unit length, area and resistance. Using a current source I and voltmeter R(kΩ) I have made a DIY ohmmeter. The voltmeter reading is in volts, but consider it to be in kohms.
l = 1; A = 1; ρ = 1: First we connect a single resistor R1. The voltmeter shows 1 V, which at a current of 1 mA corresponds to a resistance of 1 kΩ (the measured corresponds to the actual).
simulate this circuit – Schematic created using CircuitLab
As you can see from the graph below, the voltage across the resistor is proportional to the current.
l = 2; A = 1; ρ = 1: Then we increase the "length" two times by connecting another single resistor (R2) in series. The resistance increases proportionally two times.
l = 3; A = 1; ρ = 1: Finally, we increase the "length" even more by connecting another single resistor (R3) in series. The resistance increases three times.
So, we conclude that resistance is proportional to length.
l = 3; A = 2; ρ = 1: Now let's increase the "area" two times by connecting another string of resistors (R4, R5 and R6) in parallel. The resistance decreases two times.
l = 3; A = 2; ρ = 1: We can connect the same midpoints between resistors since their voltages are equal. We can even imagine that the same points inside resistors are connected.
l = 3; A = 3; ρ = 1: Finally, we increase the "area" three times by connecting another string of resistors (R7, R8 and R9) in parallel. The resistance decreases three times.
So, we conclude that resistance is inversely proportional to area.
l = 3; A = 3; ρ = 2: I have imitated the resistivity by the resistance of the single resistor. So, to increase the resistivity two times, we have to increase the resistance of each resistor two times. As a result, the total resistance increases two times.
l = 3; A = 3; ρ = 3: Finally, to increase the resistivity three times, we increase the resistance of each resistor three times. As a result, the total resistance increases three times.
Our conclusion is that resistance is proportional to resistivity.
... resistance changes/decreases. How is this possible when length, area and resistivity are constant?
Resistance is not always related to the geometric dimensions and it is not appropriate to talk about resistivity, length and area in this case. Here is an unconventional example.
I have made a (virtual) 1 kΩ resistance using an opposing behavioral voltage source whose voltage V [V] = 1000*I[mA] is controlled by the current flowing through it (ie it is a current-to-voltage converter).
Obviously, this resistance has no resistivity, no length and no area.
As you can see from the graph below, the voltage across this virtual resistor is proportional to the current like in the real resistor in Schematic 1.1.
I like to answer your question in a more general way:
In electronics, nothing is constant. Neither a supply voltage nor other related quantities like current, resistance, capacitance,... Therefore, no relation or formula which combines one or more of such quantities can really be correct.
But it is the task of an application technician/engineer to decide if - for a specific application, one particular quantity - or a corresponding formula - may be treated as constant resp. sufficiently correct.
In most cases, it is allowed/acceptable to treat the part called "resistor" with a constant value over a certain temperature range. This seems to be reasonable with respect to other uncertainties (tolerances, parasitic effects,...). However, there may be some extreme conditions where this is not the case and we must take its temperature dependence into consideration.
A similar consideration does apply to all other parts/parameters. As another example, the beta-factor of bipolar junction transistors is assumed to be constant for the majority of applcations. And this is reasonable because this factor has very large tolerances and we do not know its "actual" value (but, in fact, it depends on some other parameters).
However, of course there are many parts (also resistances) which exhibit intentionally a certain non-linearity and/or dependence on some environmental conditions (example: sensors, NTC- PTC-parts).
Therefore, when discussing this problem, we have clearly to distinguish between
cases where such a dependency is desired or
cases where this is an unwanted side effect that cannot be avoided.
Resistance is a characterisation of the \$i-v\$ curve of a two port device at some operating point. In general, it will depend on many things such as temperature, age, past history, etc.
If the device can be modeled by a suitable relationship of the form \$v= f(i)\$, then the resistance at a given current \$i\$ is given by \$ {\partial f(i) \over \partial i } \$. However, generally when we talk about 'the resistance' of a device, it is usually in the case where the relationship is modeled by \$v=Ri\$ for some fixed \$R\$.
Some devices, for example, a tunnel diode, can exhibit negative resistance at certain operating points, see https://electronics.stackexchange.com/a/460609/11869 for a nice picture.
