So my question to you will come down to this: how close to zero do you need this to be?
It does not matter what I want, I will pick a device that meets my needs based on the datasheet. This datasheet is crap because it does not operate with numbers when it comes to "true zero".
ADP1716 very clearly specifies that their tracking down to 0 V is really +/-100 mV. All other things being equal, I will at least start with a device that operates with numbers.
Some of the numbers you can (and should!) figure out yourself. How low can the SET pin of the LT3080 go? Well, stick a zero ohm resistor in there and it goes to zero. Now, what does the output do when the SET pin is at zero? Well, there's the minimum load of 1mA needed on the thing. So, if you have a 100 ohm load resistor, that means it will go down to 100mV on the output. Of course, that's based on the worst-case minimum load required by the part. Typical minimum load is actually under 0.5mA based on the data sheet. So how is the writer of the data sheet going to quote numbers when they cannot predict what someone may use on the output?
Before you label something as 'crap' maybe you should step back and look at the problem differently and try to understand what you are asking for. The overwhelming percentage of linear regulators out there are single-quadrant devices. That means that a positive linear regulator will only source current. Usually, a minimum load is needed to keep them in regulation, especially when someone is asking for the minimum complexity available (not needed as much when you have a minimum output voltage and the part can include that in its design).
The ADP1716 is a "tracking" regulator, which means you need to supply it with a voltage that it follows up to the nominal output voltage. Funny enough, I would be wary of using it because there are no numbers in the data sheet to say how much current you need to source or sink on the TRK pin. So now, how are you driving the TRK pin without knowing this?
So my question to you will come down to this: how close to zero do you need this to be? What are you running from it? You're going to have a hard time actually turning on any active circuitry below a few hundred mV. Do you need it to be 200mV? 100mV? 50mV? 1mV? 1uV? 1pV? It's all a matter of significant digits, and the more digits you throw on there the more you should be putting this in the metrology forum instead.
On my old Tektronix power supplies, the control loop is slightly biased to produce a very small negative output, and the output is pulled below zero, so the actual control range always crosses zero even if the operational amplifier offsets go the other way. The attention to detail by the designer surprised me.
It's not entirely simple; you need bias on the output device to maintain some minimum frequency response for stability and this means either a minimum load current specification (most voltage regulators) or an internal current sink (LM358, R2R opamps, maybe some regulators). Alternatively, the circuit would need some means of slowing down the control loop at low load currents, or be quite slow at all load conditions. Well, maybe the latter would be acceptable, as emitter followers need only relatively little correction by the opamp. But it could still have some risk of (relatively low frequency) oscillation if the output goes down enough: 10mV, 1mV, 1μV..
It is not that difficult, even with a common emitter output stage, but it does require attention. Usually increasing the output capacitance for dominant pole compensation is enough. This also works with many operational amplifiers.
An optocoupler or single transistor can be used to generate a small negative current, but in most applications just getting within 10s of millivolts of zero is close enough.
I would be very tempted to use an operational amplifier and transistor rather than a regulator for this application, but it does mean more complexity. Those Linear Technology emitter follower regulators are very useful.
Here's the thing that everybody seems to want to ignore. You need that negative voltage available to get down to 'true zero' if you want to do things entirely in a linear fashion off of a single input rail. If you add an extra winding or tap on your transformer from the wall, or drop in a switcher, there's no problem. Do you want simple? Or are you willing to go more complex?
If I want to make a power supply that goes to zero, I'm going to design it with a negative internal rail to ensure I can get things all the way to zero. It's not going to be simple, with one or two ICs on it but instead be more complex to ensure that I can cover the operational corners all the way. There's a positive/negative supply that I like on p. 20 of the LT3094 data sheet that uses a switcher coupled to the final output voltages to give the best efficiency across all voltages and loads while still being nice and clean on the output.
As @magic pointed out, a simple, single IC design of a power supply that can go to zero needs bias to maintain some frequency response. A common emitter stage sounds great, but frequency response at zero load is horrible as you basically now have to charge the base of the device up one Vbe just to start delivering current to the load (base capacitance is not zero). Once you have a minimum load going, frequency response is much better since you get logarithmic response from the device.
Once somebody has actually taken the time to design one of these things, they will appreciate that things are never as simple as they want.
A rail-to-rail output op-amp + a pass transistor. The problem might be capacitive loads, which would require some frequency compensation.
Again, things like this look great until you build it. In order for this one to work well, you need to make sure you keep the pass device on, which means you have to sink current in the op amp and can't get to 'true zero'. Power supplies really want capacitance to handle transient loads, otherwise you get large excursions on the output until the feedback loop can compensate, so a capacitive load is pretty much guaranteed.
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