How does the sun's surface conduct thermal energy from the convective zone to the corona? How does the surface of the photosphere conduct thermal energy from the convective zone to the corona, while remaining at such a relatively low temperature itself?
It seems odd to me that the photosphere is wedged between the convective zone and the corona, both of which are millions of degrees in temperature, while the photosphere itself is observed to be much colder by comparison - approx. 5-10 thousand degrees.
Based on my understanding of convection, this would require thermal energy to jump from the convective zone directly to the corona, and somehow skip the photosphere surface material in-between.
Is there a different model of convection that explains this?
Or is there a type of matter with thermal conductive properties that could accomplish this?
EDIT:
Perhaps the convective zone is emitting heat as a different form of energy, such as electromagnetic - thereby allowing the outer layers to cool and sink in standard convection, but without directly heating up the surface of the photosphere.
Based on my understanding of the standard model of the sun, the following layers should be present:
Predicted:


*

*The core
A central area with pressures high enough to induce hot fusion of atomic nuclei

*The radiative zone
A spinning ball of somewhat solid material surrounding the core, through which energy is conducted via radiation


Semi-observable:


*

*The convective zone
A layer of plasma surrounding the radiative zone, through which energy is conducted thermally via convection flows - millions of degrees in temperature
This is the subject of some 2012 studies by NASA in coordination with various other reputable institutions, which observed that the sun's convection appears to be "anomalously weak"
http://arxiv.org/pdf/1206.3173.pdf
Observable:


*

*The surface of the photosphere
The border between the convective zone and the corona - temperature in the ballpark of 5-10 thousand degrees

*The corona
The outer plasma atmosphere, where flares take place - millions of degrees in temperature

 A: I think I'm in a bit over my head, but I don't think it's convection.   The Corona, because of the sun's high gravity, thins out very quickly.
http://sunearthday.gsfc.nasa.gov/2008/TTT/58_hotcorona.php
from the article:    

It would be like standing in your kitchen far away from the open oven,
  but feeling temperatures almost 100 times higher than what was inside
  the oven!

That's not convection.   Convection doesn't do that.  It's probably some kind energy conversion.   Magnetic energy turns into heat energy.   
http://en.wikipedia.org/wiki/Nanoflares

According to Parker a nanoflare arises from an event of magnetic
  reconnection which converts the energy stored in the solar magnetic
  field into the motion of the plasma. The plasma motion (thought as
  fluid motion) occurs at length-scales so small that it is soon dumped
  by the turbulence and then by the viscosity. In such a way the energy
  is quickly converted into heat, and conducted by the free electrons
  along the magnetic field lines closer to the place where the nanoflare
  switches on. In order to heat a region of very high X-ray emission,
  over an area 1" x 1", a nanoflare of $10^{17}\ \mathrm{J}$ should happen every 20
  seconds, and 1000 nanoflares per second should occur in a large active
  region of $10^5 \times 10^5\ \mathrm{km^2}$. On the basis of this theory, the emission
  coming from a big flare could be caused by a series of
  micro-nanoflares, not observable individually.

It's worth pointing out that this is just one of 3 main theories and there's not absolute certainty on how it works, but the first article I posted suggested there's some evidence that micro-flares are the primary cause.
( https://www.nasa.gov/content/goddard/best-evidence-yet-for-coronal-heating-theory and http://iopscience.iop.org/0004-637X/790/2/112/ )
A: You've actually identified a key area of ongoing research known as the coronal heating problem.

First, let's get one thing out of the way. You ask:

Or is there a type of matter with thermal conductive properties that could accomplish this?

There can be no such material. Any material that passively diffuses heat from a cooler region to a warmer one would be in violation of the laws of thermodynamics.

But indeed, the temperature profile of the Sun is not monotonic, reaching a minimum near the photosphere (the surface from which a typical photon is last scattered/emitted before reaching us). In order for temperature to increase away from the source of heat, there must be some nonthermal process.
It helps somewhat to note that the corona is extremely diffuse. Thus we are not necessarily looking for a large amount of nonthermal heat transport compared to the amount of heat transported via thermal radiation through the corona. Wikipedia gives the fraction as one part in $40{,}000$. Moreover, temperature inversions in thin atmospheres are nothing new.
There are two mainstream ideas that are discussed in solving the problem, both involving magnetic fields in plasmas.
In wave heating, the idea is that waves are launched from somewhere below the corona, travel to the appropriate height, and then shock, heating the plasma. In ideal magnetohydrodynamics (MHD, the theory of perfectly conducting perfect fluids, which sounds like a lot of assumptions but is in fact quite applicable to many astrophysical plasmas), there are entropy waves (also found in plain hydrodynamics, HD), slow and fast magnetosonic waves (somewhat analogous to sounds waves in HD), and Alfvén waves (a purely magnetic phenomenon). Like in HD, a smooth MHD traveling wave (think of a sinusoid) will eventually steepen into a shock (where various quantities become discontinuous), and shocks can increase entropy/temperature as they pass.
The questions are if such waves actually propagate into the corona at all, when they would shock, and how much heat would actually be dissipated.
The other theory is that of magnetic reconnection. This is a non-ideal process, in which so-called flux-freezing is no longer valid. For background, in ideal MHD, magnetic field lines can be thought of as advecting with the motion of the fluid -- magnetic fields cannot drift relative to the fluid, since its conducting nature would have a 100% backreaction restoring the field (think Lenz's law). However, at some point, our assumptions must break down (due e.g. to the discrete nature of matter).
Reconnection can be visualized as opposing magnetic field lines cancelling each other out. Conservation of energy, though, demands the missing magnetic energy be turned into something, heat in our case. We know reconnection must happen at some level, but again it is a question of details as applied to the corona. For example, naive estimates of the rate of reconnection that simply place an electrical resistivity term into the equations are observed to be wrong by orders of magnitude.
